The cohomology of SL(3, ℤ[1/2])

Hans-Werner Henn

Abstract

We compute the cohomology of SL(3, ℤ[1/2]) with coefficients in the prime fields and in the integers. On the way we obtain the cohomology of certain mod - 2 congruence subgroups of SL(3, ℤ) with coefficients in 𝔽_{p} for p > 2. Finally we compute the cohomology of GL(3, ℤ[1/2]).

Contents

1 Introduction 2

2 Contractible spaces with actions of SL(3, ℤ) and SL(3, ℤ[1/2]) 8 2.1 The symmetric space and the Bruhat-Tits-building . . . . . . 8 2.2 Well-rounded lattices and the deformation retractions . . . . . 10 2.3 The space W_{0}/SO(3) . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Γ_{i} - equivariant cell structures on Z . . . . . . . . . . . . . . . 17

2.4.1 The case i = 0 . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2 The cases i = 1 and i = 2 . . . . . . . . . . . . . . . . 19

2.5 Symmetries of well-rounded quadratic forms . . . . . . . . . . 20 2.6 The equivalence relations ∼_{i} on the spaces Γ_{i} × D_{i} . . . . . . 23

2.6.1 3 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.2 2 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.3 1 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.4 0 - cells . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 The homology of the quotient spaces 34 3.1 Quotients of (X_{∞}, X_{∞,s}(i)) by Γ_{i} . . . . . . . . . . . . . . . . . 35 3.2 Quotients of X_{∞,s}(i) by Γ_{i} . . . . . . . . . . . . . . . . . . . . 41 3.3 Quotients of X_{∞} by Γ_{i} . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Quotients by SL(3, ℤ[1/2]) . . . . . . . . . . . . . . . . . . . . 45 4 The cohomology of SL(3, ℤ[1/2]) 49

4.1 Mod - 2 cohomology . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Mod - 3 cohomology . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Higher torsion in the integral cohomology . . . . . . . . . . . . 65

5 The cohomology of GL(3, ℤ[1/2]) 67

1 Introduction

So far there exist only very few complete computations of integral or mod - p cohomology rings of arithmetic or more generally S - arithmetic groups. Among the known results we mention the calculations for SL(2, ℤ) (which is straightforward from the well-known amalgamated product decomposition SL(2, ℤ) ≅ ℤ/6 ∗_{ℤ/2} ℤ/4), of SL(2, ℤ[1/2]) [Mi] and that of SL(3, ℤ) [So]. Soulé’s computation is already fairly involved; e.g. he obtains that the integral cohomology ring of SL(3, ℤ), after localization at the prime 2, is generated by 7 elements which are subject to 22 relations. His result suggests that the answer for SL(n, ℤ) would not be easily digestable (one should add that it also seems to be completely out of reach at this point).

From a conceptual point of view the complexity of the answer in Soulé’s calculation can also be explained by Quillen’s work [Q] which says among other things that the minimal prime ideals in the mod - p cohomology ring H^{∗}(Γ; 𝔽_{p}) of an S - arithmetic group Γ are in one to one correspondence with the conjugacy classes of maximal elementary abelian p - subgroups of Γ. (We recall that an elementary abelian p - group is a group isomorphic to (ℤ/p)^{k} for some natural number k.) From this point of view those cases in which there exists a unique conjugacy class of maximal elementary abelian p - subgroups look more favourable than others. In the case of SL(n, ℤ) or GL(n, ℤ) it is very difficult to determine the precise number of conjugacy classes of maximal elementary abelian p - subgroups (this is essentially a problem of the integral representation theory of elementary abelian p - groups) and thus the mod - p cohomology of these groups must be complicated. The situation improves if one inverts p and adjoins p - th roots of unity. In particular in the case of SL(n, ℤ[1/2]) and GL(n, ℤ[1/2]) every elementary abelian 2 - subgroup is diagonalizable and there is a unique maximal one up to conjugacy. This observation was presumably the basis of Quillen’s conjecture (p. 591 of [Q]), which in the case of H^{∗}(GL(n, ℤ[1/2]); 𝔽_{2}) claims that the inclusion of rings ℤ[1/2] ⊂ ℝ (and identifying H^{∗}(GL(n, ℤ[1/2]); 𝔽_{2}) as usual with the mod 2 - cohomology of the classifying space BGL(n, ℤ[1/2])) makes H^{∗}(GL(n, ℤ[1/2]); 𝔽_{2}) into a free, in particular a torsion free module over the polynomial ring 𝔽_{2}[w_{1}, ..., w_{n}] ≅ H^{∗}(BGL(n, ℝ); 𝔽_{2}) with w_{i} denoting as usual the i - th universal Stiefel - Whitney class. In [HLS] it was shown that torsion-freeness implies that the restriction map ρ_{n} : H^{∗}(GL(n, ℤ[1/2]); 𝔽_{2}) ⟶ H^{∗}(D_{n}; 𝔽_{2}) (with D_{n} ≅ ∏_{i=1}^{n}(ℤ[1/2])^{×} ≅ ∏_{i=1}^{n}(ℤ × ℤ/2) denoting the subgroup of diagonal matrices of GL(n, ℤ[1/2])) is injective. Quillen also remarked that with his conjecture a calculation of H^{∗}(GL(n, ℤ[1/2]); 𝔽_{2}) should be within reach. In fact, the image Im ρ_{n} has been computed by Mitchell. In order to state his result we identify the classes w_{i} with their image under restriction in H^{∗}(D_{n}; 𝔽_{2}) ≅ 𝔽[x_{1}, ..., x_{n}] ⊗ E(a_{1}, ..., a_{n}) (with E as usual denoting an exterior algebra, and with all generators of dimension 1), namely with the i - th elementary symmetric polynomial in the variables x_{i}. We also need classes e_{i} ∈ H^{2i−1}(D_{n}; 𝔽_{2}): they are the symmetrizations of the elements x_{1}^{2}...x_{i−1}^{2}a_{i} with respect to the canonical action of the symmetric group 𝔖_{n} on n letters. Now Mitchell’s result reads as follows.

Theorem 1.1 [Mi] Im ρ_{n} ≅ 𝔽_{2}[w_{1}, ..., w_{n}] ⊗ E(e_{1}, ..., e_{2n−1}) . □

Note that with this result Quillen’s conjecture would imply an isomorphism H^{∗}(GL(∞, ℤ[1/2]); 𝔽_{2}) ≅ 𝔽_{2}[w_{1}, w_{2}, ...] ⊗ E(e_{1}, e_{3}, ...) and hence the Dwyer - Friedlander version [DF] of the Lichtenbaum - Quillen conjecture at p = 2. Unfortunately Quillen’s conjecture was too optimistic. Dwyer has recently shown.

Theorem 1.2 [D] The restriction map ρ_{n} is not injective for all n. □

The only previous complete computation of H^{∗}(GL(n, ℤ[1/2]); 𝔽_{2}) was that of [Mi] for n = 2, and in this case ρ_{n} turned out to be injective. Some qualitative information on the size of the kernel of ρ_{n} as n grows is provided in [H2]. Dwyer shows, in fact, that ρ_{32} is not injective, so that the case n = 3 becomes an interesting test case in which one also has a nice candidate, namely Im ρ_{3}, for the answer. In fact, one of the main results of this paper shows that this candidate is correct.

Theorem 1.3 The restriction homomorphism maps H^{∗}(GL(3, ℤ[1/2]); 𝔽_{2}) isomorphically onto the subalgebra 𝔽_{2}[w_{1}, w_{2}, w_{3}]⊗E(e_{1}, e_{3}, e_{5}) of H^{∗}(D_{3}; 𝔽_{2}).

This result is really an easy consequence of the following companion result for SL(3, ℤ[1/2]). We denote its subgroup of diagonal matrices by SD_{3}. Note that the restriction map from H^{∗}(D_{3}; 𝔽_{2}) to H^{∗}(SD_{3}; 𝔽_{2}) kills the elements w_{1} and e_{1}. Let v_{i} be the image of w_{i} and d_{2i−1} the image of e_{2i−1}, i = 2, 3.

Theorem 1.4 The restriction homomorphism maps H^{∗}(SL(3, ℤ[1/2]); 𝔽_{2}) isomorphically onto the subalgebra 𝔽_{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5}) of H^{∗}(SD_{3}; 𝔽_{2}).

We remark that the corresponding result does not hold in the same way for n = 2, i.e. the restriction map is not an isomorphism in this case, although there is an abstract isomorphism H^{∗}(SL(2, ℤ[1/2]); 𝔽_{2}) ≅ 𝔽_{2}[v_{2}]⊗E(d_{3}) [Mi].

How can Theorem 1.4 be proved? The standard approach would be to take a suitable finite dimensional contractible space X on which Γ := SL(3, ℤ[1/2]) acts properly and with finite isotropy groups (there is a canonical such candidate, namely the product of the symmetric space SL(3, ℝ)/SO(3) and the Bruhat-Tits-building for SL(3, ℚ_{2}), see Section 2.1 below). Then one would take the Borel construction EΓ ×_{Γ} X as model for the classifying space BΓ and study its mod - 2 cohomology H_{Γ}^{∗}(X; 𝔽_{2}) via the cohomology spectral sequence of the map EΓ ×_{Γ} X ⟶ Γ\X. If X has the structure of a Γ - CW - complex then the E_{1} - term of this spectral sequence is given as E_{1}^{s,t} = ⨁_{σ} H^{t}(Γ_{σ}; 𝔽_{2}) where σ runs through a set of representatives of the Γ - orbits of s - dimensional cells of X and Γ_{σ} denotes the isotropy group of σ. This is how Soulé studied the cohomology of SL(3, ℤ) [So]. However, in our case the space X looks too complicated to make this spectral sequence manageable: in Section 2.6 we actually analyze the canonical X above and we essentially produce a Γ - equivariant deformation retract with finitely many Γ - orbits of cells; however, finite means 474 (!) orbits (see the table at the beginning of Section 3) and so this standard approach looks unfeasible. Instead we use a more manageable “centralizer spectral sequence”

E_{2}^{s,t} ≅ lim_{E∈A_{∗}(Γ)}^{s} H^{t}(C_{Γ}(E); 𝔽_{2}) ⟹ H_{Γ}^{s+t}(X_{s}; 𝔽_{2})

converging to the mod - 2 cohomology of the Borel - construction of the 2 - singular locus X_{s}, i.e. the subspace of X consisting of all points whose isotropy group contains an element of order 2. Here A_{∗}(Γ) is the category of elementary abelian 2 - subgroups of Γ, lim^{s} is the s - th derived functor of the inverse limit functor and C_{Γ}(E) is the centralizer in Γ of the elementary abelian 2 - subgroup E ⊂ Γ. This spectral sequence is based on a homotopy colimit decomposition of EΓ×_{Γ} X_{s} and was introduced in [H1]. In this paper we also evaluated this spectral sequence and obtained the following result in which Σ denotes as usual the suspension functor, e.g. Σ^{4}𝔽_{p} denotes the graded 𝔽_{p} - vectorspace which is trivial in all dimensions except in dimension 4 where it is 𝔽_{p}.

Theorem 1.5 [H1] Let Γ = SL(3, ℤ[1/2]) and let X be any mod - 2 acyclic finite dimensional Γ - CW - complex for which the stabilizers of all cells are finite. Then there is a short exact sequence

0 ⟶ Σ^{4}𝔽_{2} ⟶ H_{Γ}^{∗}(X_{s}; 𝔽_{2}) ⟶^{ρ} 𝔽_{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5}) ⟶ 0

in which ρ is an algebra homomorphism. Furthermore, if π denotes the projection map from EΓ ×_{Γ} X_{s} to the classifying space BΓ then the composition

H^{∗}(Γ; 𝔽_{2}) ⟶^{π^{∗}} H_{Γ}^{∗}(X_{s}; 𝔽_{2}) ⟶^{ρ} 𝔽_{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5}) ⊂ H^{∗}(SD_{3}; 𝔽_{2})

agrees with the restriction homomorphism of 1.4. □

Now for ∗ exceeding dim X, the dimension of X, we have isomorphisms H_{Γ}^{∗}(X_{s}; 𝔽_{2}) ≅ H_{Γ}^{∗}(X; 𝔽_{2}) ≅ H^{∗}(Γ; 𝔽_{2}) and hence Theorem 1.5 is also a computation of H^{∗}(Γ; 𝔽_{2}) in large dimensions. In fact, X can be chosen to be of dimension 5 (see [BS] or Section 2 below) and Theorem 1.5 gives encouraging evidence for Theorem 1.4.

In this paper we complete the proof of Theorem 1.4 by computing for the canonical space X mentioned above, the relative groups H_{Γ}^{∗}(X, X_{s}; 𝔽_{2}) and the boundary homomorphism of the appropriate long exact cohomology sequence. Note that, because the isotropy groups outside of X_{s} are finite of order prime to 2, we have the following isomorphisms for the relative groups: H_{Γ}^{∗}(X, X_{s}; 𝔽_{2}) ≅ H^{∗}(Γ\(X, X_{s}); 𝔽_{2}). As a byproduct of our investigations we obtain the following results which are of independent interest. In these results we abbreviate SL(3, ℤ[1/2]) by Γ, SL(3, ℤ) by Γ_{0}, and we denote the subgroup of SL(3, ℤ) consisting of all matrices whose first column agrees with the first standard basis vector modulo 2 by Γ_{1}, and the subgroup of all matrices which are upper triangular modulo 2 by Γ_{2}.

Theorem 1.6 Let X_{∞} denote the symmetric space SL(3, ℝ)/SO(3), X_{2} the Bruhat-Tits-building of SL(3, ℚ_{2}), X = X_{∞} × X_{2} and let p be any prime. Then the reduced cohomology of the quotient spaces by the obvious action of the respective groups is given as follows:

a) H̃^{∗}(Γ_{0}\X_{∞}; 𝔽_{p}) = 0

b) H̃^{∗}(Γ_{1}\X_{∞}; 𝔽_{p}) = 0

c) H̃^{∗}(Γ_{2}\X_{∞}; 𝔽_{p}) = Σ^{3}𝔽_{p}

d) H̃^{∗}(Γ\X; 𝔽_{p}) = Σ^{5}𝔽_{p} .

For p > 3 there are no elements of order p in these groups (because there are obviously no elements of order p in SL(3, ℚ)) and hence we obtain the following Corollary. For SL(3, ℤ) this was already known by [So] and for SL(3, ℤ)[1/2] by [Mo]. The results for Γ_{1} and Γ_{2} are compatible with the Euler chararacteristic computations in [Mo].

Corollary 1.7 Assume p > 3. Then

a) H̃^{∗}(Γ_{0}; 𝔽_{p}) = 0

b) H̃^{∗}(Γ_{1}; 𝔽_{p}) = 0

c) H̃^{∗}(Γ_{2}; 𝔽_{p}) = Σ^{3}𝔽_{p}

d) H̃^{∗}(Γ; 𝔽_{p}) = Σ^{5}𝔽_{p} . □

Theorem 1.8 Let X_{∞}, X_{2}, X and p be as in the previous theorem. Then we get the following relative cohomology groups (where (X_{∞,s}(i) denotes the 2 - singular locus of X_{∞} with respect to the action of Γ_{i}, and X_{s} the 2 - singular locus of X with respect to the action of Γ):

a) H^{∗}(Γ_{0}\(X_{∞}, X_{∞,s}(0)); 𝔽_{p}) = 0

b) H^{∗}(Γ_{1}\(X_{∞}, X_{∞,s}(1)); 𝔽_{p}) = Σ^{2}(𝔽_{p})^{2}

c) H^{∗}(Γ_{2}\(X_{∞}, X_{∞,s}(2)); 𝔽_{p}) = Σ^{3}𝔽_{p} ⊕ Σ^{2}(𝔽_{p})^{6}

d) H^{∗}(Γ\(X , X_{s}); 𝔽_{2}) = Σ^{5}𝔽_{2} . (Observe that we restrict to the case p = 2 for the last part of the Theorem.)

The next result together with Theorem 1.5 and the last part of Theorem 1.8 finishes the proof of Theorem 1.4.

Proposition 1.9 The boundary homomorphism

H_{Γ}^{4}(X_{s}; 𝔽_{2}) ⟶ H_{Γ}^{5}(X , X_{s}; 𝔽_{2})

is an epimorphism.

With the help of Theorem 1.6 we are also able to compute the mod - 3 cohomology. Again this was known for SL(3, ℤ) by [So].

Theorem 1.10 There are isomorphisms of 𝔽_{3} - algebras (without unit) which in the case of a), b) and d) are induced by restrictions to appropriate subgroups:

a) H̃^{∗}(Γ_{0}; 𝔽_{3}) ≅ ∏_{i=1}^{2} H̃^{∗}(𝔖_{3}; 𝔽_{3})

b) H̃^{∗}(Γ_{1}; 𝔽_{3}) ≅ ∏_{i=1}^{2} H̃^{∗}(𝔖_{3}; 𝔽_{3})

c) H̃^{∗}(Γ_{2}; 𝔽_{3}) ≅ Σ^{3}𝔽_{3}

d) H̃^{∗}(Γ; 𝔽_{3}) is isomorphic to the subalgebra of ∏_{i=1}^{2} H̃^{∗}(𝔖_{3} × ℤ; 𝔽_{3}) which can be characterized as follows: it is all of ∏_{i=1}^{2} H̃^{∗}(𝔖_{3} × ℤ; 𝔽_{3}) except in degrees 1 and 4; in degree 1 it is trivial, and in degree 4 it is of dimension 3 and is generated by the image of the Bockstein of H^{3} and one further element which restricts non-trivially to both factors.

The paper is organized as follows: In Section 2 we recall the symmetric space X_{∞} and the Bruhat - Tits building X_{2}. We discuss the Soulé - Lannes method of replacing the symmetric space by a smaller space Z for which the quotients by Γ_{0} = SL(3, ℤ) and the congruence subgroups Γ_{1} and Γ_{2} are compact. The bulk of this long section is then devoted to patiently working out an explicit cell structure of the quotients Γ_{i}\Z, i = 0, 1, 2, in fact even a Γ_{i} - equivariant cell structure on Z. This is straightforward but it is crucial for the remainder of the paper; for i = 0 it is a variation of Soulé’s investigations [So]. In Section 3 we use these cell structures to prove Theorem 1.6 and Theorem 1.8 as well as the corresponding results for the cohomology of the quotients of the singular locus X_{∞,s}(i) resp. X_{s}. This is quite an elaborate calculation but apart from the last part of Theorem 1.8 it is straightforward given the results in Section 2. The last part of Theorem 1.8 is more tricky and to settle it we use low dimensional information on H_{Γ}^{∗}(X_{s}; 𝔽_{2}) as provided by Theorem 1.5. In Section 4 we apply the results of Section 2 and Section 3 and derive the remaining results listed in this introduction. We also determine the height of torsion in H^{∗}(SL(3, ℤ[1/2]); ℤ) (Proposition 4.15). In Section 5 we compute H^{∗}(GL(3, ℤ[1/2]); 𝔽_{p}) for primes p > 2 (Proposition 5.1, 5.2) and for p = 2, i.e. we derive Theorem 1.3.

\underline{Acknowledgements:} During the research presented in this paper the author was supported by a Heisenberg fellowship of the DFG. The author is happy to acknowledge numerous discussions with Jean Lannes which stimulated his interest in the cohomology of SL(3, ℤ[1/2]). He also thanks Bob Oliver for helpful discussions in connection with the proof of Proposition 1.9.

2 Contractible spaces with actions of SL(3, ℤ) and SL(3, ℤ[1/2])

2.1 The symmetric space and the Bruhat-Tits-building

We start by recalling the contractible spaces on which our groups act with finite stabilizer groups.

The symmetric space. The space Q(n) of positive definite quadratic forms on ℝ^{n} is equipped with an action of the multiplicative group ℝ_{+} of positive real numbers, given by (rq)(x) = rq(x) for r ∈ ℝ_{+}, q ∈ Q(n) and x ∈ ℝ^{n}. The quotient will be denoted by X_{∞}(n), or simply by X_{∞} if n is clear from the context. The space X_{∞}(n) is contractible because Q(n) is a convex open cone in ℝ^{n^{2}}. Furthermore, X_{∞}(n) can be identified with the symmetric space of SL(n, ℝ), i.e. the space of left cosets SL(n, ℝ)/SO(n), via the map which sends a matrix A to the equivalence class of the positive definite quadratic form q, given by q(x) = ||A^{−1}x|| where || || denotes the euclidean norm in ℝ^{n}. The group SL(n, ℤ) acts on this coset space from the left, and this action is proper, i.e. if C ⊂ X_{∞}(n) is compact then there are only finitely many g ∈ SL(n, ℤ) for which gC ∩ C ≠ ∅; in particular the isotropy groups of the action are all finite.

The Bruhat-Tits-building. The group SL(n, ℤ[1/2]) acts on the coset space X_{∞}(n) as well. However, in this case the action is not proper. In order to get a contractible space with proper action, the space X_{∞}(n) has to be enlarged by the appropriate Bruhat-Tits-building X_{2}(n) (or simply X_{2} if n is clear from the context) for the group SL(n, ℚ_{2}). As reference for more on this bulding we recommend [B2]. We recall here only some basic properties. The space X_{2}(n) is an (n − 1)-dimensional simplicial complex which can be described as follows: an n - dimensional 2 - adic lattice L is a ℤ_{2} - submodule of ℚ_{2}^{n} which is free of rank n. The group ℚ_{2}^{×} of units in ℚ_{2} acts on the set of all such lattices via scalar multiplication, and the set of equivalence classes is the set of vertices in X_{2}(n). A finite subset {l_{0}, l_{1}, ..., l_{n}} of vertices spans an n - dimensional simplex in X_{2}(n) if and only if there are representative lattices L_{i} in the class of l_{i} for i = 0, ..., n such that L_{0} ⊊ L_{1} ⊊ ... ⊊ L_{n} ⊊ \frac{1}{2}L_{0}. The space X_{2}(n) is contractible (see Section V.8 and Theorem VI.3 in [B2]). Furthermore the set of all 2 - adic lattices can be identified with the set of left cosets GL(n, ℚ_{2})/GL(n, ℤ_{2}) via the map which sends a matrix A to the lattice A(ℤ_{2}^{n}). The natural left action of SL(n, ℚ_{2}) on this coset space induces a simplicial left action of SL(n, ℚ_{2}) on X_{2}(n) and the quotient of X_{2}(n) by the action of SL(n, ℚ_{2}) is an (n − 1) - dimensional simplex Δ^{n−1}.

Furthermore SL(n, ℤ[1/2]) is dense in SL(n, ℚ_{2}) and therefore the quotient of X_{2}(n) by the action of SL(n, ℤ[1/2]) agrees with the quotient by the group SL(n, ℚ_{2}). The group SL(n, ℤ[1/2]) embedds diagonally as a discrete subgroup into SL(3, ℝ)×SL(n, ℚ_{2}) and acts properly on the contractible space X := X_{∞} × X_{2}.

The projection maps and congruence subgroups. From now on we concentrate on the case n = 3. We will be interested in the SL(3, ℤ[1/2]) - equivariant projection map p : X ⟶ X_{2}. With respect to the action of GL(3, ℤ[1/2]) all vertices in X_{2} fall into a single orbit and hence their isotropy groups (in SL(3, ℤ[1/2])) are conjugate in the larger group GL(3, ℤ[1/2]), in particular they are abstractly isomorphic; similarly with simplices of dimension one. For the vertex l_{0} corresponding to the standard lattice L_{0} (which is spanned over ℤ_{2}^{3} by the standard basis vectors e_{1},e_{2} and e_{3}, i.e. L_{0} = ⟨e_{1}, e_{2}, e_{3}⟩), the isotropy group is SL(3, ℤ_{2}) ∩ SL(3, ℤ[1/2]) = SL(3, ℤ) =: Γ_{0}. For the edge consisting of the set {l_{0}, l_{1}} with l_{0} the class of L_{0} and l_{1} the class of the lattice L_{1} = ⟨\frac{1}{2}e_{1}, e_{2}, e_{3}⟩, the isotropy group is the subgroup Γ_{1} of Γ_{0} consisting of matrices whose first column is equal to e_{1} modulo 2; for the two-dimensional simplex spanned by the set \{l_{0}, l_{1}, l_{2}\} with l_{2} the class of the lattice L_{2} = ⟨\frac{1}{2}e_{1}, \frac{1}{2}e_{2}, e_{3}⟩, the isotropy group is the subgroup Γ_{2} of Γ_{0} consisting of all matrices which are upper triangular modulo 2. For simplicity of notation we will write Γ instead of SL(3, ℤ[1/2]). The “fibres” of the map (which is induced by p)

p̃: EΓ ×_{Γ} X ⟶ Γ\X_{2} ≅ Δ^{2}

over the 0 -, 1 - resp. 2 - dimensional simplices respectively are homotopy - equivalent to the classifying spaces BΓ_{0}, BΓ_{1} and BΓ_{2} respectively. We will have to study the mod - 2 (co)homology spectral sequence of p̃ as well as that of the map (which is also induced by p)

\overline{p} : Γ\X ⟶ Γ\X_{2} ≅ Δ^{2} .

In particular we need to understand the “fibres” of \overline{p}, i.e. the quotients Γ_{i}\X_{∞}, i = 0, 1, 2. These quotients are not compact and in the next section we recall the Soulé-Lannes method of finding a deformation retract of Γ_{i}\X_{∞} which is compact, even a finite 3 - dimensional complex (see [A]).

2.2 Well-rounded lattices and the deformation retractions

Well-rounded lattices. We note that Γ_{i}\X_{∞} ≅ Γ_{i}\(SL(3, ℝ)/SO(3)) may also be obtained as quotient of Γ_{i}\SL(3, ℝ) by the right action of SO(3). Now the right SO(3) - space GL(3, ℤ)\GL(3, ℝ) can be identified with the space of all integral lattices in ℝ^{3} (via the correspondence which sends a matrix g to the lattice g^{−1}(ℤ^{n})), and the space Γ_{0}\SL(3, ℝ) can be identified with the space of equivalence classes (with respect to scalar multiplication) of integral lattices L in ℝ^{3}, or equivalently with the space of integral lattices whose minimal vectors are of length 1, i.e. for which m(L) := min{∥x∥|x ∈ L − {0}} = 1. We will denote this latter space by L_{0}. Note that, in terms of lattices, the right action of SO(3) on Γ_{0}\SL(3, ℝ) is given by L · g := g^{−1}L for L ∈ L_{0} and g ∈ SO(3). Similarly the space Γ_{1}\SL(3, ℝ) can be identified with the space L_{1} of pairs (L_{0}, L_{1}) of lattices such that m(L_{0}) = 1 and L_{0} ⊊ L_{1} ⊊ \frac{1}{2}L_{0}, and the space Γ_{2}\SL(3, ℝ) can be identified with the space L_{2} of triples (L_{0}, L_{1}, L_{2}) of lattices such that m(L_{0}) = 1 and L_{0} ⊊ L_{1} ⊊ L_{2} ⊊ \frac{1}{2}L_{0}. We recall that a lattice L in ℝ^{3} is called well-rounded if its set of minimal vectors, i.e. {x ∈ L − {0}|∥x∥ = m(L)} spans ℝ^{3}. For i = 0, 1, 2 let W_{i} denote the subspace of L_{i} consisting of all tuples (L_{0}, ..., L_{i}) for which L_{0} is well-rounded.

The deformation retractions. There is a beautiful geometric argument which shows that W_{i} is an SO(3) - equivariant deformation retract of L_{i}, hence W_{i}/SO(3) is a deformation retract of L_{i}/SO(3) ≅ Γ_{i}\X_{∞}. We recall the construction ([A]). For i = 0, 1, 2 and 1 ≤ p ≤ 3 let W_{i}^{p} be the set of tuples (L_{0}, ..., L_{i}) of lattices such that the dimension of the subspace of ℝ^{3} spanned by the set of minimal vectors in L_{0} is at least p. Then W_{i}^{1} = L_{i}, W_{i}^{3} = W_{i} and therefore it suffices to show that W_{i}^{p+1} is an SO(3) - equivariant deformation retract of W_{i}^{p} for p = 1, 2. So assume that the set of minimal vectors in L_{0} spans a subspace U of dimension q ≥ p. If q > p then nothing happens to our tuple in the next step of the deformation. Otherwise, consider a radial contracting homotopy in the subspace U^{⊥} of ℝ^{3} perpendicular to U, and extend linearly to a deformation of ℝ^{3} by leaving U fixed. This defines a deformation L_{j}(t), 0 < t ≤ 1 of lattices (for 0 ≤ j ≤ i) with L_{j}(1) = L_{j} and there will be a maximal t_{0} with 0 < t_{0} < 1 for which L_{0}(t_{0}) has a new vector of minimal length 1. The corresponding tuple (L_{0}(t_{0}), ..., L_{i}(t_{0})) of lattices lies in W_{i}^{p+1} and is the image under the next step in the deformation. It is easy to see that these constructions describe continuous SO(3) - equivariant maps which combine to give an SO(3) - equivariant deformation retraction from L_{i} to W_{i} and induce a deformation retraction from Γ_{i}\X_{∞} ≅ L_{i}/SO(3) to W_{i}/SO(3). We will see in the next section that the spaces W_{i}/SO(3) are compact and of dimension 3.

We can do even a bit better: the SO(3) - equivariant deformation retraction of L_{0} can be lifted to give a left SL(3, ℤ) - equivariant and right SO(3) - equivariant deformation retraction of SL(3, ℝ) onto the subspace Y := {g ∈ SL(3, ℝ)|g^{−1}(ℤ^{n}) is a wellrounded lattice}. Dividing out by the SO(3) - action gives a left SL(3, ℤ) - space Z and an SL(3, ℤ) - equivariant deformation retraction from X_{∞} to Z. The space Z will also be called the space of (equivalence classes of) well - rounded quadratic forms. The remainder of Section 2 is devoted to a detailed analysis of the spaces Γ_{i}\Z ≅ W_{i}/SO(3), in particular we will exhibit explicit finite cell structures on them.

2.3 The space W_{0}/SO(3)

Our first task is to understand the space Γ_{0}\Z ≅ W_{0}/SO(3). This space agrees with Soulé’s deformation retract of the space Γ_{0}\SL(3, ℝ)/SO(3) [So]; however, our point of view is a bit different in so far as we emphasize lattices rather than quadratic forms, i.e. we prefer to think in terms of W_{0}/SO(3), the space of wellrounded 3 - dimensional lattices L with m(L) = 1, modulo the action of SO(3). We will see in a moment that in dimension 3 (unlike in higher dimensions) the sublattice spanned by any set of 3 linearly independent vectors of minimal length in a well-rounded lattice L is all of L, and therefore L is (up to the action of SO(3)) determined by m(L) and the 3 scalar products between these vectors. We will analyze which of these 3 - tuples occur in this way and which tuples give the same lattice, up to the action of SO(3). This analysis will lead to an explicit description of the spaces W_{i}/SO(3). In this section we will first concentrate on the case i = 0. Our first step is given by the following Lemma.

Lemma 2.1 Suppose L ⊂ ℝ^{3} is a well-rounded lattice and let v_{1}, v_{2} and v_{3} be linearly independent vectors of minimal length m(L) in L. Then the sublattice L′ spanned by these vectors is all of L.

Proof. By scaling and rotating L we may assume that m(L) = 1, v_{1} = (1, 0, 0) and v_{2}, v_{3} have the form: v_{2} = (a, x, 0) and v_{3} = (b, y, z). Assume there exists w = (w_{1}, w_{2}, w_{3}) ∈ L − L′. By adding a suitable vector in L′ we may assume that |w_{3}| ≤ \frac{1}{2}|z| ≤ \frac{1}{2}, |w_{2}| ≤ \frac{1}{2}|x| ≤ \frac{1}{2} and |w_{1}| ≤ \frac{1}{2}. But then ||w|| < 1 and we obtain a contradiction to the assumption that m(L) = 1. □

The next two results will enable us to give an explicit description of the space W_{0}/SO(3). They will be proved together.

Proposition 2.2 Suppose v_{1}, v_{2} and v_{3} are linearly independent vectors of length 1 in ℝ^{3} with scalar products a = ⟨v_{1}, v_{2}⟩, b = ⟨v_{1}, v_{3}⟩ and c = ⟨v_{2}, v_{3}⟩. Assume that a ≥ 0 and b ≥ 0. Then the lattice L spanned by v_{1}, v_{2} and v_{3} is well-rounded with m(L) = 1 if and only if

1. c ≥ 0 and a, b, c ≤ \frac{1}{2}, or

2. c ≤ 0, a, b, |c| ≤ \frac{1}{2} and a + b − c ≤ 1.

Clearly, the assumption on a and b can be assured by replacing, if necessary, one of the vectors v_{i} by its negative.

Proposition 2.3 Suppose v_{1}, v_{2} and v_{3}, a, b, c and L are as in Proposition 2.2. Furthermore assume a ≥ b ≥ |c|. Then the set of minimal vectors in L contains ±v_{1}, ±v_{2}, ±v_{3} and in addition only the following vectors:

1. ±(v_{1} − v_{2}) if a = \frac{1}{2}, b ≠ \frac{1}{2} and a + b − c ≠ 1.

2. ±(v_{1} − v_{2} − v_{3}) if a ≠ \frac{1}{2}, b ≠ \frac{1}{2} and a + b − c = 1.

3. ±(v_{1} − v_{2}) and ±(v_{1} − v_{3}) if a = b = \frac{1}{2} and c ≠ 0, \frac{1}{2}.

4. ±(v_{1} − v_{2}) and ±(v_{1} − v_{2} − v_{3}) if a = \frac{1}{2}, b ≠ \frac{1}{2} and a + b − c = 1.

5. ±(v_{1} − v_{2}), ±(v_{1} − v_{3}) and ±(v_{2} − v_{3}) if a = b = c = \frac{1}{2}.

6. ±(v_{1} − v_{2}), ±(v_{1} − v_{3}) and ±(v_{1} − v_{2} − v_{3}) if a = b = \frac{1}{2} and c = 0.

Again the assumption on a, b and c can always be assured by permuting the vectors v_{i} and passing to negatives if necessary.

Proof. 1. Let us first consider the case c ≥ 0. Consider a vector w in L and write

w = n_{1}v_{1} + n_{2}v_{2} + n_{3}v_{3}, n_{i} ∈ ℤ, i = 1, 2, 3 .

Then

||w||^{2} = n_{1}^{2} + n_{2}^{2} + n_{3}^{2} + 2an_{1}n_{2} + 2bn_{1}n_{3} + 2cn_{2}n_{3} ,  (2.1)

or equivalently

||w||^{2} = a(n_{1} + n_{2})^{2} + b(n_{1} + n_{3})^{2} + c(n_{2} + n_{3})^{2}
+(1 − a − b)n_{1}^{2} + (1 − a − c)n_{2}^{2} + (1 − b − c)n_{3}^{2}. (2.2)

If a > \frac{1}{2} then n_{1} = −n_{2} = 1, n_{3} = 0 gives a vector w with ||w|| = 2 − 2a < 1. The same argument for b and c shows that, if m(L) = 1, then b, c ≤ \frac{1}{2}. Now assume that a, b, c ≤ \frac{1}{2}. We distinguish different cases.

1.1. At least one n_{i} = 0, w.l.o.g. n_{3} = 0. Then we obtain

||w||^{2} = n_{1}^{2} + n_{2}^{2} + 2an_{1}n_{2} = a(n_{1} + n_{2})^{2} + (1 − a)n_{1}^{2} + (1 − a)n_{2}^{2}. (2.3)

Because 1 − a ≥ \frac{1}{2} and 1 − b ≥ \frac{1}{2} it is clear from (2.3) that ||w||^{2} ≥ 1 unless w = 0. We also observe that the only vectors of length 1 in L with n_{3} = 0 are the vectors ±v_{1}, ±v_{2}, and if a = \frac{1}{2}, the vector ±(v_{1} − v_{2}). Similarly, the only vectors with n_{2} = 0 are the vectors ±v_{1}, ±v_{3}, and if b = \frac{1}{2}, the vector ±(v_{1} − v_{3}). The only vectors with n_{1} = 0 are the vectors ±v_{2}, ±v_{3}, and if c = \frac{1}{2}, the vector ±(v_{2} − v_{3}). 1.2. We may now assume that all n_{i} ≠ 0. Then at least one of the sums n_{1} +n_{2}, n_{1} +n_{3}, n_{2} +n_{3} must be different from 0. If precisely one of the sums is non-zero, say n_{2} + n_{3}, then n_{1} = −n_{3}, n_{1} = −n_{2} and |n_{2} + n_{3}| ≥ 2 and (2.2) yields ||w||^{2} ≥ 3 − 2a − 2b + 2c ≥ 1; equality holds iff c = 0, a = b = \frac{1}{2}, n_{1} = −n_{2} = −n_{3} = ±1, i.e. w = ±(v_{1} − v_{2} − v_{3}). If at least two of the sums are non-zero, say n_{1} + n_{3} and n_{2} + n_{3}, then |n_{1} + n_{3}| ≥ 2 and |n_{2} + n_{3}| ≥ 2 and (2.2) yields ||w||^{2} ≥ 3 − 2a + 2b + 2c > 1, in particular there are no such vectors of length 1.

2. Now consider the case c ≤ 0. Then we write

||w||^{2} = a(n_{1} + n_{2})^{2} + b(n_{1} + n_{3})^{2} − c(n_{2} − n_{3})^{2}
+(1 − a − b)n_{1}^{2} + (1 − a + c)n_{2}^{2} + (1 − b + c)n_{3}^{2}. (2.4)

As before we see that a, b, |c| ≤ \frac{1}{2} is necessary for L to satisfy m(L) = 1. Now assume these inequalities hold. Again we distinguish different cases. 2.1. If at least one n_{i} = 0 and w ≠ 0, then we see as above that ||w||^{2} ≥ 1 and we only obtain additional vectors of length 1 iff a = \frac{1}{2} resp. b = \frac{1}{2} resp. c = −\frac{1}{2}, namely the vectors ±(v_{1} − v_{2}) resp. ±(v_{1} − v_{3}) resp. ±(v_{2} + v_{3}). 2.2. We may now assume that all n_{i} ≠ 0. Consider the sums n_{1}+n_{2}, n_{1}+n_{3}, n_{2} − n_{3}. We subdivide into further cases. In case all sums are zero we have n_{1} = −n_{2} = −n_{3} and from (2.4) we obtain again ||w||^{2} ≥ 3 − 2a − 2b + 2c. By taking n_{1} = −n_{2} = −n_{3} = ±1 we see that the condition a + b − c ≤ 1 is necessary for L to satisfy m(L) = 1, and there are further vectors of length 1 iff a + b − c = 1, namely the vectors ±(v_{1} − v_{2} − v_{3}). If two sums are zero, then the third one is as well, hence we may next assume that at most one sum is zero, hence at least two of the terms |n_{1} + n_{2}| and |n_{1} + n_{3}|, |n_{2} − n_{3}| are ≥ 2. In case |n_{1} + n_{2}| and |n_{1} + n_{3}| are ≥ 2, (2.4) yields ∥w∥^{2} ≥ 3 + 2a + 2b + 2c > 1, in particular there are no such vectors of length 1. The other two cases are analogous. □

After these preparations we can now describe the space W_{0}/SO(3). Consider the following subspace D_{0} of ℝ^{3} (see figure 1):

D_{0} := \{(a, b, c) ∈ ℝ^{3}| |c| ≤ b ≤ a ≤ \frac{1}{2}, and a + b − c ≤ 1 if c ≤ 0\} .

We define a map

γ : D_{0} ⟶ Y = {g ∈ SL(3, ℝ)|g^{−1}(ℤ^{n}) is a wellrounded lattice}

by sending the triple (a, b, c) to the unique matrix γ(a, b, c) with the following properties: γ(a, b, c) is (up to a scalar multiple guaranteeing γ(a, b, c) ∈ SL(3, ℝ)) the inverse of the matrix whose i - th column is the basis vector v_{i}, where v_{1} = (1, 0, 0), v_{2} = (a, x, 0), v_{3} = (b, y, z) and x, y and z are uniquely determined by the requirements x ≥ 0, ab + xy = c, z ≥ 0 and ||v_{i}|| = 1 for i = 1, 2, 3. By construction and Proposition 2.2 the lattice γ(a, b, c)^{−1}(ℤ^{n}) is well - rounded, hence γ(a, b, c) ∈ Y. Let λ : D_{0} ⟶ W_{0} denote the composition of γ with the canonical projection Y ⟶ W_{0}; then λ(a, b, c) is the well - rounded lattice spanned by the vectors v_{1}, v_{2} and v_{3}. Note that by construction a = ⟨v_{1}, v_{2}⟩, b = ⟨v_{1}, v_{3}⟩ and c = ⟨v_{2}, v_{3}⟩. Finally let φ : D_{0} ⟶ W_{0}/SO(3) be the composition of λ with the canonical projection W_{0} ⟶ W_{0}/SO(3). Clearly all these maps are continuous. Finally we define an equivalence relation ∼ on D_{0} by declaring the points (\frac{1}{2}, b, c) with 0 ≤ c ≤ \frac{1}{2}b equivalent to (\frac{1}{2}, b, b − c) and equivalent to (\frac{1}{2}, b − c, −c) (cf. figure 1).

Theorem 2.4 The map φ : D_{0} ⟶ W_{0}/SO(3) is onto and induces a homeomorphism φ̃: D_{0}/∼ ⟶ W_{0}/SO(3).

In the proof we will make repeated use of the following elementary fact.

Lemma 2.5 Assume v_{1}, v_{2}, v_{3} and v_{1}′, v_{2}′, v_{3}′ are two sets of linearly independent vectors of length 1 in ℝ^{3} such that ⟨v_{i}, v_{j}⟩ = ⟨v_{i}′, v_{j}′⟩ for all 1 ≤ i < j ≤ 3. Then there exist unique rotations R, S ∈ O(3) such that Rv_{i} = v_{i}′ and Sv_{i} = −v_{i}′ for i = 1, 2, 3, and either R or S is in SO(3). □

Proof of Theorem. That φ is onto can be seen as follows. Assume we are given a well-rounded lattice L with minimal vectors of length 1. By Lemma 2.1 we can find spanning vectors w_{1}, w_{2} and w_{3} in L of length 1, and after a suitable permutation (and passing to additive inverses, if necessary) we may assume that the scalar products a, b and c are as in 2.2 and 2.3. Then Lemma 2.5 implies φ(a, b, c) = [L] where [L] denotes the image of L in W_{0}/SO(3).

[figure: a; b; c; C = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}); O = (0, 0, 0); D = (\frac{1}{2}, \frac{1}{2}, \frac{1}{4}); B = (\frac{1}{2}, \frac{1}{2}, 0); A = (\frac{1}{2}, 0, 0); E = (\frac{1}{2}, \frac{1}{4}, −\frac{1}{4}); F = (\frac{1}{3}, \frac{1}{3}, −\frac{1}{3})]

Figure 1: The space D_{0} and the equivalence relation ∼ . The equivalence relation is given by identifying the triangle ΔABD with the triangles ΔACD and ΔABE via reflections at the edges AD and AB.

Next we show that equivalent triples have the same image under φ so that φ induces a continuous map φ̃. So assume 0 ≤ c ≤ \frac{1}{2}b and consider the lattice λ(\frac{1}{2}, b, c). This has (at least) 4 pairs of minimal vectors, namely ±v_{1} = ±(1, 0, 0), ±v_{2} = ±(a, x, 0), ±v_{3} = ±(b, y, z) and ±(v_{1} − v_{2}). (Here x, y and z are as before.) Then it is straightforward to check that the scalar products between the vectors v′_{1} := v_{1}, v′_{2} := v_{1} − v_{2} and v′_{3} := v_{3} are (\frac{1}{2}, b, b − c) and those between the vectors v′′_{1} := v_{2} − v_{1}, v′′_{2} := v_{2} and v′′_{3} := −v_{3} are (\frac{1}{2}, b − c, −c) and Lemma 2.5 implies again that the image under φ of these triples agree.

Now we turn to injectivity of φ̃. As D_{0}/ ∼ is compact and W_{0}/SO(3) is Hausdorff, this will show that φ̃ is a homeomorphism and finish the proof. So assume φ(a, b, c) = φ(a′, b′, c′). By assumption the corresponding lattices L := λ(a, b, c) and L′ := λ(a′, b′, c′) agree up to a rotation R ∈ SO(3), i.e. L = RL′. In particular, L and L′ have the same number of minimal vectors, the vectors Rv′_{1}, Rv′_{2}, Rv′_{3} form a set of linearly independent vectors of length 1 in L and the triple (a′, b′, c′) occurs as a triple of scalar products between three linearly independent vectors of length 1 of L. We have to show that this happens only if (a, b, c) and (a′, b′, c′) are equivalent under the relation ∼. In the “generic” case, i.e. if a ≠ \frac{1}{2} and a+b−c ≠ 1, L has only the minimal vectors ±v_{1}, ±v_{2} and ±v_{3} (cf. Proposition 2.3), and in this case it is obvious that the triple of scalar products is uniquely determined by L and by the condition a ≥ b ≥ |c|. Now assume (a, b, c) ≠ (a′, b′, c′) and we have 6 pairs of minimal vectors. By Proposition 2.3 this can only happen if w.l.o.g. (a, b, c) = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}) and (a′, b′, c′) = (\frac{1}{2}, \frac{1}{2}, 0). However, these points are clearly equivalent under ∼. Next assume we have precisely 5 pairs of minimal vectors in L. By Proposition 2.3 and because φ is constant on ∼ - equivalence classes we may assume that our triples are of the form (\frac{1}{2}, \frac{1}{2}, c) and (\frac{1}{2}, \frac{1}{2}, c′) with 0 < c, c′ ≤ \frac{1}{4} and we have to show c = c′. The lattice L = λ(\frac{1}{2}, \frac{1}{2}, c) has the following pairs of minimal vectors of length 1: ±v_{1}, ±v_{2}, ±v_{3}, ±(v_{1} − v_{2}) and ±(v_{1} − v_{3}), and the triple (\frac{1}{2}, \frac{1}{2}, c′) must occur as a triple of scalar products of 3 linearly independent vectors taken from those 5 pairs. It is now straightforward to check that this can happen only if c = c′. Finally assume we have exactly 4 pairs of linearly independent vectors of length 1 in L = λ(a, b, c). Again by Proposition 2.3 and because φ is constant on ∼ - equivalence classes we may assume that the triple (a, b, c) satisfies either a = \frac{1}{2} and 0 ≤ c ≤ \frac{1}{2}b < \frac{1}{4}, or a ≠ \frac{1}{2}, c ≤ 0 and a + b − c = 1. We have to show that L determines uniquely a triple of this form. In the first case the set of minimal vectors consists of ±v_{1}, ±v_{2}, ±v_{3}, ±(v_{1} − v_{2}), in the second case of ±v_{1}, ±v_{2}, ±v_{3}, ±(v_{1} − v_{2} − v_{3}). Again it is straightforward to see that all triples of linearly independent vectors taken from those sets which lead to scalar products of the required form, lead indeed to the same scalar products, and thus the proof is complete. □

2.4 Γ_{i} - equivariant cell structures on Z

2.4.1 The case i = 0

We recall the map γ : D_{0} ⟶ Y which sends d = (a, b, c) ∈ D_{0}, up to a scalar multiple, to the inverse of the matrix whose i - th colum is the vector v_{i} specified in the last section (cf. the discussion before Theorem 2.4). The composition of the map γ : D_{0} ⟶ Y with the quotient map π_{Z} : Y ⟶ Y/SO(3) ≅ Z will be denoted by ψ_{0}. Note that ψ_{0}(d) is the equivalence class of the positive definite quadratic form for which the scalar products between the standard basis vectors e_{1}, e_{2}, e_{3} are given by ⟨e_{i}, e_{j}⟩ = ⟨v_{i}, v_{j}⟩ for i ≤ i, j ≤ 3, i.e. ⟨e_{i}, e_{i}⟩ = 1 for i = 1, 2, 3, ⟨e_{1}, e_{2}⟩ = a, ⟨e_{1}, e_{3}⟩ = b and ⟨e_{2}, e_{3}⟩ = c. In particular this map is injective and a homeomorphism from D_{0} to ψ_{0}(D_{0}). The Γ_{0} - equivariant extension Γ_{0} × D_{0} ⟶ Z which sends (g, d) to gψ_{0}(d) will still be denoted by ψ_{0}. Let ∼_{0} be the equivalence relation on Γ_{0} × D_{0} induced by the map ψ_{0}, i.e. defined by (g, d) ∼_{0} (g′, d′) iff ψ_{0}(g, d) = ψ_{0}(g′, d′). Then ∼_{0} is Γ_{0} - equivariant, i.e. if (g, d) ∼_{0} (g′, d′) then (hg, d) ∼_{0} (hg′, d′) for every h ∈ Γ_{0}. Let g^{AD} ∈ Γ_{0} be given by g^{AD}(e_{1}) = −e_{1}, g^{AD}(e_{2}) = −e_{1}+e_{2} and g^{AD}(e_{3}) = −e_{3}, and let g^{AB} ∈ Γ_{0} be given by g^{AB}(e_{1}) = e_{2} − e_{1}, g^{AB}(e_{2}) = e_{2} and g^{AB}(e_{3}) = −e_{3}. Then we have the following result which is a refinement of Theorem 2.4.

Theorem 2.6 The equivalence relation ∼_{0} on Γ_{0} × D_{0} induced by the map ψ_{0} : Γ_{0} × D_{0} ⟶ Z is the smallest Γ_{0} - equivariant equivalence relation generated by the following elementary relations: (g, d) and (g′, d′) are elementary equivalent if either

1. g′ = 1, d = d′ and g belongs to the isotropy group H_{d} ⊂ Γ_{0} of the (class of the) quadratic form ψ_{0}(d).

2. g′ = 1, d = (\frac{1}{2}, b, c), 0 ≤ c ≤ \frac{1}{2}b, and either

d′ = (\frac{1}{2}, b, b − c), g = g^{AD} , or

d′ = (\frac{1}{2}, b − c, −c), g = g^{AB} .

Furthermore the induced map Ψ_{0} : Γ_{0} × D_{0}/ ∼_{0}⟶ Z is a homeomorphism of Γ_{0} - spaces.

Proof. First we observe that points of Γ_{0} × D_{0} which are elementary equivalent are mapped to the same point in Z under ψ_{0}. This is trivial for the first elementary relation. For the second one it follows because by definition of g^{AD} and Lemma 2.5 we have g^{AD}γ(\frac{1}{2}, b, c) ∈ γ(\frac{1}{2}, b, b − c)SO(3), i.e. g^{AD}ψ_{0}(\frac{1}{2}, b, c) = ψ_{0}(\frac{1}{2}, b, b − c). Similarly, g^{AB} is defined such that g^{AB}γ(\frac{1}{2}, b, c) ∈ γ(\frac{1}{2}, b − c, −c)SO(3), i.e. g^{AB}ψ_{0}(\frac{1}{2}, b, c) = ψ_{0}(\frac{1}{2}, b − c, −c). It follows that ψ_{0} induces a map Ψ_{0} as claimed. Furthermore ψ_{0} induces (on passing to the quotients with respect to the actions of Γ_{0}) the surjection φ of Theorem 2.4. In particular, if follows that ψ_{0} and hence Ψ_{0} is surjective. Next assume that ψ_{0}(g, d) = ψ_{0}(g′, d′). Then Theorem 2.4 shows that d ∼ d′ and by definition of ∼_{0} we may therefore assume that d = d′. But then we clearly have g^{−1}g′ ∈ H_{d} and by Γ_{0} - equivariance of ∼_{0} we see that (g, d) ∼_{0} (g′, d′) and injectivity of Ψ_{0} follows. Finally it is easy to see that the map Ψ_{0} is an open map and hence a homeomorphism (e.g. by using that the actions of Γ_{0} on Γ_{0} × D_{0}/ ∼_{0} and Z are proper, and that the induced map on the quotient spaces is a homeomorphism by Theorem 2.4). □

Cell structures on D_{0}, D_{0}/ ∼ and a Γ_{0} - equivariant cell structure on Z. Theorem 2.6 allows us to establish a Γ_{0} - equivariant cell structure on Z in terms of a cell structure on D_{0} resp. on D_{0}/ ∼. We start with cell structures on D_{0} and D_{0}/ ∼ (see figure 1).

0. The 0 - dimensional cells of D_{0} are the vertices O = (0, 0, 0), A = (\frac{1}{2}, 0, 0), B = (\frac{1}{2}, \frac{1}{2}, 0), C = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}), D = (\frac{1}{2}, \frac{1}{2}, \frac{1}{4}), E = (\frac{1}{2}, \frac{1}{4}, −\frac{1}{4}) and F = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}). On D_{0}/ ∼ this gives 5 cells which will still be labelled O, A, B ∼ C, D ∼ E and F.

1. The 1 - dimensional cells of D_{0} are the edges OC, OF, OA, EF, BF, AB, AC, AD, AE, BD, CD and BE. On D_{0}/ ∼ this gives 8 cells labelled OC, OF, OA, EF, BF, AB ∼ AC, AD ∼ AE and BD ∼ CD ∼ BE.

2. The 2 - dimensional cells of D_{0} are the quadrangles OAEF (characterized by b = −c) and OCBF (a = b), and the triangles OAC (b = c), BEF (a + b − c = 1) and ABD, ACD and ABE. On D_{0}/ ∼ this gives 5 cells labelled OAEF, OCBF, OAC, BEF and ABD ∼ ACD ∼ ABE.

3. D_{0} has one cell of dimension 3, namely the interior of D_{0}, and this gives also one cell for D_{0}/ ∼.

It follows easily from Proposition 2.3 (see also Section 2.5 below) that the isotropy groups H_{d} of the action of Γ_{0} on Z at ψ_{0}(d) are constant within the interior of each cell e of D_{0} and this is the reason for the choice of our cell structure on D_{0}. If we denote this isotropy group by H_{e} then Theorem 2.6 shows that Z has an equivariant cell structure with one orbit (Γ_{0}/H_{e}) × e of cells for each equivalence class of cells in D_{0}/ ∼. The attaching maps can be read off from figure 1.

2.4.2 The cases i = 1 and i = 2

We consider the right Γ_{0} - spaces Γ_{i}\Γ_{0}. In case i = 1 this coset space can be identified with the set S_{1} of non-zero vectors in (𝔽_{2}^{3} − {0}) and in case i = 2 with the set S_{2} of pairs consisting of a line in 𝔽_{2}^{3} and a plane in 𝔽_{2}^{3} containing the line, i.e. with the set of complete flags in 𝔽_{2}^{3}. In fact, there is a canonical left action of Γ_{0} on the sets S_{i}, and if we convert this into a right action in the usual way via s · g := g^{−1}s, then the map Γ_{0} ⟶ S_{1}, g ↦ g^{−1}(e_{1}) mod 2 induces an isomorphism of right Γ_{0} - spaces Γ_{1}\Γ_{0} ⟶ S_{1}; similarly the map Γ_{0} ⟶ S_{2} which sends g to the flag (⟨g^{−1}(100)⟩ ⊂ ⟨g^{−1}(100), g^{−1}(010)⟩ mod 2) induces an isomorphism of right Γ_{0} - spaces Γ_{2}\Γ_{0} ⟶ S_{2}. (Here ⟨ ⟩ denotes the subgroup generated by the elements within the brackets and 100, 010 are standard basis vectors in 𝔽_{2}^{3}.) The sets D_{0} × S_{i} will be denoted by D_{i}. Now we choose representatives for the right cosets of Γ_{i} in Γ_{0}. Such a choice of a representative g_{s} for each s ∈ S_{i} gives an explicit Γ_{i} - equivariant homeomorphism

Γ_{i} × D_{0} × S_{i} ⟶ Γ_{0} × D_{0}, (g, d, s) ↦ (gg_{s}, d) .

In order to obtain a Γ_{i} - equivariant cell structure on Z we will carry over the Γ_{0} - equivariant equivalence relation ∼_{0} on Γ_{0} × D_{0} to a Γ_{i} - equivariant equivalence relation ∼_{i} on Γ_{i} × D_{i}. For this we note that the isotropy groups H_{d} act from the right on the coset spaces S_{i}. Likewise the matrices g^{AB} and g^{AD} act from the right on S_{i}. The following result is now a straightforward consequence of Theorem 2.6.

Theorem 2.7 The equivalence relation ∼_{i} on Γ_{i} × D_{i} induced by the map

ψ_{i} : Γ_{i} × D_{i} ⟶ Z, (g, d, s) ↦ gg_{s}ψ_{0}(d)

is the smallest Γ_{i} - equivariant equivalence relation generated by the following elementary relations: (g, d, s) and (g′, d′, s′) are elementary equivalent if either

1. g′ = 1, d = d′, there exists an element h ∈ H_{d} with s = s′h (in particular s and s′ belong to the same H_{d} - orbit with respect to the right action of H_{d} on the set S_{i}) and g is determined by gg_{s} = g_{s′}h.

2. g′ = 1, d = (\frac{1}{2}, b, c), 0 ≤ c ≤ \frac{1}{2}b and either

d′ = (\frac{1}{2}, b, b − c), s = s′g^{AD}, gg_{s} = g_{s′}g^{AD} , or

d′ = (\frac{1}{2}, b − c, −c), s = s′g^{AB}, gg_{s} = g_{s′}g^{AB} .

Furthermore the induced map Ψ_{i} : Γ_{i}×D_{i}/ ∼_{i}⟶ Z is a homeomorphism of Γ_{i} - spaces, the Γ_{i} - equivariant equivalence relation ∼_{i} induces an equivalence relation (denoted by ∼_{(i)}) on the quotient D_{i} of Γ_{i} × D_{i} such that the induced map Ψ̃ : D_{i}/ ∼_{(i)}⟶ Γ_{i}\Z is a homeomorphism. □

Γ_{i} - equivariant cell structures on Z and cell structures on Γ_{i}\Z. Theorem 2.7 yields Γ_{i} - equivariant cell strucures on the space Z (and then ordinary cell structures on the quotients Γ_{i}\Z). The indexing set for the Γ_{i} - orbits of cells on Z (resp. the cells on the quotients Γ_{i}\Z) are equivalence classes of pairs (e, s) with e a cell in D_{0} and s ∈ S_{i}, with the equivalence relation generated by the following elementary relations: (e, s) ∼_{i} (e′, s′) iff either

1. e = e′ and s and s′ are in the same H_{e} - orbit, or 2a. e is ABD or a face of it, e′ is ACD or the corresponding face of it and s = s′g^{AD}, or 2b. e is ABD or a face of it, e′ is ABE or the corresponding face of it and s = s′g^{AB}.

The Γ_{i} - orbits of cells of Z are then of the form Γ_{i}/H_{(e,s)} × (e, s) where (e, s) runs through a set of representatives of equivalence classes of such pairs and the isotropy group H_{(e,s)} of the cell (e, s) is given by Γ_{i} ∩ g_{s}H_{e}g_{s}^{−1}, i.e agrees up to conjugation by g_{s} with g_{s}^{−1}Γ_{i}g_{s} ∩H_{e} which is the isotropy group of s with respect to the right action of H_{e} on S_{i}. The attaching maps can again be read off from figure 1. In the next two sections we will make this concrete, i.e. we will describe in explicit form the isotropy groups H_{e}, their actions on the sets S_{i}, and also the effect of the action of g^{AB} and g^{AD} on S_{i}.

2.5 Symmetries of well-rounded quadratic forms

In order to make the equivariant cell structure of the spaces Γ_{i}\Z concrete we need to determine the isotropy groups H_{(a,b,c)} of the action of Γ_{0} on Z at ψ_{0}(a, b, c). Of course, H_{(a,b,c)} preserves the length of vectors and the scalar products between them (both taken, of course, with respect to a representative quadratic form of (the equivalence class of quadratic forms) ψ_{0}(a, b, c)), and hence H_{(a,b,c)} acts on the set of minimal vectors in the standard lattice. These sets have been determined in Theorem 2.4 (we just have to replace the letter v by e everywhere). The standard basis vectors are always minimal vectors and so H_{(a,b,c)} is determined by this action. It is clear that the groups H_{(a,b,c)} are constant in the interior of each cell of D_{0} and this gives the justification for the choice of our cell structure on D_{0}. The case of the 3 - dimensional cell is particularly simple. If (a, b, c) is in its interior then we have only the 3 standard basis vectors and their negatives in the set of minimal vectors and it is easy to check that H_{(a,b,c)} = {1}. Tables 1, 2 and 3 below give the isotropy groups on the open cells of D_{0} of dimension 2, 1 and 0. In fact it will be enough for us to take one cell from each ∼ - equivalence class of cells. The first column lists the name of the cell, the second one the set of minimal vectors on the standard lattice with respect to (a representative quadratic form of) ψ_{0}(a, b, c) if (a, b, c) is an interior point of the appropriate cell and the third column gives the isotropy group. The last column describes the action of the isotropy group on the tuple (e_{1}, e_{2}, e_{3}) of minimal vectors explicitly; the 3 - tuples in this column are the images of the tuple (e_{1}, e_{2}, e_{3}) under the action of appropriate generators. The proofs are straightforward and are left to the reader. We use the following notation in these tables: for the symmetric group on n - letters we write 𝔖_{n}, ≀ denotes the wreath product construction, and the dihedral group with n elements is denoted by D_{n}.

Table 1: Symmetries on the 2-dimensional cells

Cell Minimal vectors Isotropy

Generators

ABD ±e_{1}, ±e_{2}, ±e_{3}, trivial
±(e_{1} − e_{2})
OAC ±e_{1}, ±e_{2}, ±e_{3}  ℤ/2  (−e_{2}, −e_{1}, −e_{3})
OAEF ±e_{1}, ±e_{2}, ±e_{3}  ℤ/2  (e_{2}, e_{1}, −e_{3})
OCBF ±e_{1}, ±e_{2}, ±e_{3}  ℤ/2  (−e_{1}, −e_{3}, −e_{2})
BEF  ±e_{1}, ±e_{2}, ±e_{3} ℤ/2 × ℤ/2 (−e_{2}, −e_{1}, e_{1} − e_{2} − e_{3})
±(e_{1} − e_{2} − e_{3})  (−e_{3}, e_{1} − e_{2} − e_{3}, −e_{1})

22 Hans-Werner Henn

Table 2: Symmetries on the 1-dimensional cells

Cell Minimal vectors Isotropy

Generators

OC  ±e_{1}, ±e_{2}, ±e_{3}  𝔖_{3}  (−e_{1}, −e_{3}, −e_{2})

(−e_{2}, −e_{1}, −e_{3})

OF  ±e_{1}, ±e_{2}, ±e_{3}  𝔖_{3}  (−e_{1}, −e_{3}, −e_{2})

(e_{2}, e_{1}, −e_{3})

OA  ±e_{1}, ±e_{2}, ±e_{3}  ℤ/2 × ℤ/2  (−e_{2}, −e_{1}, −e_{3})

(e_{2}, e_{1}, −e_{3})

AB  ±e_{1}, ±e_{2}, ±e_{3},  ℤ/2  (e_{2} − e_{1}, e_{2}, −e_{3})

±(e_{1} − e_{2})

AD  ±e_{1}, ±e_{2}, ±e_{3},  ℤ/2  (−e_{1}, e_{2} − e_{1}, −e_{3})

±(e_{1} − e_{2})

BD  ±e_{1}, ±e_{2}, ±e_{3},  ℤ/2 × ℤ/2  (−e_{1}, −e_{3}, −e_{2})
±(e_{1} − e_{2}), ±(e_{1} − e_{3})  (−e_{1}, e_{3} − e_{1}, e_{2} − e_{1})

BF ±e_{1}, ±e_{2}, ±e_{3},

D_{8}

BEF symmetries,

±(e_{1} − e_{2} − e_{3})  (−e_{1}, −e_{3}, −e_{2})

EF ±e_{1}, ±e_{2}, ±e_{3}

D_{8}

BEF symmetries,

±(e_{1} − e_{2} − e_{3})  (e_{2}, e_{1}, −e_{3})

Table 3: Symmetries on the 0-dimensional cells

Cell Minimal vectors Isotropy

Description

O ±e_{1}, ±e_{2}, ±e_{3} 𝔖_{4} ≅ (ℤ/2)^{2} ≀ 𝔖_{3} symmetry of a cube;

index 2 in (ℤ/2)^{3} ≀ 𝔖_{3}

permuting the set of pairs

{±e_{1}}, {±e_{2}}, {±e_{3}}

C ±e_{1}, ±e_{2}, ±e_{3} 𝔖_{4} ≅ (ℤ/2)^{2} ≀ 𝔖_{3} ℤ/2 × ℤ/2 generated by:

±(e_{1} − e_{2})  (e_{2} − e_{3}, e_{1} − e_{3}, −e_{3})
±(e_{1} − e_{3})  (e_{3} − e_{2}, −e_{2}, e_{1} − e_{2});

±(e_{2} − e_{3})

𝔖_{3} symmetry as on OC

F ±e_{1}, ±e_{2}, ±e_{3} 𝔖_{4} ≅ (ℤ/2)^{2} ≀ 𝔖_{3} ℤ/2 × ℤ/2 action as on BEF;

±(e_{1} − e_{2} − e_{3})

𝔖_{3} symmetry as on OF

A ±e_{1}, ±e_{2}, ±e_{3} D_{12} {±e_{3}} is invariant.

±(e_{1} − e_{2})

Standard action on the regular planar hexagon formed by

±e_{1}, ±e_{2}, ±(e_{1} − e_{2})

D ±e_{1}, ±e_{2}, ±e_{3} D_{8} ℤ/2 × ℤ/2 action as on BD;

±(e_{1} − e_{2})

additional generator:

±(e_{1} − e_{3})  (−e_{1}, e_{2} − e_{1}, −e_{3})

2.6 The equivalence relations ∼_{i} on the spaces Γ_{i} × D_{i}

By Theorem 2.7 the equivalence relations ∼_{i}, as well as Γ_{i} - equivariant cell structures on Z and ordinary cell structures on Γ_{i}\Z, are determined by the right actions of the isotropy groups H_{d} ⊂ Γ_{0}, d ∈ D, on the sets S_{i} together with the right action of g^{AB} and g^{AD} on these sets. Clearly, the associated left action of H_{d} on the sets S_{i} has identical orbits and isotropy groups as the right action; in the case of g^{AB} and g^{AD} the left and right actions are even identical because both elements agree with their own inverses; we prefer to work with the left actions. As remarked above the isotropy groups and hence their actions are constant on the cells of D and we consider cell by cell separately. In our analysis the elements in S_{2} will be labelled by pairs consisting of a non-zero vector in 𝔽_{2}^{3} and a plane in 𝔽_{2}^{3} containing this vector, e.g. 010x = z, 011x = 0, …. The plane with equation x + y + z = 0 will be abbreviated by Σ = 0. We will also abbreviate (ABD, 100y = z) by ABD100y = z and so on. The proofs in this section are all straightforward and are left to the reader.

2.6.1 3 - cells

By Theorem 2.7 and by Section 2.5 there are no identifications in the interior of the 3 - cells. As there is only one 3 - cell in D_{0}, the 3 - cells in Γ_{1}\Z will be labelled just by the non-zero vectors 𝔽_{2}^{3}. So there are seven 3 - cells in Γ_{1}\Z, labelled:

100, 010, 001, 110, 101, 011, 111 .

Similarly there are 21 cells of dimension 3 in Γ_{1}\Z which will be labelled:

\begin{matrix}100y = 0, & 100z = 0, & 100y = z \\ 010x = 0, & 010z = 0, & 010x = z, \\ 001x = 0, & 001y = 0, & 001x = y, \\ 110z = 0, & 110x = y, & 110Σ = 0, \\ 101y = 0, & 101x = z, & 101Σ = 0, \\ 011x = 0, & 011y = z, & 011Σ = 0, \\ 111x = y, & 111x = z, & 111y = z. \end{matrix}

The isotropy group of the 3 - cell is trivial, so there is nothing else to do in this case.

2.6.2 2 - cells

1. ABD By Section 2.5 and Theorem 2.7 all the relations involving these 2 - cells are of type 2, i.e. the following cells become equivalent.

ABDs∼_{i}ACDg^{AD}s∼_{i}ABEg^{AB}s .  (2.5)

Clearly the isotropy groups are trivial for all these cells. We will now make the maps g^{AD} and g^{AB} explicit. We recall that by definition these matrices induce the linear maps on 𝔽_{2}^{3} given by

g^{AD}(100) = 100, g^{AD}(010) = 110, g^{AD}(001) = 001 ,
g^{AB}(100) = 110, g^{AB}(010) = 010, g^{AB}(001) = 001 .

Explicit knowledge of the action of these maps on the sets S_{i} will be used repeatedly later on and therefore these maps are explicitly described in tables 4 and 5 below.

Table 4: Action of g^{AD} and g^{AB} on S_{1}

\begin{matrix}s & 100 & 010 & 001 & 110 & 101 & 011 & 111 \\ g^{AD}s & 100 & 110 & 001 & 010 & 101 & 111 & 011 \\ g^{AB}s & 110 & 010 & 001 & 100 & 111 & 011 & 101 \end{matrix}

Table 5: Action of g^{AD} and g^{AB} on S_{2}

s  100y = 0 100z = 0 100y = z 010x = 0 010z = 0 010x = z
g^{AD}s 100y = 0 100z = 0 100y = z 110x = y 110z = 0 110Σ = 0
g^{AB}s 110x = y 110z = 0 110Σ = 0 010x = 0 010z = 0 010x = z
s  001x = 0 001y = 0 001x = y 110z = 0 110x = y 110Σ = 0
g^{AD}s 001x = y 001y = 0 001x = 0 010z = 0 010x = 0 010x = z
g^{AB}s 001x = 0 001x = y 001y = 0 100z = 0 100y = 0 100y = z
s  101y = 0 101x = z 101Σ = 0 011x = 0 011y = z 011Σ = 0
g^{AD}s 101y = 0 101Σ = 0 101x = z 111x = y 111y = z 111x = z
g^{AB}s 111x = y 111x = z 111y = z 011x = 0 011Σ = 0 011y = z
s  111x = y 111x = z 111y = z
g^{AD}s 011x = 0 011Σ = 0 011y = z
g^{AB}s 101y = 0 101x = z 101Σ = 0

2. OAC In the interior of these cells all relations are of type 1. In other words we have to determine the action of the group H_{OAC} ≅ ℤ/2 on the vector space 𝔽_{2}^{3}. By table 1 the action of the non-trivial element h ∈ H_{OAC} on 𝔽_{2}^{3} is given by

h100 = 010, h010 = 100, h001 = 001 .

Hence we get the following orbits and isotropy groups for the action on S_{1}:

Isotropy groups {1} {1} ℤ/2 ℤ/2 ℤ/2
Orbits  100, 010 101, 011 001 110 111

For the action on S_{2} we obtain:

Isotropy groups {1}  {1}  {1}  {1}  {1}
Orbits  100y = 0 100z = 0 100y = z 101y = 0 101x = z
010x = 0 010z = 0 010x = z 011x = 0 011y = z
Isotropy groups {1}  {1}  {1}  ℤ/2  ℤ/2
Orbits  101Σ = 0 001x = 0 111x = z 001x = y 111x = y
011Σ = 0 001y = 0 111y = z
Isotropy groups ℤ/2  ℤ/2  ℤ/2
Orbits  110x = y 110z = 0 110Σ = 0

3. OAEF Again all relations are of type 1. Furthermore the action of H_{OAEF} on 𝔽_{2}^{3} is the same as in the case of OAC. Therefore we obtain the same list of orbits and isotropy groups.

4. OCBF Once again all relations are of type 1. By table 1 the action of the non-trivial element h ∈ H_{OCBF} on 𝔽_{2}^{3} is given by

h100 = 100, h010 = 001, h001 = 010

and we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups {1} {1} ℤ/2 ℤ/2 ℤ/2
Orbits  010, 001 110, 101 100 011 111

Isotropy groups {1}  {1}  {1}  {1}  {1}
Orbits  010x = 0 010z = 0 010x = z 110z = 0 110x = y
001x = 0 001y = 0 001x = y 101y = 0 101x = z
Isotropy groups {1}  {1}  {1}  ℤ/2  ℤ/2
Orbits  110Σ = 0 100y = 0 111x = y 100y = z 111y = z
101Σ = 0 100z = 0 111x = z
Isotropy groups ℤ/2  ℤ/2  ℤ/2
Orbits  011x = 0 011y = z 011Σ = 0

5. BEF All relations are of type 1. By table 1 the action of two generators h_{1} and h_{2} of H_{BEF} ≅ ℤ/2 × ℤ/2 on 𝔽_{2}^{3} is given by

h_{1}100 = 010, h_{1}010 = 100, h_{1}001 = 111 ,
h_{2}100 = 001, h_{2}010 = 111, h_{2}001 = 100 .

Hence we get the following orbits and (types of) isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups  {1}  ℤ/2 ℤ/2 ℤ/2
Orbits  100, 111, 010, 001 110 101 011

Isotropy groups {1}  {1}  {1}  ℤ/2  ℤ/2
Orbits  100y = 0 100z = 0 100y = z 110x = y 101x = z
111x = z 111x = y 111y = z 110z = 0 101y = 0
010x = z 010z = 0 010x = 0
001y = 0 001x = y 001x = 0

Isotropy groups ℤ/2 ℤ/2 × ℤ/2 ℤ/2 × ℤ/2 ℤ/2 × ℤ/2

Orbits  011y = z 110Σ = 0 101Σ = 0 011Σ = 0
011x = 0

2.6.3 1 - cells

1. OC There are only relations of type 1. By table 1 the action of two generators h_{1} and h_{2} of H_{OC} ≅ 𝔖_{3} on 𝔽_{2}^{3} is given by

h_{1}100 = 100, h_{1}010 = 001, h_{1}001 = 010,
h_{2}100 = 010, h_{2}010 = 100, h_{2}001 = 001,

and we get the following orbits and isotropy groups for the actions on S_{1} resp. S_{2}:

Isotropy groups  ℤ/2  ℤ/2  𝔖^{3}
Orbits  100, 010, 001 110, 101, 011 111

Isotropy groups  {1}  ℤ/2  ℤ/2
Orbits  100y = 0, 100z = 0 100y = z 110x = y
010x = 0, 010z = 0 010x = z 101x = z
001x = 0, 001y = 0 001x = y 011y = z
Isotropy groups  ℤ/2  ℤ/2  ℤ/2
Orbits  110z = 0  110Σ = 0 111x = y
101y = 0  101Σ = 0 111x = z
011x = 0  011Σ = 0 111y = z

2. OF Again there are only relations of type 1, and furthermore the action of H_{OF} on 𝔽_{2}^{3} is the same as in the case of OC. Therefore we obtain the same list of orbits and isotropy groups.

3. OA There are only relations of type 1. By table 1 the action of two generators h_{1} and h_{2} of H_{OA} ≅ ℤ/2 × ℤ/2 on 𝔽_{2}^{3} is given by

h_{1}100 = 100, h_{1}010 = 010, h_{1}001 = 001 ,
h_{2}100 = 010, h_{2}010 = 100, h_{2}001 = 001 .

Hence we get the same orbits as in the case of the 2 - cells OAC resp. OAEF. However, the isotropy groups are now larger: the trivial ones get replaced by ℤ/2 generated by h_{1}h_{2}, the ℤ/2 gets replaced by ℤ/2 × ℤ/2.

4. AB and AC There are relations of both types. Those of type 2 lead to ABs ∼_{i} ACg^{AD}s and are described by tables 4 and 5. As far as relations of type 1 are concerned we can concentrate on the edge AB. By table 2 the action of the non-trivial element h ∈ H_{AB} on 𝔽_{2}^{3} is here given by

h100 = 110, h010 = 010, h001 = 001 .

Hence we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2} (the orbits for AB and those for AC in the same column correspond to each other via g^{AD}; the same conventions will hold in later tables of this section):

Isotropy groups {1} {1} ℤ/2 ℤ/2 ℤ/2 Orbits for AB 100, 110 101, 111 010 001 011 Orbits for AC 100, 010 101, 011 110 001 111

Isotropy groups {1}  {1}  {1}  {1}  {1}

Orbits for AB 100y = 0 100z = 0 100y = z 101y = 0 101Σ = 0

110x = y 110z = 0 110Σ = 0 111x = y 111y = z

Orbits for AC 100y = 0 100z = 0 100y = z 101y = 0 101x = z

010x = 0 010z = 0 010x = z 011x = 0 011y = z
Isotropy groups {1}  {1}  {1}  ℤ/2  ℤ/2

Orbits for AB 101x = z 001x = y 011Σ = 0 010x = 0 010z = 0

111x = z 001y = 0 011y = z

Orbits for AC 101Σ = 0 001x = 0 111x = z 110x = y 110z = 0

011Σ = 0 001y = 0 111y = z
Isotropy groups ℤ/2  ℤ/2  ℤ/2

Orbits for AB 010x = z 001x = 0 011x = 0 Orbits for AC 110Σ = 0 001x = y 111x = y

5. AD and AE There are again relations of both types. Those of type 2 lead to ADs ∼_{i} AEg^{AB}s and are described by tables 4 and 5. As far as relations of type 1 are concerned we can concentrate on the edge AD. By table 2 the action of the non-trivial element h ∈ H_{AD} on 𝔽_{2}^{3} is here given by

h100 = 100, h010 = 110, h001 = 001 .

Hence we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups {1} {1} ℤ/2 ℤ/2 ℤ/2 Orbits for AD 110, 010 111, 011 001 100 101 Orbits for AE 100, 010 101, 011 001 110 111

Isotropy groups {1}  {1}  {1}  {1}  {1}

Orbits for AD 110x = y 110z = 0 110Σ = 0 111x = y 111x = z

010x = 0 010z = 0 010x = z 011x = 0 011Σ = 0

Orbits for AE 100y = 0 100z = 0 100y = z 101y = 0 101x = z

010x = 0 010z = 0 010x = z 011x = 0 011y = z
Isotropy groups {1}  {1}  {1}  ℤ/2  ℤ/2

Orbits for AD 111y = z 001x = 0 101x = z 100y = 0 100z = 0

011y = z 001x = y 101Σ = 0

Orbits for AE 101Σ = 0 001x = 0 111x = z 110x = y 110z = 0

011Σ = 0 001y = 0 111y = z
Isotropy groups ℤ/2  ℤ/2  ℤ/2

Orbits for AD 100y = z 001y = 0 101y = 0 Orbits for AE 110Σ = 0 001x = y 111x = y

6. BD, CD and BE There are again relations of both types. Those of type 2 lead to BDs ∼_{i} BEg^{AB}s resp. BDs ∼_{i} CDg^{AD}s and are described by tables 4 and 5. As far as relations of type 1 are concerned we can concentrate on the edge BD. By table 2 the action of two generators h_{1} and h_{2} of H_{BD} ≅ ℤ/2 × ℤ/2 on 𝔽_{2}^{3} is here given by

h_{1}100 = 100, h_{1}010 = 001, h_{1}001 = 010 ,
h_{2}100 = 100, h_{2}010 = 101, h_{2}001 = 110 .

Hence we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups {1} ℤ/2 × ℤ/2 ℤ/2 × ℤ/2 ℤ/2 × ℤ/2 Orbits for BD 010, 001 100 011 111

101, 110

Orbits for CD 110, 001 100 111 011

101, 010

Orbits for BE 010, 001 110 011 101 111, 100

Isotropy groups {1}  {1}  {1}  ℤ/2  ℤ/2

Orbits for BD 010x = 0 010z = 0 010x = z 100y = 0 011x = 0

001x = 0 001y = 0 001x = y 100z = 0 011Σ = 0
101Σ = 0 101y = 0 101x = z
110Σ = 0 110z = 0 110x = y

Orbits for CD 110x = y 110z = 0 110Σ = 0 100y = 0 111x = y

001x = y 001y = 0 001x = 0 100z = 0 111x = z
101x = z 101y = 0 101Σ = 0
010x = z 010z = 0 010x = 0

Orbits for BE 010x = 0 010z = 0 010x = z 110x = y 011x = 0

001x = 0 001x = y 001y = 0 110z = 0 011y = z
111y = z 111x = y 111x = z
100y = z 100z = 0 100y = 0

Isotropy groups ℤ/2 ℤ/2 × ℤ/2 ℤ/2 × ℤ/2 ℤ/2 × ℤ/2 Orbits for BD 111x = y 100y = z 011y = z 111y = z

111x = z

Orbits for CD 011x = 0 100y = z 111y = z 011y = z

011Σ = 0

Orbits for BE 101y = 0 110Σ = 0 011Σ = 0 101Σ = 0 101x = z

7. BF There are only relations of type 1. By table 1 and table 2 we know the action of three generating involutions of H_{BF} ≅ D_{8} on 𝔽_{2}^{3}

h_{1}100 = 010, h_{1}010 = 100, h_{1}001 = 111 ,
h_{2}100 = 001, h_{2}010 = 111, h_{2}001 = 100 ,
h_{3}100 = 100, h_{2}010 = 001, h_{2}001 = 010 .

Hence we get the following orbits and isotropy groups for the action on S_{1} resp S_{2}:

Isotropy groups  ℤ/2  ℤ/2 × ℤ/2 D^{8}
Orbits  100, 111, 010, 001 110, 101 011

Isotropy groups  {1}  ℤ/2  ℤ/2
Orbits  100y = 0, 111x = z 100y = z 110x = y
010x = z, 001y = 0 111y = z 110z = 0
100z = 0, 111x = y 010x = 0 101x = z
001x = y, 010z = 0 001x = 0 101y = 0
Isotropy groups  ℤ/2 × ℤ/2  ℤ/2 × ℤ/2 D^{8}
Orbits  110Σ = 0  011y = z 011Σ = 0
101Σ = 0  011x = 0

8. EF Again there are only relations of type 1 and by table 1 and table 2 we know the action of three generating involutions of H_{EF} ≅ D_{8} on 𝔽_{2}^{3}

h_{1}100 = 010, h_{1}010 = 100, h_{1}001 = 111 ,
h_{2}100 = 001, h_{2}010 = 111, h_{2}001 = 100 ,
h_{3}100 = 010, h_{3}010 = 100, h_{3}001 = 001 .

Hence we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups  ℤ/2  ℤ/2 × ℤ/2 D^{8}
Orbits  100, 111, 010, 001 101, 011 110

Isotropy groups  {1}  ℤ/2  ℤ/2
Orbits  100y = 0, 111x = z 100z = 0 101x = z
010x = z, 001y = 0 111x = y 101y = 0
010x = 0, 111y = z 010z = 0 011y = z
100y = z, 001x = 0 001x = y 011x = 0
Isotropy groups  ℤ/2 × ℤ/2  ℤ/2 × ℤ/2 D^{8}
Orbits  101Σ = 0  110x = y 110Σ = 0
011Σ = 0  110z = 0

2.6.4 0 - cells

1. O There are only relations of type 1. By table 3 the action of H_{O} ≅ 𝔖_{4} factors through an action of 𝔖_{3}, and this action agrees with that in the case of the edge OC. Therefore we get the same orbits; the isotropy groups “grow” by ℤ/2 × ℤ/2, more precisely the trivial isotropy group gets replaced by ℤ/2 × ℤ/2, ℤ/2 gets replaced by D_{8} and 𝔖_{3} by 𝔖_{4}.

2. B and C There are relations of both types. Those of type 2 lead to Bs ∼_{i} Cg^{AD}s and are described by table 4 and 5. As far as relations of type 1 are concerned we can concentrate on C. By table 3 the action of H_{C} ≅ 𝔖_{4} ≅ ℤ/2 × ℤ/2 ≀ 𝔖_{3} is described as follows: for the generators h_{1} and h_{2} of ℤ/2 × ℤ/2 we have

h_{1}100 = 011, h_{1}010 = 101, h_{1}001 = 001 ,
h_{2}100 = 011, h_{2}010 = 010, h_{2}001 = 110 .

The action of 𝔖_{3} is again as in the case of the edge OC. Hence we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups  ℤ/2 × ℤ/2  𝔖^{4}
Orbits for C  100, 010, 001, 110, 101, 011 111
Orbits for B  100, 110, 001, 010, 101, 111 011

Isotropy groups  ℤ/2  ℤ/2 × ℤ/2  D^{8}
Orbits for C  100y = 0, 100z = 0 100y = z 111x = y
010x = 0, 010z = 0 010x = z 111x = z
001x = 0, 001y = 0 001x = y 111y = z
110z = 0, 110Σ = 0 110x = y
101y = 0, 101Σ = 0 101x = z
011x = 0, 011Σ = 0 011y = z
Orbits for B  100y = 0, 100z = 0 100y = z 011x = 0
010x = z, 010z = 0 010x = 0 011y = z
001y = 0, 001x = y 001x = 0 011Σ = 0
110z = 0, 110x = y 110Σ = 0
101y = 0, 101x = z 101Σ = 0
111x = y, 111x = z 111y = z

3. F Here there are only relations of type 1. By table 3 the action of H_{F} ≅ 𝔖_{4} ≅ ℤ/2 × ℤ/2 ≀ 𝔖_{3} is described as follows: for the generators h_{1} and h_{2} of ℤ/2 × ℤ/2 we have

h_{1}100 = 010, h_{1}010 = 100, h_{1}001 = 111 ,
h_{2}100 = 001, h_{2}010 = 111, h_{2}001 = 100 .

The action of 𝔖_{3} is again as in the case of the edge OC resp. OF. Hence we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups  𝔖^{3}  D^{8}
Orbits  100, 010, 001, 111 110, 101, 011

Isotropy groups  ℤ/2  ℤ/2 × ℤ/2  D^{8}
Orbits  100y = 0, 010x = 0 110x = y 110Σ = 0
100z = 0, 010z = 0 110z = 0 101Σ = 0
100y = z, 010x = z 101x = z 011Σ = 0
001x = 0, 111x = y 101y = 0
001y = 0, 111x = z 011x = 0
001x = y, 111y = z 011y = z

4. A Again there are only relations of type 1. By table 3 the action of H_{A} ≅ D_{12} factors through an action of 𝔖_{3} and permutes the elements 100, 010 and 110 while 001 is fixed under the action. Therefore we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups ℤ/2 × ℤ/2 ℤ/2 × ℤ/2 D^{12}
Orbits  100, 110, 010 101, 111, 011 001

Isotropy groups  ℤ/2  ℤ/2 × ℤ/2 ℤ/2 × ℤ/2
Orbits  101x = z, 101Σ = 0 100y = z 100y = 0
011y = z, 011Σ = 0 110Σ = 0 110x = y
111x = z, 111y = z 010x = z 010x = 0
Isotropy groups  ℤ/2 × ℤ/2  ℤ/2 × ℤ/2 ℤ/2 × ℤ/2
Orbits  100z = 0  101y = 0 001x = 0
110z = 0  011x = 0 001x = y
010z = 0  111x = y 001y = 0

5. D and E There are relations of both types. Those of type 2 lead to Ds ∼_{i} Eg^{AB}s and are described by table 4 and 5. As far as relations of type 1 are concerned we can concentrate on D. By table 3 we know the action of three generating involutions of H_{D} ≅ D_{8}

h_{1}100 = 100, h_{1}010 = 001, h_{1}001 = 010 ,
h_{2}100 = 100, h_{2}010 = 101, h_{2}001 = 110 ,
h_{3}100 = 100, h_{3}010 = 110, h_{3}001 = 001 .

Hence we get the following orbits and isotropy groups for the action on S_{1} resp. S_{2}:

Isotropy groups  ℤ/2  ℤ/2 × ℤ/2 D^{8}
Orbits for D  110, 101, 010, 001 111, 011 100
Orbits for E  100, 111, 010, 001 101, 011 110

Isotropy groups  {1}  ℤ/2  ℤ/2
Orbits for D  010x = 0, 010x = z 010z = 0 011x = 0
001x = 0, 001x = y 001y = 0 011Σ = 0
101x = z, 101Σ = 0 110z = 0 111x = y
110x = y, 110Σ = 0 101y = 0 111x = z
Orbits for E  100y = 0, 100y = z 100z = 0 101y = 0
010x = 0, 010x = z 010z = 0 101x = z
001x = 0, 001y = 0 001x = y 011x = 0
111x = z, 111y = z 111x = y 011y = z
Isotropy groups  ℤ/2 × ℤ/2  ℤ/2 × ℤ/2  D^{8}
Orbits for D  100y = 0  011y = z 100y = z
100z = 0  111y = z
Orbits for E 110z = 0 101Σ = 0 110Σ = 0
110x = y  011Σ = 0

3 The homology of the quotient spaces

Let p be any prime. In this section we will compute the mod - p cohomology resp. homology of the quotients of X_{∞}, X_{∞,s}(i) and the pair (X_{∞}, X_{∞,s}(i)) by the groups Γ_{i}, and also the cohomology of the quotients of X, X_{s} and (X , X_{s}) by Γ := SL(3, ℤ[1/2]); in particular we prove Theorem 1.6, Corollary 1.7 and Theorem 1.8. In Sections 2.4 and 2.6 we described cell structures on the spaces Γ_{i}\Z ≃ Γ_{i}\X_{∞}. Let Z_{s}(i) be the 2 - singular locus of Z with respect to the action of Γ_{i}, i = 0, 1, 2, so that Γ_{i}\Z_{s}(i) ≃ Γ_{i}\X_{∞,s}(i). We will use the results of Section 2 to give the boundary homomorphisms of the chain complexes C_{∗}(Γ_{i}\(Z, Z_{s}(i))) and C_{∗}(Γ_{i}\Z_{s}(i)) (with integral coefficients) in an explicit form. Then we compute the homology groups of interest from these complexes. As these complexes are quite big our computations will be simplified by Euler characteristic considerations. We summarize the discussion of Section 2 relevant for the Euler characteristic χ in the following table.

0-cells 1-cells 2-cells 3-cells

number of all cells χ

Γ_{0}\Z  5  8  5  1  19  1
Γ_{0}\Z_{s}(0)  5  8  4  0  17  1
Γ_{0}\(Z, Z_{s}(0)) 0  0  1  1  2  0
Γ_{1}\Z  13  31  26  7  77  1
Γ_{1}\Z_{s}(1)  13  26  12  0  51  −1
Γ_{1}\(Z, Z_{s}(1)) 0  5  14  7  26  2
Γ_{2}\Z  24  72  69  21  186  0
Γ_{2}\Z_{s}(2)  23  49  21  0  93  −5
Γ_{2}\(Z, Z_{s}(2)) 1  23  48  21  93  5

In order to determine the incidence matrices, i.e. the boundary homomorphisms in the relevant cellular chain complexes, we will have to choose orientations for our cells. We will choose the orientation of the edges and triangles in D_{0} in accordance with the ordering of the vertices in their names so that for example [ACD] = −[ADC] and for the boundary of [ABD] we obtain [AB] + [BD] − [AD]. (Here [ACD], [ABD] etc. denote the basis elements in the chain complex given by the cells ACD, ABD etc.; similar notation will be used below.) The 2 - dimensional cell OAEF is oriented such that its boundary is [OA] + [AE] + [EF] − [OF]; likewise with OCBF. The 3 - dimensional cell in D can then be oriented such that its boundary is given by [OAEF] − [ABD] − [ADC] − [AEB] − [BEF] − [OCBF] − [OAC]. Then we get an orientation of the cells (e, s) in D_{0} × S_{i} (by choosing the orientation of e) and finally we get induced orientations for the cells in Γ_{i}\Z. For example in C_{∗}(Γ_{0}\Z) we obtain [ABD] = [ACD] = −[ADC].

3.1 Quotients of (X_{∞}, X_{∞,s}(i)) by Γ_{i}

We will compute the homology of the homotopy-equivalent quotients of the pairs (Z, Z_{s}(i)) by Γ_{i}.

1. Γ_{0} : There is only one 2 - and one 3 - dimensional cell in Γ_{0}\(Z, Z_{s}(0)) and it is clear that the boundary map ∂_{3} : C_{3} ⟶ C_{2} in the cellular chain complex C_{∗}(Γ_{0}\(Z, Z_{s}(0)) is an isomorphism. This implies part a) of Theorem 1.8.

2. Γ_{1} : Using the description of Γ_{1}\Z that we gave in Section 2.4 and 2.6 it is straightforward to check that the boundary maps ∂_{2} and ∂_{3} in the cellular complex C_{∗}(Γ_{1}\(Z, Z_{s}(1)) are given by the matrices in tables 6 and 7 below. In these matrices the columns and rows are labelled by cells in Γ_{1}\(Z, Z_{s}(1)), i.e. by equivalence classes of “nonsingular” cells in D_{1} (cf. Section 2.4.2), and we have chosen representatives from equivalence classes where necessary. Furthermore all zero entries in these matrices have been omitted. One sees at once that ∂_{3} has trivial kernel, i.e. H_{3}(Γ_{1}\(Z, Z_{s}(1)); 𝔽_{p}) = 0 and that ∂_{2} is onto, i.e. H_{1}(Γ_{1}\(Z, Z_{s}(1)); 𝔽_{p}) = 0. Then the Euler characteristic argument implies that H_{2}(Γ_{1}\(Z, Z_{s}(1)); 𝔽_{p}) ≅ (𝔽_{p})^{2} and we obtain part b) of Theorem 1.8. For later use we specify two 2 - dimensional cycles which form a basis of H_{2}. We can take the cycles

[ABD100] − [OAC100] and [ABD011] + [OAEF 101] .  (3.1)

3. Γ_{2} : Now consider the complex C_{∗}(Γ_{2}\(Z, Z_{s}(2))). First of all it is clear that ∂_{1} : C_{1} ⟶ C_{0} is onto and hence we obtain H_{0}(Γ_{2}\(Z, Z_{s}(2)); 𝔽_{p}) = 0. Furthermore, using our description in Section 2.6 again, it is straightforward to check that ∂_{2} and ∂_{3} are given by the matrices in tables 8 - 11 below. These matrices show that the kernel of ∂_{3} is of dimension 1 and is generated by the cycle:

[100y = 0]−[100z = 0]−[010x = 0]+[010z = 0]+[001x = 0]−[001y = 0] .

(3.2)

In particular we get H_{3}(Γ_{2}\(Z, Z_{s}(2)); 𝔽_{p}) ≅ 𝔽_{p}. (Here [100y = 0] etc. denote the 3 - dimensional cells in Γ_{2}\Z corresponding to the elements 100y = 0 etc. in S_{2}.) Furthermore, the image of ∂_{2} is of dimension 22, i.e. H_{1}(Γ_{2}\(Z, Z_{s}(2)); 𝔽_{p}) = 0, and then the Euler characteristic argument implies H_{3}(Γ_{2}\(Z, Z_{s}(2)); 𝔽_{p}) ≅ (𝔽_{p})^{6} and hence part c) of Theorem 1.8. Again for later use we specify six 2 - dimensional cycles whose homology classes form a basis of H_{2}. We can take the cycles

c_{1} : = [OAC100y = z] − [ABD100y = z]  (3.3)

c_{2} : = [OAC100z = 0] + [ABD100y = 0] − [ABD100z = 0] −
−[OAC100y = 0]  (3.4)
c_{3} : = −[ABD001y = 0] + [BEF 100z = 0] + [OAC001x = 0] +
+[ABD100y = 0] − [OAC100y = 0] + [OCBF 111x = y] (3.5)
c_{4} : = [ABD011x = 0] + [OAEF 101y = 0]  (3.6)
c_{5} : = [ABD011y = z] − [OAC111x = z] + [OAEF 101Σ = 0] (3.7)
c_{6} : = [ABD011Σ = 0] − [OAC111x = z] + [OAEF 101x = z] . (3.8)

Table 6: The boundary homomorphism C_{2} → C_{1} for Γ_{1}\(Z, Z_{s}(1))

ABD  BEF OAC  OAEF OCBF
100 010 001 110 101 011 111 100 100 101 100 101 010 110
BD 010  1 1 1 1  1
AB 100 1  1  1
101  1  1  1
AD 110  −1  −1  1
111  −1 −1  1

Table 7: The boundary homomorphism C_{3} → C_{2} for Γ_{1}\(Z, Z_{s}(1))

100 010 001 110 101 011 111
ABD 100  1
010  1
001  1

110 1 1 −1 101  1 011  1 111 1 1 −1 BEF 100 −1 −1 −1  −1 OAC 100 −1 −1

101  −1 −1 OAEF 100 1  1

101 1 1 OCBF 010  −1 −1

110  −1 −1

Table 8: The boundary homomorphism C_{2} → C_{1} for Γ_{2}\(Z, Z_{s}(2)); Part I

ABD  BEF
100  010  001  110  101  011  111  100
y = 0 z = 0 y = z x = 0 z = 0 x = z x = 0 y = 0 x = y z = 0 x = y Σ = 0 y = 0 x = z Σ = 0 x = 0 y = z Σ = 0 x = y x = z y = z y = 0 z = 0 y = z
BD 010 x = 0  1  1  1  1  1
010 z = 0  1  1  1  1  1
010 x = z  1  1  1  1  1
AB 001 y = 0  1  1
011 y = z  1  1
100 y = 0 1  1
100 z = 0  1  1
100 y = z  1  1
101 y = 0  1  1
101 Σ = 0  1  1
101 x = z  1  1
AD 001 x = 0  −1  −1
101 x = z  −1  −1
110 x = y  −1  −1
110 z = 0  −1  −1
110 Σ = 0  −1  −1
111 x = y  −1  −1
111 x = z  −1  −1
111 y = z  −1  −1
OC 100 y = 0
OF 100 y = 0
EF 100 y = 0  1  1
BF 100 y = 0  −1  −1

Table 9: The boundary homomorphism C_{2} → C_{1} for Γ_{2}\(Z, Z_{s}(2)); Part II

OAC  OAEF  OCBF
100  101  001 111  100  101  001 111  010  110  100 111
y = 0 z = 0 y = z y = 0 x = z Σ = 0 x = 0 x = z y = 0 z = 0 y = z y = 0 x = z Σ = 0 x = 0 x = z x = 0 z = 0 x = z z = 0 x = y Σ = 0 y = 0 x = y
BD 010 x = 0  −1  1  1  −1
010 z = 0
010 x = z  1  −1  −1  1
AB 001 y = 0  1
011 y = z  1
100 y = 0 1
100 z = 0  1
100 y = z  1
101 y = 0  1
101 Σ = 0  1
101 x = z  1
AD 001 x = 0  1
101 x = z  1
110 x = y  1
110 z = 0  1
110 Σ = 0  1
111 x = y  1
111 x = z  1
111 y = z  1
OC 100 y = 0 −1  −1  −1  1  1  1
OF 100 y = 0  −1  −1  −1  −1  −1  −1
EF 100 y = 0  1  1  1  1
BF 100 y = 0  1  1  1  1

Table 10: The boundary homomorphism C_{3} → C_{2} for Γ_{2}\(Z, Z_{s}(2)); Part I
100  010  001  110  101  011  111
y = 0 z = 0 y = z x = 0 z = 0 x = z x = 0 y = 0 x = y z = 0 x = y Σ = 0 y = 0 x = z Σ = 0 x = 0 y = z Σ = 0 x = y x = z y = z
ABD 100 y = 0  1
100 z = 0  1
100 y = z  1
010 x = 0  1
010 z = 0  1
010 x = z  1
001 x = 0  1
001 y = 0  1
001 x = y  1  1  −1
110 z = 0  1  1  −1
110 x = y 1  1  −1
110 Σ = 0  1  1  −1
101 y = 0  1
101 x = z  −1  1  1
101 Σ = 0  1  −1  1
011 x = 0  1
011 y = z  −1  1  1
011 Σ = 0  1  −1  1
111 x = y  1  1  −1
111 x = z  1  1  −1
111 y = z  1  1  −1
BEF 100 y = 0 −1  −1  −1  −1
100 z = 0  −1  −1  −1  −1
100 y = z  −1  −1  −1  −1

Table 11: The boundary homomorphism C_{3} → C_{2} for Γ_{2}\(Z, Z_{s}(2)); Part II
100  010  001  110  101  011  111
y = 0 z = 0 y = z x = 0 z = 0 x = z x = 0 y = 0 x = y z = 0 x = y Σ = 0 y = 0 x = z Σ = 0 x = 0 y = z Σ = 0 x = y x = z y = z
OAC 100 y = 0 −1  −1
100 z = 0  −1  −1
100 y = z  −1  −1
101 y = 0  −1  −1
101 x = z  −1  −1
101 Σ = 0  −1  −1
001 x = 0  −1  −1
111 x = z  −1  −1
OAEF 100 y = 0 1  1
100 z = 0  1  1
100 y = z  1  1
101 y = 0  1  1
101 x = z  1  1
101 Σ = 0  1  1
001 x = 0  1  1
111 x = z  1  1
OCBF 010 x = 0  −1  −1
010 z = 0  −1  −1
010 x = z  −1  −1
110 z = 0  −1  −1
110 x = y  −1  −1
110 Σ = 0  −1  −1
100 y = 0 −1  −1
111 x = y  −1  −1

3.2 Quotients of X_{∞,s}(i) by Γ_{i}

We will derive Theorem 1.6 from Theorem 1.8 and from the following result.

Theorem 3.1 Let p be any prime. Then the reduced cohomology of the quotients of X_{∞,s}(i) by the action of the respective groups is given as follows.

a) H̃^{∗}(Γ_{0}\X_{∞,s}(0); 𝔽_{p}) = 0

b) H̃^{∗}(Γ_{1}\X_{∞,s}(1); 𝔽_{p}) = Σ(𝔽_{p})^{2}

c) H̃^{∗}(Γ_{2}\X_{∞,s}(2); 𝔽_{p}) = Σ(𝔽_{p})^{6}

Proof. We will compute the mod - p (co-)homology from the cell complexes of the homotopy equivalent spaces Γ_{i}\Z_{s}(i). First of all we note that in all cases we have H̃_{0}(Γ_{i}\Z_{s}(i); 𝔽_{p}) = 0 (e.g. because Z is connected and because of Theorem 1.8).

a) Because of Euler characteristic considerations it suffices in the case of Γ_{0} to show that the boundary map ∂_{2} : C_{2} ⟶ C_{1} in the cellular complex C_{∗}(Γ_{0}\Z_{s}(0)) is a monomorphism. This can be easily seen from figure 1.

b) In the case of Γ_{1} the Euler characteristic argument shows that is enough to verify H̃_{2}(Γ_{1}\Z_{s}(1); 𝔽_{p}) = 0. For this we need to show that the boundary homomorphism ∂_{2} is injective. This boundary homomorphism can be easily determined by the information provided in Section 2.6 and is described in table 12 below. Injectivity is now easily checked.

c) Finally we consider the case of Γ_{2}. Again by the Euler characteristic argument it suffices to show H̃_{2}(Γ_{2}\Z_{s}(2); 𝔽_{p}) = 0. The boundary map ∂_{2} is now described in tables 13 and 14 and again injectivity is easily checked. □

3.3 Quotients of X_{∞} by Γ_{i}

In order to determine H̃_{∗}(Γ_{i}\X_{∞}; 𝔽_{p}) ≅ H̃_{∗}(Γ_{i}\Z_{∞}; 𝔽_{p}) it remains to compute the relevant connecting homomorphismsm in the long exact sequence for the homology of the pair Γ_{i}\(Z, Z_{s}(i)). In case i = 0 we obtain clearly H̃_{∗}(Γ_{0}\Z; 𝔽_{p}) = 0 and in the other two cases one checks easily that under ∂_{2} : C_{2}(Γ_{i}\Z) ⟶ C_{1}(Γ_{i}\Z) the images of the relative cycles in (3.1) resp. (3.3) ff. are linearly independent in the quotient of C_{1}(Γ_{i}\Z_{s}) by the image of ∂_{2} : C_{2}(Γ_{i}\Z_{s}(i)) ⟶ C_{1}(Γ_{i}\Z_{s}(i)); in fact, to see this it is enough to determine the “OA” - part of the total boundary of these relative cycles and compare with tables 12 resp. 13 and 14. In other words, the corresponding connecting homomorphism in the long exact sequence is injective and then even an isomorphism because of dimension reasons. Part a), b) and c) of Theorem 1.6 follow.

Table 12: The boundary homomorphism C_{2} → C_{1} for Γ_{1}\Z_{s}(1)

OAC  OAEF  OCBF  BEF
001 110 111 001 110 111 100 011 111 110 101 011
BD 100  1
011  −1 1  1
111  1 −1  1
AC 001 1
110  1
111  1
AE 001  1
110  1
111  1
OC 100 −1  1
110  −1  1
111  −1  1
OF 100  −1  −1
110  −1  −1
111  −1  −1
EF 110  1  1
101  1 1
100  1  1
BF 011  1  −1
110  −1 −1
100 1 1
OA 001 1  1
110  1  1
111  1  1

100

Table 13: The boundary homomorphism C_{2} → C_{1} for Γ_{2}\Z_{s}(2); Part I
OAC  OAEF  OCBF  BEF
001 111  110  001 111  110  100 111  011  101  110  011
x = y x = y x = y z = 0 Σ = 0 x = y x = y x = y z = 0 Σ = 0 y = z y = z y = z x = 0 Σ = 0 x = z Σ = 0 x = y Σ = 0 y = z Σ = 0
BD 100 y = 0  1
100 y = z  1
011 x = 0  −1  −1  1
011 y = z  1  −1  1
111 x = y  1  1  1
111 y = z  −1  1  1
AC 001 x = y 1
111 x = y  1
110 x = y  1
110 z = 0  1
110 Σ = 0  1
AE 001 x = y  1
111 x = y  1
110 x = y  1
110 z = 0  1
110 Σ = 0  1
OC 100 y = z −1  1
111 x = y  −1  1
110 x = y  −1  1
110 z = 0  −1  1
110 Σ = 0  −1  1
OF 100 y = z  −1  −1
111 x = y  −1  −1
110 x = y  −1  −1
110 z = 0  −1  −1
110 Σ = 0  −1  −1

Table 14: The boundary homomorphism C_{2} → C_{1} for Γ_{2}\Z_{s}(2); Part II
OAC  OAEF  OCBF  BEF
001 111  110  001 111  110  100 111  011  101  110  011
x = y x = y x = y z = 0 Σ = 0 x = y x = y x = y z = 0 Σ = 0 y = z y = z y = z x = 0 Σ = 0 x = z Σ = 0 x = y Σ = 0 y = z Σ = 0
EF 100 z = 0  1  1
110 x = y  1  1  1
110 Σ = 0  1  1
101 x = z  1  1
101 Σ = 0  1  1
BF 100 y = z  1  1
110 x = y  −1  −1
110 Σ = 0  −1  −1
011 y = z  1  1  −1
011 Σ = 0  1  −1
OA 100 y = 0
100 z = 0
100 y = z
001 x = 0
001 x = y 1  1
101 y = 0
101 x = z
101 Σ = 0
110 x = y  1  1
110 z = 0  1  1
110 Σ = 0  1  1
111 x = y  1  1
111 x = z

3.4 Quotients by SL(3, ℤ[1/2])

As before we abbreviate SL(3, ℤ[1/2]) by Γ. In this section we are concerned with the proof of part d) of Theorem 1.6 and Theorem 1.8, i.e. with the computation of the mod - p cohomology of Γ\X and the mod - 2 cohomology of the pair (Γ\X , Γ\X_{s}). For this we consider the Γ - equivariant projection map p : X ⟶ X_{2} and the spectral sequences (which arise from the skeletal filtrations of the bases) of the following associated maps

\overline{p}_{X} : Γ\X ⟶ Γ\X_{2} ≅ Δ^{2} ,

\overline{p}_{(X ,X_{s})} : Γ\(X , X_{s}) ⟶ Γ\X_{2} ≅ Δ^{2} ,

and also that of

\overline{p}_{X_{s}} : Γ\X_{s} ⟶ Γ\X_{2} ≅ Δ^{2} .

The fibres of these maps over the i - simplices in Δ^{2} are homeomorphic to the spaces Γ_{i}\X_{∞} resp. to Γ_{i}\(X_{∞}, X_{∞,s}(i)) resp. to Γ_{i}\X_{∞,s}(i) (cf. Section 2.1). Therefore Theorem 3.1 and the already proven parts a), b) and c) of Theorem 1.6 and Theorem 1.8 immediately give the following E_{1} - terms for the cohomology spectral sequences converging to H^{∗}(Γ\X; 𝔽_{p}), H^{∗}(Γ\(X , X_{s}); 𝔽_{p}) resp. H^{∗}(Γ\X_{s}; 𝔽_{p}).

[figure: t; 0; 1; 2; 3; s; 0; 1; 2; 3; 3; 1; 1; E_{1}^{s,t}(Γ\X); t; 0; 1; 2; 3; s; 0; 1; 2; 6; 6; 1; E_{1}^{s,t}(Γ\(X , X_{s})); t; 0; 1; 2; 3; s; 0; 1; 2; 3; 3; 1; 6; 6; E_{1}^{s,t}(Γ\X_{s})]

The numbers in these diagramms give the dimension of E_{1}^{s,t} as an 𝔽_{p} - vector space. Missing numbers are to be interpreted as 0. The differential d_{1} on the line t = 0 (in the first and the third case) is as in the case of the simplicial chains on Δ^{2}, in particular we get in these cases E_{0,0}^{2} ≅ 𝔽_{p} and E_{s,0}^{2} = 0 if s > 0 . In particular, we immediately obtain H̃^{∗}(Γ\X; 𝔽_{p}) ≅ Σ^{5}𝔽_{p}, i.e. part d) of Theorem 1.6. We also see that H^{i}(Γ\(X , X_{s}); 𝔽_{p}) = 0 for i ≤ 2, independent of the precise behaviour of the spectral sequences. What remains to be calculated is the differential d_{1}^{1,2} : E_{1}^{1,2} ⟶ E_{1}^{2,2} in the second case and the differential d_{1}^{1,1} : E_{1}^{1,1} ⟶ E_{1}^{2,1} in the third case. The connecting homomorphisms of the long exact sequences associated to the pairs Γ_{i}\(X_{∞}, X_{∞,s}(i)) induce (by Section 3.3) isomorphisms between E_{1}^{i,1}(Γ\X_{s}) and E_{1}^{i,2}(Γ\(X , X_{s})) for i = 1, 2, hence it suffices to do the calculation in one case. We will show that in the third case d_{1,1}^{1} is an isomorphism if p = 2 and this will finish the proof of Theorem 1.8.

For this consider the mod - 2 cohomology spectral sequences (arising from a skeletal filtration of the base) of the maps

EΓ ×_{Γ} X_{s} ⟶ Γ\X_{s}

and

EΓ ×_{Γ} X ⟶ Γ\X .

(For a discussion of the existence of a cellular structure on these bases we refer to the remark at the end of this section.) The E_{1} - terms of both spectral sequences agree except on the line t = 0 because the mod - 2 cohomology of a fibre outside of Γ\X_{s} vanishes in positive dimensions. Consequently the E_{2} - terms of both spectral sequences also agree except on the line t = 0 and there we get E_{2}^{s,0} ≅ H^{s}(Γ\X_{s}; 𝔽_{2}) resp. E_{2}^{s,0} ≅ H^{s}(Γ\X; 𝔽_{2}).

Claim 1: E_{2}^{0,1} = 0 in both spectral sequences.

Proof. We consider the spectral sequence converging to H^{∗}(EΓ ×_{Γ} X; 𝔽_{2}). As we have seen above the groups E_{2}^{i,0} ≅ H^{i}(Γ\X , 𝔽_{2}) are trivial for i = 1, 2. Therefore we have E_{2}^{0,1} = 0 if and only if H^{1}(EΓ ×_{Γ} X; 𝔽_{2}) = 0. From Theorem 1.5 we know that H^{1}(EΓ ×_{Γ} X_{s}; 𝔽_{2}) = 0 and from the discussion above we know that H^{1}(EΓ ×_{Γ} (X , X_{s}); 𝔽_{2}) ≅ H^{1}(Γ\(X , X_{s}); 𝔽_{2}) = 0. Then the long exact sequence of the pair EΓ×_{Γ}(X , X_{s}) shows H^{1}(EΓ×_{Γ}X; 𝔽_{2}) = 0 and we are done. □

Now consider the class v_{2} ∈ H^{2}(EΓ ×_{Γ} X_{s}; 𝔽_{2}) which is pulled back from the second universal Stiefel Whitney class in H^{∗}(BSL(3, ℝ); 𝔽_{2}) under the induced map of the composition EΓ ×_{Γ} X_{s}⟶^{π}BΓ⟶^{i}BSL(3, ℝ) (where π is given by sending X_{s} to a point and i by the canonical inclusion Γ ↪ SL(3, ℝ)).

Claim 2: In the spectral sequence converging to H^{∗}(EΓ ×_{Γ} X_{s}; 𝔽_{2}) the class v_{2} is detected on E_{1}^{0,2}.

Proof. We have E_{1}^{0,2} ≅ ⨁_{e} H^{2}(Γ_{e}; 𝔽_{2}) where e runs through a set of Γ - orbits of 0 - dimensional cells in X_{s} and Γ_{e} denotes the isotropy group of the cell e. We may write e = (e_{2}, e_{∞}); if the X_{2} - component e_{2} is given by the vertex l_{0} defined by the standard ℤ_{2} - lattice in ℚ_{2} then we have Γ_{e} = Γ_{e_{∞}}, the isotropy group of the cell e_{∞} ⊂ X_{∞,s} with respect to the action of SL(3, ℤ). For any such cell the class v_{2} restricts to the Stiefel-Whitney class of the representation of Γ_{e} arising from the embedding Γ_{e} ↪ SL(3, ℤ[1/2]) ↪ SL(3, ℝ). Now Γ_{e} contains at least one element of order 2 and all such elements are conjugate in SL(3, ℝ). It follows that v_{2} restricts non-trivially to any subgroup of order 2 in Γ_{e} and hence the claim is proved. □

Now assume that the differential d_{1}^{1,1}(Γ\X_{s}) is not an isomorphism. Then H^{2}(Γ\X_{s}; 𝔽_{2}) ≠ 0 and in the spectral sequence converging to H^{∗}(EΓ ×_{Γ} X_{s}; 𝔽_{2}) we have E_{2}^{2,0} ≅ H^{2}(Γ\X_{s}; 𝔽_{2}) ≠ 0. Because of Claim 1 we conclude that all of E_{2}^{2,0} survives to E_{∞}, and because of Claim 2 we see that the assumption implies that H^{2}(EΓ ×_{Γ} X_{s}; 𝔽_{2}) is a vector space of dimension bigger than 1 in contradiction to Theorem 1.5. This finishes the proof of part d) of Theorem 1.8. □

Remark. a) Our approach to the computation of H^{∗}(Γ\(X , X_{s}); 𝔽_{2}) is rather indirect and one might wonder why we did not analyze the differential

E_{1}^{1,2} ≅ ⨁_{i=1}^{3} H^{2}(Γ_{1}\(X_{∞}, X_{∞,s}(1)); 𝔽_{2}) ⟶ H^{2}(Γ_{2}\(X_{∞}, X_{∞,s}(2)); 𝔽_{2}) ≅ E_{1}^{2,2}

directly? The reason is that the three summands in the source (corresponding to the three Γ - orbits of 1 - dimensional cells in X_{2} resp. the three edges in Δ^{2}) are mapped differently under this differential; only on one summand is the map induced by the inclusion Γ_{2} ⊂ Γ_{1}, on the other two summands it is induced by the inclusion of Γ_{2} into the isotropy groups of the edges {l_{0}, l_{2}} resp. {l_{1}, l_{2}} where as in Section 2.1 l_{0}, l_{1}, l_{2} are the classes of the ℤ_{2} - lattices L_{0} = ⟨e_{1}, e_{2}, e_{3}⟩, L_{1} = ⟨\frac{1}{2}e_{1}, e_{2}, e_{3}⟩ and L_{2} = ⟨\frac{1}{2}e_{1}, \frac{1}{2}e_{2}, e_{3}⟩ respectively. The component of the differential corresponding to the inclusion Γ_{2} ⊂ Γ_{1} (corresponding to the edge {l_{0}, l_{1}}) is straightforward to determine: with respect to our cell structures on the spaces Γ_{i}\Z it is induced by a cellular map which is determined by the forgetful map S_{2} ⟶ S_{1}. The component corresponding to the edge {l_{0}, l_{2}} can also be worked out on the level of the spaces Γ_{i}\Z. In contrast the isotropy group H_{{l_{1},l_{2}}} of the edge {l_{1}, l_{2}} is not contained in Γ_{0}, hence the deformation retraction X_{∞} ⟶ Z is not H_{{l_{1},l_{2}}} - equivariant and this makes this component of the differential much harder to evaluate. In terms of integral lattices in ℝ^{3} and the spaces W_{i}/SO(3) this last component is induced by the map which associates to the triple (L_{0}, L_{1}, L_{2}) of lattices (with L_{0} being well-rounded and m(L_{0}) = 1) the pair (L_{1}, L_{2}). Because L_{1} need not be well-rounded one has to work out the effect of the deformation retraction L_{1}/SO(3) ⟶ W_{1}/SO(3) of Section 2.2 explicitly. Although one would not expect that this could cause unsurmountable problems it would be at the very least very laborious and the author found his initial attempts to carry this out very frustrating.

b) We have tacitly used above that X is a Γ - equivariant CW - complex and we will use it again, namely in the final step of the proof of 1.4 which combines Proposition 1.9 and Theorem 1.5. It is quite likely that there is such a stucture but we do not know of a suitable reference. However, it is easy to show that X has the equivariant homotopy type of a Γ - CW complex, and this will be enough: for example, we can do induction on the skeleta X_{2}^{k} of the evident Γ - equivariant cell structure of the simplicial complex X_{2} using that the preimages of the space X_{2}^{k} and X_{2}^{k} − X_{2}^{k−1} with respect to the projection map X ⟶ X_{2} are understood by Section 2.4.

4 The cohomology of SL(3, ℤ[1/2])

4.1 Mod - 2 cohomology

In this section we will complete the proof of Theorem 1.4. Because of Theorem 1.5 and Theorem 1.8 it is enough to prove Proposition 1.9, i.e. that the connecting homomorphism

H_{Γ}^{4}(X_{s}; 𝔽_{2}) ⟶ H_{Γ}^{5}(X , X_{s}; 𝔽_{2}) ≅ 𝔽_{2}

in the long exact sequence of the pair EΓ ×_{Γ} (X , X_{s}) is an epimorphism, or equivalently that the natural map

H_{Γ}^{5}(X , X_{s}; 𝔽_{2}) ⟶ H_{Γ}^{5}(X; 𝔽_{2})

is trivial. For this consider the following commutative diagramm in which the horizontal maps are induced by inclusions and the vertical maps by projections:

H_{Γ}^{5}(X , X_{s}; 𝔽_{2}) ⟶ H_{Γ}^{5}(X; 𝔽_{2})
≅↑  ^{q^{∗}}↑
H^{5}(Γ\(X , X_{s}); 𝔽_{2}) ⟶^{≅} H^{5}(Γ\X; 𝔽_{2})

The indicated arrows are isomorphisms because of Section 3.4 resp. because the isotropy groups in X − X_{s} are of odd order. Therefore we have to show that the map q^{∗} : H^{5}(Γ\X; 𝔽_{2}) ⟶ H_{Γ}^{5}(X; 𝔽_{2}) is trivial.

This will be a consequence of the following two results.

Lemma 4.1 If the map q_{2}^{∗} : H^{3}(Γ_{2}\X_{∞}; 𝔽_{2}) ⟶ H_{Γ_{2}}^{3}(X_{∞}; 𝔽_{2}) (induced by projection) is trivial then so is q^{∗} : H^{5}(Γ\X; 𝔽_{2}) ⟶ H_{Γ}^{5}(X; 𝔽_{2}).

Lemma 4.2 The map q_{2}^{∗} : H^{3}(Γ_{2}\X_{∞}; 𝔽_{2}) ⟶ H_{Γ_{2}}^{3}(X_{∞}; 𝔽_{2}) is trivial.

Proof of Lemma 4.1 This follows immediately from naturality applied to the following situation. If X^{1} denotes the preimage of the 1-skeleton ∂Δ^{2} of Δ^{2} under the projection map X ⟶ X_{2} ⟶ Γ\X_{2} ≅ Δ^{2} then consider the following commutative diagram in which the vertical maps are induced by projections and the horizontal maps by inclusions:

H_{Γ}^{5}(X , X^{1}; 𝔽_{2}) ⟶ H_{Γ}^{5}(X; 𝔽_{2})
↑  ↑
H^{5}(Γ\(X , X^{1}); 𝔽_{2}) ⟶ H^{5}(Γ\X; 𝔽_{2}) .

It is clear from Section 3.4 that the horizontal arrow on the bottom line of the diagram is an isomorphism. By excision we see

H^{5}(Γ\(X , X^{1}); 𝔽_{2}) ≅ H^{5}((Δ^{2}, ∂Δ^{2}) × (Γ_{2}\X_{∞}); 𝔽_{2}) ≅ Σ^{2}H^{3}(Γ_{2}\X_{∞}; 𝔽_{2})

and

H_{Γ}^{5}(X , X^{1}; 𝔽_{2}) ≅ H_{Γ_{2}}^{5}((Δ^{2}, ∂Δ^{2}) × X_{∞}; 𝔽_{2}) ≅ Σ^{2}H_{Γ_{2}}^{3}(X_{∞}; 𝔽_{2})

and the claim follows. □

Proof of Lemma 4.2 This is more involved. We prefer to work in homology and there we have to show that the map q_{2∗} : H_{3}^{Γ_{2}}(X_{∞}; 𝔽_{2}) ⟶ H_{3}(Γ_{2}\X_{∞}; 𝔽_{2}) is trivial, or equivalently that the non-trivial element (cf. Theorem 1.6) of H_{3}(Γ_{2}\X_{∞}; 𝔽_{2}) is not a permanent cycle in the homology spectral sequence of the projection map. For this consider the short exact sequence of 𝔽_{2}Γ_{2} - modules

0 ⟶ C_{3} ⟶^{∂_{3}} C_{2} ⟶^{∂_{2}} I ⟶ 0  (4.1)

in which C_{i} denotes the i-th cellular chain group (with coefficients in 𝔽_{2}) of the contractible Γ_{2} - space Z and I is the image of the boundary map ∂_{2} : C_{2} ⟶ C_{1}. Note that C_{i} can be identified with ⨁_{τ} 𝔽_{2}[Γ_{2}/Γ_{τ}] where τ runs through a set of representatives of Γ_{2} - orbits of i - cells in Z and Γ_{τ} is the isotropy subgroup of the chosen representative τ. The E^{1} - term of the spectral sequence of the projection map is given as E_{s,t}^{1} ≅ H_{t}(Γ_{2}; C_{s}) and the differential d_{3,0}^{2} : E_{3,0}^{2} ⟶ E_{1,1}^{2} can be described as follows: The group E_{3,0}^{2} is given as

E_{3,0}^{2} ≅ H_{3}(Γ_{2}\Z) ≅ Ker (H_{0}(Γ_{2}; C_{3}) ⟶ H_{0}(Γ_{2}; C_{2}))

while E_{1,1}^{2} is given as quotient

E_{1,1}^{2} ≅ \frac{Ker (H_{1}(Γ_{2}; C_{1}) ⟶ H_{1}(Γ_{2}; C_{0}))}{Im (H_{1}(Γ_{2}; C_{2}) ⟶ H_{1}(Γ_{2}; C_{1}))} .

All maps are, of course, induced by the differentials in the chain complex C_{∗}. If z is an element in E_{3,0}^{2} ⊂ H_{0}(Γ_{2}, C_{3}) then z = ∂y for some y ∈ H_{1}(Γ_{2}; I) (with ∂ denoting the connecting homomorphism associated to the exact sequence (4.1)) and d_{3,0}^{2}z is represented by i_{∗}y ∈ H_{1}(Γ_{2}; C_{1}) (with i denoting the inclusion of I into C_{1}. In particular we see that the following two conditions are equivalent:

1. d_{3,0}^{2} : E_{3,0}^{2} ≅ 𝔽_{2} ⟶ E_{1,1}^{2} is non-trivial.

2. Im (H_{1}(Γ_{2}; I) ⟶^{i_{∗}} H_{1}(Γ_{2}; C_{1})) is strictly larger than Im (H_{1}(Γ_{2}; C_{2}) ⟶^{d_{2,1}^{1}} H_{1}(Γ_{2}; C_{1})).

We will verify the second condition and this will complete the proof of Lemma 4.2. In fact, it suffices to verify the second condition after projecting off to a suitable summand in H_{1}(Γ_{2}; C_{1}) ≅ ⨁_{τ} H_{1}(Γ_{2}; 𝔽_{2}[Γ_{2}/Γ_{τ}]) where as above τ runs through a set of representatives of Γ_{2} - orbits of 1 - cells in Z. We choose the 1 - dimensional cells τ_{1} resp. τ_{2} given by 1 · (AC, 001x = y) ⊂ Γ_{2} × D_{2}/ ∼_{2} and 1 · (OA, 001x = y) ⊂ Γ_{2} × D_{2}/ ∼_{2} (where 1 ∈ Γ and our conventions for labelling the cells in Z ≅ Γ_{2} × D_{2}/ ∼_{2} are those of Section 2.4.2). We will denote the corresponding projections by π_{1} and π_{2} respectively. Lemma 4.2 will then follow from the following two results. □

Lemma 4.3 If y ∈ H_{1}(Γ_{2}; C_{2}) is mapped non-trivially by the map H_{1}(Γ_{2}; C_{2})

⟶^{d_{2,1}^{1}} H_{1}(Γ_{2}; C_{1}) ⟶^{π_{1}} H_{1}(Γ_{2}; 𝔽_{2}[Γ_{2}/Γ_{τ_{1}}]) then y is also mapped non-trivially by

H_{1}(Γ_{2}; C_{2}) ⟶^{d_{2,1}^{1}} H_{1}(Γ_{2}; C_{1}) ⟶^{π_{2}} H_{1}(Γ_{2}; 𝔽_{2}[Γ_{2}/Γ_{τ_{2}}]).

Lemma 4.4 There is an element u ∈ H_{1}(Γ_{2}; I) which maps non-trivially by the map H_{1}(Γ_{2}; I) ⟶^{i_{∗}} H_{1}(Γ_{2}; C_{1}) ⟶^{π_{1}} H_{1}(Γ_{2}; 𝔽_{2}[Γ_{2}/Γ_{τ_{1}}]) and trivially by

H_{1}(Γ_{2}; I) ⟶^{i_{∗}} H_{1}(Γ_{2}; C_{1}) ⟶^{π_{2}} H_{1}(Γ_{2}; 𝔽_{2}[Γ_{2}/Γ_{τ_{2}}]).

Proof of Lemma 4.3. The differential d_{s,∗}^{1} can be described as follows (cf. chapter VII.8 of [B1]): via the identifications of C_{s} with ⨁_{σ} 𝔽_{2}[Γ_{2}/Γ_{σ}] and of C_{s−1} with ⨁_{τ} 𝔽_{2}[Γ_{2}/Γ_{τ}] the στ component of d_{s,∗}^{1} is induced by the

corresponding component 𝔽_{2}[Γ_{2}/Γ_{σ}] ⟶^{∂_{σ,τ}} 𝔽_{2}[Γ_{2}/Γ_{τ}] of the boundary map C_{s} ⟶ C_{s−1}. By equivariance this component is determined by the image of the coset 1 · Γ_{σ} ∈ 𝔽_{2}[Γ_{2}/Γ_{σ}], i.e. by understanding the incidence numbers [σ : gτ] between the cell σ and all cells in the Γ_{2} - orbit of τ. We obtain ∂_{σ,τ}(1 · Γ_{σ}) = ∑_{gΓ_{τ}} [σ : gτ]gΓ_{τ} where the sum is over the Γ_{2} - orbit of τ. Because we project off via π_{1} and π_{2} and we are interested in homology in degree 1 only it suffices to consider “singular” 2 - dimensional cells σ ⊂ Z_{s}(2) for which [σ : gτ_{i}] is non-trivial for some cell in the orbit of τ_{i}; in particular all cells of the form g · (ACD, s), g · (ABD, s) and g · (ABE, s) in Γ_{2}\Z ≅ Γ_{2} × D_{2}/ ∼_{2} are “non-singular” and can be ignored. By going through the discussion of the relevant 2 - cells in Section 2.6 and using Theorem 2.7 we see that we only get contributions to ∂_{σ,τ_{i}}(1·Γ_{σ}) for σ = σ_{1} := 1·(OAC001, x = y) in the case of τ_{1}, and σ = σ_{1} or σ = σ_{2} := 1 · (OAEF, 001x = y) in the case of τ_{2}. Furthermore σ_{1} and σ_{2} are the only cells in their Γ_{2} - orbit for which the incidence numbers are non-trivial, namely equal to 1. Therefore it suffices to consider the following situation in which we identify H_{1}(Γ_{2}; 𝔽_{2}[Γ_{2}/Γ_{σ}]) with H_{1}(Γ_{σ}; 𝔽_{2}) for σ ∈ {σ_{1}, σ_{2}, τ_{1}, τ_{2}} and drop the coefficients from the notation; the maps i_{1} resp. i_{2} denote the inclusions of Γ_{σ_{1}} resp. Γ_{σ_{2}} into Γ_{τ_{2}} (cf. table 1 and table 2 for the isotropy groups and their inclusions).

\begin{matrix}H_{1}(Γ_{σ_{1}}) ⊕ H_{1}(Γ_{σ_{2}}) ≅ H_{1}(ℤ/2) ⊕ H_{1}(ℤ/2) & (a, b) \\ ↓ & ↓ \\ H_{1}(Γ_{τ_{1}}) ⊕ H_{1}(Γ_{τ_{2}}) ≅ H_{1}(ℤ/2) ⊕ H_{1}(ℤ/2 ⊕ ℤ/2) & (a, i_{1}_{∗}a + i_{2}_{∗}b) . \end{matrix}

The proof of the Lemma is now reduced to showing that a ≠ 0 implies i_{1}_{∗}a + i_{2}_{∗}b ≠ 0, and this is clear. □

Proof of Lemma 4.4. Of course, the connecting homomorphism associated to the exact sequence (4.1) has to send the element u in question to the element in H_{3}(Γ_{2}\Z; 𝔽_{2}) ⊂ H_{0}(Γ_{2}; C_{3}) given by the cycle of (3.2):

[100y = 0] + [100z = 0] + [010x = 0] + [010z = 0] + [001x = 0] + [001y = 0] .

Consider now the following chain in C_{3} whose class in H_{0}(Γ_{2}; C_{3}) agrees with this cycle:

z : = [1 · (100y = 0)] + [1 · (100z = 0)] + [1 · (010x = 0)] +
+[1 · (010z = 0)] + [1 · (001x = 0)] + [1 · (001y = 0)] .

Let σ denote the 2 - dimensional cell 1·(ABD, 001x = 0) in Γ_{2}×D_{2}/ ∼_{2}≅ Z. This cell generates a free 𝔽_{2}[Γ_{2}] - submodule that we denote by 𝔽_{2}[Γ_{2}]⟨σ⟩. Let π_{3} : C_{2} ⟶ 𝔽_{2}[Γ_{2}]⟨σ⟩ denote the projection map. Then Section 2.4.2 and inspection of table 5 in Section 2.6 yield π_{3}∂_{3}z = λσ where λ = (1 + g_{s}g^{AB}g_{s}^{−1}) ∈ 𝔽_{2}[Γ_{2}], s ∈ S_{2} ≅ Γ_{2}\Γ_{0} is the element 001x = 0, g_{s} ∈ Γ_{0} is a fixed chosen coset representative of s and g^{AB} ∈ Γ_{0} is as in Section 2.4. Note that because of sg^{AB} = s we have g_{s}g^{AB}g_{s}^{−1} ∈ Γ_{2}, and in fact, by Section 2.4.2, the element g_{s}g^{AB}g_{s}^{−1} is the unique non-trivial element in the isotropy group of the cell 1 · (AB, 001x = 0). In Z the cell g · (AB, 001x = 0) gets identified with the cell 1 · (AC, 001x = y) = τ_{1} if g is determined by the equation gg_{s} = g_{s′}g^{AD}, s′ ∈ S_{2} is the element 001x = y, g_{s′} is a fixed chosen coset representative of s′ and g^{AD} is again as in Section 2.4.2. It follows that the assignment σ ↦ g^{−1}τ_{1} induces an isomorphism 𝔽_{2}[Γ_{2}]⟨σ⟩/λ𝔽_{2}[Γ_{2}]⟨σ⟩ ≅ 𝔽_{2}[Γ_{2}/Γ_{g^{−1}τ_{1}}] of 𝔽_{2}[Γ_{2}] - modules.

Now let 𝔽_{2}[Γ_{2}]⟨z⟩ be the 𝔽_{2}[Γ_{2}] - submodule of C_{3} generated by the cycle z; obviously this is a free 𝔽_{2}[Γ_{2}] - module. Let C_{2}^{∂z} be the 𝔽_{2}[Γ_{2}] - submodule of C_{2} which is generated by all cells appearing in ∂_{3}z if this is written as linear combination of cells; observe that C_{2}^{∂z} is a direct summand of C_{2} as a 𝔽_{2}[Γ_{2}]

- module. Next let I^{∂z} be the quotient of C_{2}^{∂z} by the 𝔽_{2}[Γ_{2}] - submodule generated by ∂_{3}z. Then we get the following diagramm of exact sequences of 𝔽_{2}[Γ_{2}] - modules:

0 ⟶  C_{3}  ⟶^{∂_{3}}  C_{2}  ⟶  I  ⟶ 0
↑  ^{i}↑  ^{j}↑
0 ⟶ 𝔽_{2}[Γ_{2}]⟨z⟩ ⟶  C_{2}^{∂z}  ⟶  I^{∂z}  ⟶ 0
↓  ^{π_{3}}↓  π̃_{3}↓
0 ⟶ λ𝔽_{2}[Γ_{2}]⟨σ⟩ ⟶ 𝔽_{2}[Γ_{2}]⟨σ⟩ ⟶ 𝔽_{2}[Γ_{2}/Γ_{g^{−1}τ_{1}}] ⟶ 0

where the left hand vertical arrow in the upper half of the diagram is an inclusion, i is the inclusion of a direct summand, the upper left hand rectangle commutes and induces j. The lower left hand vertical arrow is given by z ↦ λσ, hence the lower left hand rectangle commutes by the formula for π_{3}∂_{3}(z) and induces the map π̃_{3}. By going through table 5 in Section 2.6 one checks that π̃_{3} agrees with the composition I^{∂z} ⟶^{j} I ⟶^{i} C_{1} \underrightarrow{π̃_{1}} 𝔽_{2}[Γ_{2}/Γ_{g^{−1}τ_{1}}] where π̃_{1} denotes the composition of π_{1} with left multiplication by g^{−1}. Now the upper half of the diagramm shows that there is an element u′ ∈ H_{1}(Γ_{2}; I^{∂z}) such that ∂j_{∗}u′ = z. (Use that z ∈ H_{0}(Γ_{2}; C_{3}) comes from an element, still denoted by z, in H_{0}(Γ_{2}; 𝔽_{2}[Γ_{2}]⟨z⟩) whose image in H_{0}(Γ_{2}; C_{2}^{∂z}) vanishes because i is split injective.) Pick any such u′ and let u := j_{∗}u′. Then the lower half of the diagramm shows that the connecting homomorphism maps π̃_{3}_{∗}(u′) to the non-trivial element in H_{0}(Γ_{2}; λ𝔽_{2}[Γ_{2}]⟨σ⟩) ≅ 𝔽_{2}, in particular π̃_{3}_{∗}(u′) ≠ 0 and hence π_{1}_{∗}i_{∗}u ≠ 0. Finally, using Section 2.6 once more, it is straightforward to check that π_{2}∂_{2} : C_{2} ⟶ 𝔽_{2}[Γ_{2}/Γ_{τ_{2}}] vanishes on C_{2}^{∂z} and hence π_{2}ij : I^{∂z} ⟶ 𝔽_{2}[Γ_{2}/Γ_{τ_{2}}] is the zero map and the second statement of the Lemma follows. □

4.2 Mod - 3 cohomology

In this section we prove Theorem 1.10. We take advantage of our investigations in Sections 2.4, 2.5 and 2.6. In particular we will make use of the description of the Γ_{j} - space Z given by Theorem 2.6 resp. Theorem 2.7, i.e. we will identify Z with Γ_{j} × D_{j}/ ∼_{j} and write A, B, … for the points of this space given by the class of (1, A), (1, B), . . ..

We break the proof into two parts. Proof of Theorem 1.10 (a)-(c). Let Z_{s,3}(j) denote the 3 - singular locus of Z with respect to the action of Γ_{j}.

Part c) of the Theorem follows immediately from Section 2.6 because in this case Z_{s,3}(2) = ∅, and therefore we get H̃^{∗}(Γ_{2}; 𝔽_{3}) ≅ H̃_{Γ_{2}}^{∗}(Z; 𝔽_{3}) ≅ H̃^{∗}(Γ_{2}\Z; 𝔽_{3}) ≅ Σ^{3}𝔽_{3} by Theorem 1.6.

In the cases of Γ_{0} and Γ_{1} we derive from Section 2.6 that the space Γ_{j}\Z_{s,3}(j) consists of two components. In the case of Γ_{0} one of the components consists of the image of the Γ_{0} - orbit of the 0 - cell A in Z_{s,3}(0) (with isotropy group isomorphic to D_{12}). The other one consists of the image of the Γ_{0} - orbit of the subcomplex with the two 1 - simplices OC and OF in Z_{s,3}(0); the Γ_{0} - orbits of the 1 - dimensional cells have isotropy isomorphic to 𝔖_{3} and the Γ_{0} - orbits of the three 0 - dimensional cells have isotropy group isomorphic to 𝔖_{4}. In the case of Γ_{1} one component in Γ_{1}\Z_{s,3}(1) comes from the 0 - cell A001 (where we use the convention of Section 2.6 for labelling the cells), again with isotropy group isomorphic to D_{12}; the other one comes from the subcomplex with 1 - cells OC111 and OF 111, with isotropy groups isomorphic to 𝔖_{3} for the 1 - dimensional cells and the 0 - dimensional cell F 111, and isomorphic to 𝔖_{4} for the 0 - dimensional cells O111 and C111. Now we consider the spectral sequences associated to the maps

EΓ_{j} ×_{Γ_{j}} Z_{s}(j) ⟶ Γ_{j}\Z_{s,3}(j), j = 0, 1 .

Because the inclusions of 𝔖_{3} into 𝔖_{4} and of 𝔖_{3} into D_{12} induce isomorphisms in mod - 3 cohomology we find in both cases an isomorphism

H_{Γ_{j}}^{∗}(Z_{s,3}(j); 𝔽_{3}) ≅ ∏_{i=1}^{2} H^{∗}(𝔖_{3}; 𝔽_{3}) .

Furthermore it is clear that H^{∗}(Γ_{j}\Z_{s}(j); 𝔽_{3}) = 𝔽_{3}⊕𝔽_{3}; from Theorem 1.6 we know H̃^{∗}(Γ_{j}\Z; 𝔽_{3}) = 0 and therefore we conclude H^{∗}(Γ_{j}\(Z, Z_{s,3}(j)); 𝔽_{3}) = Σ𝔽_{3}. Finally the long exact sequence for the Borel cohomology of the pair (Z, Z_{s,3}(j)) gives part a) and b). □

To prove part (d) of Theorem 1.10 one could now consider the spectral sequence of the map EΓ ×_{Γ} X ⟶ Δ^{2} and we will actually make some use of this spectral sequence. However, both for the final description of the result as well as for the proofs centralizers of elementary abelian 3 - subgroups turn out to be helpful again. In fact, we will combine information derived from the knowledge of these centralizers with information coming from the analysis of the spectral sequence of the map EΓ×_{Γ}X ⟶ Δ^{2} . We start by analyzing the elementary abelian 3 - subgroups of GL(3, ℤ[1/2]). First it is clear that there are no elementary abelian 3 - subgroups of rank 2 (isomorphic to ℤ/3 × Z/3) because there are none in GL(3, ℝ).

Proposition 4.5 In GL(3, ℤ[1/2]) there are precisely two conjugacy classes of subgroups isomorphic to ℤ/3.

Proof. These conjugacy classes are in one to one correspondence with the isomorphism classes of modules M over the group algebra ℤ[1/2][ℤ/3] which are free of rank 3 as ℤ[1/2] - modules and on which ℤ/3 acts faithfully. Such modules are classified by the (obvious modification for the ring ℤ[1/2] of the) Diederichsen-Reiner Theorem (cf. Theorem (74.3) of [CR]); one of the two classes corresponds to the free ℤ[1/2][ℤ/3] - module F on one generator, the other one to T ⊕ R, the direct sum of the trivial module T and the module R := ℤ[1/2][ζ_{3}] where a fixed chosen generator of ℤ/3 acts by multiplication with ζ_{3}, a fixed chosen third root of unity. □

We pick a subgroup E_{1} corresponding to the module F and a subgroup E_{2} corresponding to T ⊕ R. If E is a subgroup of a group G we write C_{G}(E) for the centralizer of E in G and N_{G}(E) for the normalizer of E in G. The units in a ring R will be denoted by R^{×}. We will now analyze the centralizers and normalizers of E_{i}. We start with the case of E_{2}.

Proposition 4.6 The centralizer C_{GL(3,ℤ[1/2])}(E_{2}) is isomorphic to ℤ/3 × ℤ[1/2]^{×} × ℤ[1/2]^{×}.

Proof. The centralizer is isomorphic to the group of automorphisms of the corresponding ℤ[1/2][ℤ/3] - module, i.e. to the group of units in its endomorphism ring. After tensoring with ℚ both F and R ⊕ T become isomorphic; both R ⊗ ℚ and T ⊗ ℚ are irreducible, in particular there are no ℤ[1/2][ℤ/3] - module maps between R and T. Therefore we obtain C_{GL(3,ℤ[1/2])}(E_{2}) ≅ ℤ[1/2]^{×} × (ℤ[1/2][ζ_{3}])^{×} and it is easy to check (say by using the norm map from the cyclotomic extension ℚ[ζ_{3}] down to ℚ) that the map

ℤ/3 × ℤ[1/2]^{×} → (ℤ[1/2][ζ_{3}])^{×}
(a, b) ↦ ζ_{3}^{a}b

is an isomorphism. □

Remark. The norm can also be used to show that ℤ[ζ_{3}] is a Euclidean ring and therefore a principal ideal domain. This simplifies the proof and statement of the Diederichsen-Reiner Theorem for modules over ℤ[ℤ/3] and ℤ[1/2][ℤ/3].

Corollary 4.7 E_{2} is contained in SL(3, ℤ[1/2]) and there is a unique conjugacy class of elementary abelian 3 - subgroups in SL(3, ℤ[1/2]) which maps to the GL(3, ℤ[1/2]) - conjugacy class of E_{2}. Furthermore

C_{SL(3,ℤ[1/2])}(E_{2}) ≅ ℤ/3 × ℤ[1/2]^{×} ,
N_{SL(3,ℤ[1/2])}(E_{2}) ≅ C_{SL(3,ℤ[1/2])}(E_{2}) ⋊ ℤ/2

and the isomorphism can be chosen such that the conjugation action of ℤ/2 on ℤ/3 is non-trivial while it is trivial on ℤ[1/2]^{×}.

Proof. It is clear that E_{2} is contained in SL(3, ℤ[1/2]) and also that C_{SL(3,ℤ[1/2])}(E_{2}) is isomorphic to ℤ/3 × ℤ[1/2]^{×}. Furthermore, if σ denotes the Galois automorphism of ℤ[1/2][ζ_{3}] then σ ⊕(−id) normalizes E_{2} and this shows N_{SL(3,ℤ[1/2])}(E_{2}) ≅ C_{SL(3,ℤ[1/2]}(E_{2}) ⋊ ℤ/2 with the conjugation action as claimed. Now assume E′ is another subgroup of SL(3, ℤ[1/2]) which becomes conjugate in GL(3, ℤ[1/2]) to E_{2}, say be an element g. Now the determinant from GL(3, ℤ[1/2]) to ℤ[1/2]^{×} remains onto when restricted to C_{GL(3,ℤ[1/2])}(E_{2}), hence we can write g = g_{1}g_{2} with g_{1} ∈ C_{GL(3,ℤ[1/2])}(E_{2}) and g_{2} ∈ SL(3, ℤ[1/2]) and this implies that E and E′ are already conjugate in SL(3, ℤ[1/2]). □

Proposition 4.8 The centralizer C_{GL(3,ℤ[1/2])}(E_{1}) is isomorphic to ℤ/3 × ℤ[1/2]^{×} × ℤ.

Proof. The module F contains the direct sum of the submodules Ker(g − 1) (generated as abelian group by 1 + g + g^{2}) and Ker(1 + g + g^{2}) (generated by 1 − g and 1 − g^{2}) with quotient isomorphic to ℤ/3 (observe that 3 = (1 + g + g^{2}) + (1 − g) + (1 − g^{2})). These submodules are isomorphic to T resp. R and are preserved by any automorphism of F. In other words we get a homomorphism

Aut(F) ⟶ Aut(R ⊕ T) ≅ ℤ/3 × ℤ[1/2]^{×} × ℤ[1/2]^{×} .

This is obviously injective and we claim that its image is isomorphic to ℤ/3× ℤ[1/2]^{×} × ℤ. To see this note that the subgroup ℤ/3 is clearly in the image; scalar automorphisms show that the diagonal of ℤ[1/2]^{×} × ℤ[1/2]^{×} is also in the image. Therefore it suffices to determine which of the automorphisms α_{ϵ,n} : R ⊕ T → R ⊕ T, (r, t) ↦ (r, ϵ2^{n}t) (with ϵ ∈ {0, 1} and n ∈ ℤ) extends to one of F. Because of 3 = (1 + g + g^{2}) + (1 − g) + (1 − g^{2}) an extension exists iff ϵ2^{n}(1 + g + g^{2}) + (1 − g) + (1 − g^{2}) = (g + g^{2})(ϵ2^{n} − 1) + (ϵ2^{n} + 2) is divisible by 3. This happens iff ϵ2^{n} − 1 is divisible by 3. In other words, n may be chosen arbitrarily but ϵ is then determined by n. □

Corollary 4.9 E_{1} is contained in SL(3, ℤ[1/2]) and there is a unique conjugacy class of elementary abelian 3 - subgroups in SL(3, ℤ[1/2]) which maps to the GL(3, ℤ[1/2]) - conjugacy class of E_{1}. Furthermore

C_{SL(3,ℤ[1/2])}(E_{1}) ≅ ℤ/3 × ℤ ,
N_{SL(3,ℤ[1/2])}(E_{1}) ≅ C_{SL(3,ℤ[1/2])}(E_{1}) ⋊ ℤ/2

and the isomorphism can be chosen such that the conjugation action of ℤ/2 on ℤ/3 is non-trivial while it is trivial on ℤ.

Proof. The proof is analogous to that of Proposition 4.7. One only has to check that the restriction of the determinant to C_{GL(3,ℤ[1/2])}(E_{1}) remains onto and that the automorphism −σ ⊕ (-id) of R ⊕ T (with σ the Galois automorphism of ℤ[1/2][ζ_{3}]) extends to an automorphism of F. □

We can now use the “centralizer spectral sequence”

E_{2}^{s,t} ≅ lim_{E}^{s} H^{t}(C_{Γ}(E); 𝔽_{3}) ⟹ H_{Γ}^{s+t}(X_{s,3}; 𝔽_{3})

of [H1] to compute H_{Γ}^{∗}(X_{s,3}; 𝔽_{3}) where as before Γ = SL(3, ℤ[1/2]), X is the space X_{∞} × X_{2}, but now X_{s,3} denotes the 3 - singular locus of X with respect to the action of Γ; the limit is here taken over the category of elementary abelian 3 - subgroups of Γ. Because the 3 - rank of Γ is equal to 1, the spectral sequence degenerates into an isomorphism

H_{Γ}^{∗}(X_{s,3}; 𝔽_{3}) ≅ ∏_{(E)}(H^{∗}(C_{Γ}(E); 𝔽_{3}))^{N_{Γ}(E)/C_{Γ}(E)} ≅ ∏_{(E)} H^{∗}(N_{Γ}(E); 𝔽_{3})

where the product is indexed by conjugacy classes of elementary abelian 3 - subgroups of Γ (see 3.3.1 of [H1]). In our case there are two conjugacy classes whose normalizers are isomorphic to 𝔖_{3} × ℤ × ℤ/2 resp. 𝔖_{3} × ℤ resp., so we obtain the following result.

Proposition 4.10

H_{Γ}^{∗}(X_{s,3}; 𝔽_{3}) ≅ ∏_{i=1}^{2} H̃^{∗}(𝔖_{3} × ℤ; 𝔽_{3})

We now turn towards the proof of part (d) of Theorem 1.10. By Proposition 4.10 it suffices to prove the following result.

Proposition 4.11 a) H_{Γ}^{∗}(X , X_{s,3}; 𝔽_{3}) ≅ Σ𝔽_{3} ⊕ Σ^{2}(𝔽_{3})^{2} ⊕ Σ^{5}(𝔽_{3}).

b) The boundary map

H_{Γ}^{∗}(X_{s,3}; 𝔽_{3}) ⟶ H_{Γ}^{∗+1}(X , X_{s,3}; 𝔽_{3})

is surjective. Its kernel in degree 4 is of dimension 3 and is generated by the image of the Bockstein of H^{3} and one further element which has non-trivial restriction to the two factors in Proposition 4.10.

The proof of Proposition 4.11 depends on another result whose proof we give at the end of this section.

Proposition 4.12 The restriction map H^{∗}(Γ_{1}; 𝔽_{3}) ⟶ H^{∗}(Γ_{2}; 𝔽_{3}) is onto, and with respect to the isomorphism H^{3}(Γ_{1}; 𝔽_{3}) ≅ H^{3}(𝔖_{3}; 𝔽_{3}) ⊕ H^{3}(𝔖_{3}; 𝔽_{3}) of part (b) of Theorem 1.10 the kernel in degree 3 restricts non-trivially to both factors.

Proof of Proposition 4.11. We consider the E_{1} - term of the spectral sequence of the map

EΓ ×_{Γ} (X , X_{s,3}) → Δ^{2} ≅ Γ\X_{2} .

By the proof of part (a) - (c) of Theorem 1.10 we get E_{1}^{0,1} ≅ E_{1}^{1,1} ≅ (𝔽_{3})^{3}, E_{1}^{2,0} ≅ E_{1}^{2,3} ≅ 𝔽_{3} and E_{1}^{s,t} = 0 in all other cases. In particular, we see that H_{Γ}^{3}(X , X_{s,3}; 𝔽_{3}) = H_{Γ}^{4}(X , X_{s,3}; 𝔽_{3}) = 0.

Now one could try to directly compute the differentials in order to prove (a). This can presumably be done directly, but we proceed in a different way which at the same time turns out to be quite useful for the proof of part (b). We use the spectral sequence of the map

EΓ ×_{Γ} X → Δ^{2} ≅ Γ\X_{2} .

By Theorem 1.10 its E_{1} - term is given by

E_{1}^{s,∗} ≅ ∏_{i=1}^{3} H^{∗}(Γ_{s}; 𝔽_{3}) if s = 0, 1 and E_{1}^{2,∗} ≅ H^{∗}(Γ_{2}; 𝔽_{3}) ≅ 𝔽_{3} ⊕ Σ^{3}𝔽_{3} .

By Proposition 4.10 we already know H^{∗}(Γ; 𝔽_{3}) in large dimensions. This together with the multiplicative structure of the spectral sequence forces the behaviour of d_{1} : E_{1}^{0,∗} ⟶ E_{1}^{1,∗} and gives for all ∗ > 0 with ∗ ≡ 3, 4 mod 4 that the kernel and cokernel of the map

d_{1} : (𝔽_{3})^{6} ≅ E_{1}^{0,∗} ⟶ E_{1}^{1,∗} ≅ (𝔽_{3})^{6}

is of dimension 2. By Proposition 4.12 the restriction map H^{∗}(Γ_{1}; 𝔽_{3}) ⟶ H^{∗}(Γ_{2}; 𝔽_{3}) is onto and therefore d_{1} : E_{1}^{1,∗} ⟶ E_{1}^{2,∗} is onto as well. In particular we find that the spectral sequence collapses at E_{2} and with some extra effort one could probably also determine the multiplicative structure. Here we need only the additive result in dimensions up to 5 where we find H^{0}(Γ; 𝔽_{3}) ≅ 𝔽_{3}, H^{1}(Γ; 𝔽_{3}) = H^{2}(Γ; 𝔽_{3}) = 0, H^{3}(Γ; 𝔽_{3}) ≅ H^{5}(Γ; 𝔽_{3}) = (𝔽_{3})^{2}, H^{4}(Γ; 𝔽_{3}) ≅ (𝔽_{3})^{3}. Part (a) and the surjectivity in part (b) of the proposition follow now immediately from the long exact sequence of the pair (X , X_{s,3}) together with the knowledge that H_{Γ}^{3}(X , X_{s,3}; 𝔽_{3}) = H_{Γ}^{4}(X , X_{s,3}; 𝔽_{3}) = 0. It is also clear that the kernel in degree 4 is of dimension 3 and contains the image of the Bockstein of H^{3}. The following proposition finishes the proof. □

Proposition 4.13 The restriction maps

H^{∗}(Γ; 𝔽_{3}) ⟶ H^{∗}(N_{Γ}(E_{i}); 𝔽_{3}) ≅ H^{∗}(𝔖_{3} × ℤ); 𝔽_{3})

are surjective for i = 1, 2 except in degree 1.

Proof of Proposition 4.13. We abbreviate N_{Γ}(E_{i}) by N_{i}. By Smith theory the space X^{E_{i}} is mod 3 - acyclic, so we try to understand the N_{i} - space X^{E_{i}} and for this we consider the canonical N_{i} - equivariant map X^{E_{i}} ⟶ X_{2}^{E_{i}} induced by the Γ - equivariant projection X ⟶ X_{2}. The quotient of X_{2} by the action of Γ is Δ^{2}. It is an elementary exercise to verify that the quotient of X_{2}^{E_{i}} by N_{i} is the 1 - skeleton ∂Δ^{2} of Δ^{2}, and furthermore that the isotropy group of the j - simplices in ∂Δ^{2} are isomorphic to N_{i} ∩ Γ_{j} for j = 0, 1. Now we compare the mod - p cohomology spectral sequences of the maps

EΓ ×_{Γ} X → Δ^{2} ≅ Γ\X_{2}

and

EΓ ×_{N_{i}} X^{E_{i}} → ∂Δ^{2} ≅ N_{i}\X_{2}^{E_{i}} .

As observed before the first spectral spectral sequence has as E_{1} - terms

E_{1}^{s,∗} ≅ (H^{∗}(Γ_{s}; 𝔽_{3}))^{⊕3} if s = 0, 1 and E_{1}^{2,∗} ≅ H^{∗}(Γ_{2}; 𝔽_{3}) ≅ 𝔽_{3} ⊕ Σ^{3}𝔽_{3} ,

while the second has

Ẽ_{1}^{s,∗} ≅ (H^{∗}(Γ_{s} ∩ N_{i}; 𝔽_{3}))^{⊕3} if s = 0, 1 and Ẽ_{1}^{2,∗} = 0 .

The map on E_{1} - terms is induced by the restriction maps H^{∗}(Γ_{s}; 𝔽_{3}) ⟶ H^{∗}(Γ_{s} ∩ N_{i}; 𝔽_{3}) for s = 0, 1. The groups Γ_{s} ∩ N_{i} are easily identified with 𝔖_{3} (for E_{1}) resp. 𝔖_{3} × ℤ/2 (for E_{2}). The map on E_{1}^{s,∗} corresponds for s = 0, 1 to the projection onto the i-th factor, i = 1, 2 (with respect to the product decomposition of the source, cf. Theorem 1.10(a+b)). Now we use Proposition 4.12 to finish the proof. □

Finally we turn towards the proof of Proposition 4.12.

Proof of Proposition 4.12. We dualize and work in homology. Furthermore we use Theorem 1.10(a+b) to identify H_{∗}(Γ_{0}; 𝔽_{3}) with H_{∗}(Γ_{1}; 𝔽_{3}) via the map induced by inclusion. Therefore we may consider the homomorphism H_{∗}(Γ_{2}; 𝔽_{3}) ⟶ H_{∗}(Γ_{0}; 𝔽_{3}) again induced by inclusion. By Shapiro’s lemma this homomorphism can be identified with the map

H_{∗}(Γ_{0}; 𝔽_{3}[Γ_{0}/Γ_{2}]) ⟶ H_{∗}(Γ_{0}; 𝔽_{3})

induced by the canonical map of Γ_{0} - modules 𝔽_{3}[Γ_{0}/Γ_{2}] ⟶ 𝔽_{3}. We denote the kernel of this map by K. The following lemma is the main step in the proof.

Lemma 4.14 H_{2}(Γ_{0}; K) ≅ 𝔽_{3}.

We continue with the proof of Proposition 4.12. The lemma immediately implies the first part of 4.12. For the second part we consider the two nonconjugate elementary abelian 3 - subgroups E_{1} and E_{2} of Γ_{0}; they are the 3 - Sylow subgroups of the two 𝔖_{3}’s which detect H^{∗}(Γ_{0}; 𝔽_{3}). The proof of the proposition will be complete once we have seen that the inclusions of both 𝔖_{3}’s into Γ_{0} induce isomorphisms H_{2}(𝔖_{3}; K) ⟶ H_{2}(Γ_{0}; K). (We note that 𝔽_{3}[Γ_{0}/Γ_{2}] is projective as 𝔽_{3}[𝔖_{3}] - module, and hence H_{2}(𝔖_{3}; K) ≅ H_{3}(𝔖_{3}; 𝔽_{3}) ≅ 𝔽_{3}.) In fact, this follows at once from the following observation: via mod - 2 reduction both 𝔖_{3}’s map monomorphically to SL(3, 𝔽_{2}) and there they agree with the normalizer of a 3 - Sylow subgroup; therefore the composition

H_{2}(𝔖_{3}; K) ⟶ H_{2}(Γ_{0}; K) ⟶ H_{2}(SL(3, 𝔽_{2}); K)

(the second arrow being induced by mod - 2 reduction) is an isomorphism. □

We turn towards the proof of Lemma 4.14. We could explicitly work out a projective resolution of the trivial 𝔽_{3}[Γ_{0}] - module 𝔽_{3} from the resolution provided by the cellular chains of Z, and then compute H_{2}(Γ_{0}; K) from this projective resolution. As this would be quite involved we construct just as much of this resolution as necessary.

Proof of Lemma 4.14. We consider the mod - 3 cellular chain complex of Z and break it apart into the following exact sequences of Γ_{0} - modules where i_{1}δ_{2} = ∂_{2} and i_{0}δ_{1} = ∂_{1}:

0 ⟶ C_{3} ⟶^{∂_{3}} C_{2} ⟶^{δ_{2}} I_{2} ⟶ 0  (4.2)

0 ⟶ I_{2} ⟶^{i_{1}} C_{1} ⟶^{δ_{1}} I_{1} ⟶ 0  (4.3)

0 ⟶ I_{1} ⟶^{i_{0}} C_{0} ⟶^{ε} 𝔽_{3} ⟶ 0  (4.4)

The lemma will follow from the long exact sequence in Tor_{∗}^{Γ_{0}}(−; K) associated to the exact sequence (4.4) and the following claims. (Here and in the sequel we abbreviate Tor_{∗}^{𝔽_{3}[Γ_{0}]}(−; K) by Tor_{∗}^{Γ_{0}}(−; K).) Claim 1: Tor_{2}^{Γ_{0}}(C_{0}; K) ≅ (𝔽_{3})^{4}. Claim 2: Tor_{1}^{Γ_{0}}(I_{1}; K) = 0 and Tor_{2}^{Γ_{0}}(I_{1}; K) ≅ (𝔽_{3})^{3}. Claim 3: The map Tor_{2}^{Γ_{0}}(I_{1}; K) ⟶ Tor_{2}^{Γ_{0}}(C_{0}; K) induced by i_{0} is injective.

Proof of Claim 1. The Γ_{0} - modules C_{i} are direct sums ⨁_{σ} 𝔽_{3}[Γ_{0}/Γ_{σ}] where σ runs through the set of Γ_{0} - orbits of i - dimensional cells in Z and Γ_{σ} is the isotropy group of a chosen representative of the orbit σ. By Sections 2.4 and 2.5 we have 5 orbits of 0 - cells in Z corresponding to the vertices A, O, C, F and D in Γ_{0}\Z, with respective isotropy groups D_{12} (for A), 𝔖_{4} (for O, C and F) and D_{8} (for D). By Shapiro’s lemma we have therefore isomorphisms

Tor_{2}^{Γ_{0}}(C_{0}; K) ≅ H_{2}(D_{12}, K) ⊕ (H_{2}(𝔖_{4}; K))^{⊕3} ⊕ H_{2}(D_{8}, K) .

The contribution coming from D_{8} is trivial because the order of D_{8} is prime to 3. Furthermore Γ_{2} has no 3 - torsion, hence the 3 - Sylow subgroup of all the other finite subgroups acts freely on 𝔽_{3}[Γ_{0}/Γ_{2}] and hence this module is projective when restricted to the other finite subgroups. As a consequence we obtain H_{2}(D_{12}; K) ≅ H_{3}(D_{12}; 𝔽_{3}) ≅ 𝔽_{3} and H_{2}(𝔖_{4}; K) ≅ H_{3}(𝔖_{4}; 𝔽_{3}) ≅ 𝔽_{3} and the claim follows.

□

Proof of Claim 2. Here we use the exact sequences (4.2) and (4.3). Just as above we find

Tor_{i}^{Γ_{0}}(C_{s}; K) ≅ ⨁_{σ} H_{i+1}(Γ_{σ}; 𝔽_{3})  (4.5)

if s = 0, 1, 2, 3 and all i > 0. If i = 0, we observe that for all σ we have H_{1}(Γ_{σ}; 𝔽_{3}) = 0, and hence the functors Tor_{0}^{Γ_{0}}(C_{s}; −) carry the short exact sequence

0 ⟶ K ⟶ 𝔽_{3}[Γ_{0}/Γ_{2}] ⟶ 𝔽_{3} ⟶ 0

into short exact sequences. Hence we have a short exact sequence of complexes

0 ⟶ Tor_{0}^{Γ_{0}}(C_{∗}; K) ⟶ Tor_{0}^{Γ_{0}}(C_{∗}; 𝔽_{3}[Γ_{2}/Γ_{0}]) ⟶ Tor_{0}^{Γ_{0}}(C_{∗}; 𝔽_{3}) ⟶ 0 (4.6)

for which the homology is known in the middle and on the right by Theorem 1.6. From the exact sequence (4.2), formula (4.5) and our analysis of the cell structures and their symmetries in Sections 2.4, 2.5 and 2.6 we deduce that Tor_{i}^{Γ_{0}}(I_{2}; K) = 0 if i > 1. For i = 1 it is isomorphic to the homology in dimension 3 of the complex Tor_{0}^{Γ_{0}}(C_{∗}; K); this in turn is isomorphic to the homology in dimension 3 of the complex Tor_{0}^{Γ_{0}}(C_{∗}; 𝔽_{3}[Γ_{2}/Γ_{0}]), i.e. to H_{3}(Γ_{2}\X_{∞}; 𝔽_{3}) ≅ 𝔽_{3} by Theorem 1.6. For i = 0 we obtain

Tor_{0}^{Γ_{0}}(I_{2}; K) ≅ Coker(Tor_{0}^{Γ_{0}}(C_{3}; K) ⟶^{∂_{3}} Tor_{0}^{Γ_{0}}(C_{2}; K)) .

Now we can compute Tor_{i}^{Γ_{0}}(I_{1}; K) for i = 1, 2 from the long exact sequence in Tor which is associated to the exact sequence (4.3). Using once more our analysis in section 2.4, 2.5 and 2.6 we see that Tor_{1}^{Γ_{0}}(C_{1}; K) = 0 and we obtain a short exact sequence

0 ⟶ (𝔽_{3})^{2} ≅ Tor_{2}^{Γ_{0}}(C_{1}; K) ⟶ Tor_{2}^{Γ_{0}}(I_{1}; K) ⟶ Tor_{1}^{Γ_{0}}(I_{2}; K) ≅ 𝔽_{3} ⟶ 0

(4.7)

where the contribution to Tor_{2}^{Γ_{0}}(C_{1}; K) comes from the two Γ_{0} - orbits of 1 - dimensional cells corresponding to OC and OF with symmetry group isomorphic to 𝔖_{3} in both cases. For Tor_{1}^{Γ_{0}}(I_{1}; K) we use again that Tor_{1}^{Γ_{0}}(C_{1}; K) = 0 and that the map Tor_{0}^{Γ_{0}}(I_{2}; K) ⟶ Tor_{0}^{Γ_{0}}(C_{1}; K) is injective (because the complex Tor_{0}^{Γ_{0}}(C_{∗}; K) has no homology in degree 2 by Theorem 1.6). □

Proof of Claim 3. We proceed in several steps. In a first step we reduce the evaluation of

i_{0}_{∗} : Tor_{2}^{Γ_{0}}(I_{1}; K) ⟶ Tor_{2}^{Γ_{0}}(C_{0}; K) ≅ H_{2}(D_{12}, K) ⊕ (H_{2}(𝔖_{4}; K))^{⊕3}

to the study of a particular chain map. In a second step we descroibe this chain map and in a final step we finish the computation of i_{0}_{∗}.

\underline{First step.} We consider the restriction of the map i_{0}_{∗} to the subgroup Tor_{2}^{Γ_{0}}(C_{1}; K) ≅ (H_{2}(𝔖_{3}; K))^{⊕2} ≅ (𝔽_{3})^{2} (cf. (4.7)). This restriction is induced by injections of the isotropy groups (isomorphic to 𝔖_{3}) of the edges OC and OF into the isotropy groups (isomorphic to 𝔖_{4}) of the vertices O, C and F. Each of these injections induces an isomorphism of cohomology in H_{2}(−; K) ≅ H_{3}(−; 𝔽_{3}), hence these injections map (H_{2}(𝔖_{3}; K))^{⊕2} monomorphically to the summand (H_{2}(𝔖_{4}; K))^{⊕3} of Tor_{2}^{Γ_{0}}(C_{0}; K). It suffices therefore to show that the composition of i_{0} : I_{1} ⟶ C_{0} with the projection π : C_{0} ⟶ 𝔽_{3}[Γ_{0}/D_{12}] induces a non-trivial map in Tor_{2}^{Γ_{0}}(−; K). To this end we should construct 𝔽_{3}[Γ_{0}] - projective resolutions P_{∗} of I_{1} and Q_{∗} of 𝔽_{3}[Γ_{0}/D_{12}] and lift πi_{0} to a chain map α : P_{∗} ⟶ Q_{∗}. In the case of Q_{∗} we start with the minimal projective resolution Q′_{∗} for 𝔽_{3} as a 𝔽_{3}[D_{12}] - module and take Q_{∗} = 𝔽_{3}[Γ_{0}] ⊗_{𝔽_{3}[D_{12}]} Q′_{∗}. In the case of I_{1} we obtain a projective resolution P_{∗} by taking first projective resolutions R_{2}_{∗} of I_{2} and R_{1}_{∗} of C_{1}; then we lift i_{1} : I_{2} ⟶ C_{1} to a chain map ĩ_{1} : R_{2}_{∗} ⟶ R_{1}_{∗} and obtain a double complex R_{∗∗} whose total complex gives the desired projective resolution P_{∗}. More concretely, we can take the exact sequence 0 ⟶ C_{3} ⟶ C_{2} ⟶ I_{2} ⟶ 0 as a projective resolution R_{2}_{∗} of I_{2}. For C_{1} we use the direct sum decomposition C_{1} ≅ ⨁_{σ} 𝔽_{3}[Γ_{0}/Γ_{σ}] and, if Rσ 1_{∗} is a minimal projective resolution of the trivial 𝔽_{3}[Γ_{σ}] - module 𝔽_{3}, then we take R_{1}_{∗} = ⨁_{σ} 𝔽_{3}[Γ_{0}] ⊗_{𝔽_{3}[Γ_{σ}]} Rσ 1_{∗}. In these terms, the projective resolution P_{∗} of I_{1} looks as follows:

⋯ ⟶ R_{14} ⟶^{∂_{3}^{P}} R_{13} ⟶^{∂_{2}^{P}} R_{12} ⊕ C_{3} ⟶^{∂_{1}^{P}} R_{11} ⊕ C_{2} ⟶^{∂_{0}^{P}} R_{10}

with ∂_{3}^{P} = ∂_{3}^{R_{1}}, ∂_{2}^{P} = ( \begin{matrix}∂_{2}^{R_{1}} \\ 0 \end{matrix} ), ∂_{1}^{P} = ( \begin{matrix}∂_{1}^{R_{1}} & ĩ_{11} \\ 0 & −∂_{0}^{R_{2}} \end{matrix} ) and ∂_{0}^{P} = (∂_{0}^{R_{1}} ĩ_{10}).

We denote the direct summands of C_{2} resp. R_{1} corresponding to the 2 - dimensional cells BEF and OCBF resp. the 1 - dimensional faces of these 2 - dimensional cells by \overline{C_{2}} resp. \overline{R_{1}}. It is clear that the lift ĩ_{1} of i_{1} can be chosen in such a way that ĩ_{10}(\overline{C_{2}}) ⊂ \overline{R_{10}} so that

⋯ ⟶ \overline{R_{14}} ⟶^{∂_{3}^{P}} \overline{R_{13}} ⟶^{∂_{2}^{P}} \overline{R_{12}} ⟶^{∂_{1}^{P}} \overline{R_{11}} ⊕ \overline{C_{2}} ⟶^{∂_{0}^{P}} \overline{R_{10}}

is a subcomplex of P_{∗}. Furthermore the lift α of πi_{0} can be chosen such that this subcomplex maps trivially to Q_{∗}. Therefore, if we denote the direct summands of C_{2} resp. C_{1} correponding to the other 2 - dimensional resp. 1 - dimensional cells by C̃_{2} resp. C̃_{1} and if we use that C̃_{1} is projective (because all isotropy groups are of even order), i.e. R̃_{10} = C̃_{1} and R̃_{1∗} = 0 for ∗ > 0, we obtain a factorization of α to a map α̃ of complexes P̃_{∗} ⟶ Q_{∗} as follows (with i_{10} being induced by ∂_{1}^{C}) :

0 ⟶ C_{3} ⟶^{−∂_{0}^{R_{2}}} C̃_{2} \underrightarrow{ĩ_{10}} C̃_{1}
α_{2}↓  ^{α_{1}}↓  ^{α_{0}}↓
Q_{3} ⟶^{∂_{3}^{Q}} Q_{2} ⟶^{∂_{2}^{Q}} Q_{1} ⟶^{∂_{1}^{Q}} Q_{0}

\underline{Second step.} Now we construct α̃ in detail. First we observe that the complexes P̃_{∗} resp. Q are induced from 𝔽_{3}[D_{12}] - complexes P′_{∗} resp. Q′_{∗} and that α̃ can also be constructed as a map induced from a chain map α′ of D_{12} - chain complexes. So it is enough to construct α′. We denote the natural 𝔖_{3} - modules 𝔽_{3}[𝔖_{3}/𝔖_{2}] by S and the tensor product of S with the non-trivial one-dimensional 𝔖_{3} - module by S(−1). With respect to the action of the 3 - Sylow subgroup of 𝔖_{3} the two module structures agree, so if we denote a fixed generator of this 3 - Sylow subgroup by t, we have well defined linear maps S(−1) ⟶ S and S(−1) ⟶ S which deserve to be labelled t^{2} − t. We consider all these modules as D_{12} - modules via the natural homomorphism D_{12} ⟶ 𝔖_{3} and leave it to the reader to verify that the minimal resolution Q′_{∗} of the trivial 𝔽_{3}[D_{12}] - module 𝔽_{3} is periodic of order 4 and can be described as follows:

⋯ S(−1) ⟶^{t^{2}−t} S ⟶^{1+t+t^{2}} S ⟶^{t^{2}−t} S(−1) ⟶^{1+t+t^{2}} S(−1) ⟶^{t^{2}−t} S .

Now we define maps α′_{i} for i = 1, 2, 3 and leave it to the reader to verify that they fit together to give a D_{12} - equivariant chain map α′ : P′_{∗} ⟶ Q′_{∗} as desired. As before we denote the isotropy groups of the cells OA, AB, … by Γ_{OA}, Γ_{AB}, . . .. Then the map

α′_{0} : 𝔽_{3}[D_{12}/Γ_{OA}] ⊕ 𝔽_{3}[D_{12}/Γ_{AB}] ⊕ 𝔽_{3}[D_{12}/Γ_{AD}] ⟶ S

is given as follows (we choose the letter e as a generic letter for the generators of the various modules while g^{AB} and g^{AD} are the elements in Γ_{A} = D_{12} which have been introduced in section 2.4):

α′_{0}(e_{OA}) = e_{S}, α′_{0}(e_{AB}) = −g^{AD}e_{S}, α′_{0}(e_{AD}) = −g^{AB}e_{S} .

This is D_{12} - equivariant if the subgroup 𝔖_{2} which occurs in the definition of the module S is chosen as the subgroup generated by the image of g^{AB}g^{AD}g^{AB} with respect to the projection D_{12} ⟶ 𝔖_{3}; for t we take the image of g^{AD}g^{AB}. The map

α′_{1} : 𝔽_{3}[D_{12}/Γ_{OAC}] ⊕ 𝔽_{3}[D_{12}/Γ_{OAF}] ⊕ 𝔽_{3}[D_{12}/Γ_{ABD}] ⟶ S(−1)

may then be given by

α′_{1}(e_{OAC}) = 0, α′_{1}(e_{OAF}) = 0, α′_{1}(e_{ABD}) = −e_{S(−1)} .

Finally we have

α′_{2} : 𝔽_{3}[D_{12}] ⟶ S(−1), α′_{2}(e) = e_{S(−1)} .

\underline{Third step.} We are now ready to finish the calculation of i_{0}_{∗}. The following element of K ⊂ 𝔽_{3}[Γ_{0}/Γ_{2}] (cf. formula (3.2) in section 3.1)

[100y = 0] − [100z = 0] − [010x = 0] + [010z = 0] + [001x = 0] − [001y = 0]

represents a class in H_{2}(P′_{∗} ⊗_{𝔽_{3}[D_{12}]} K) and it suffices to show that its image via

α′_{2} ⊗ id_{K} : K ≅ 𝔽_{3}[D_{12}] ⊗_{𝔽_{3}[D_{12}]} K ⟶ S(−1) ⊗_{𝔽_{3}[D_{12}]} K

is non-trivial in H_{2}(Q′_{∗} ⊗_{𝔽_{3}[D_{12}]} K), i.e. is not in the image of

S ⊗_{𝔽_{3}[D_{12}]} K ⟶^{t^{2}−t} S(−1) ⊗_{𝔽_{3}[D_{12}]} K .

We leave this verification to the patient reader with the hint that the calulation can be significantly simplified by making use of the decomposition of 𝔽_{3}[Γ_{0}/Γ_{2}] as a 𝔽_{3}[D_{12}] - module (cf. section 2.6). □

4.3 Higher torsion in the integral cohomology

It is clear from Corollary 1.7 that the p - torsion in H^{∗}(SL(3, ℤ[1/2]); ℤ) is trivial for primes p > 3. Furthermore the mod - 3 Bockstein spectral sequence and Theorem 1.10 shows that the 3 - torsion is all of order 3 and is easily understood from the results in the last section. Therefore we restrict attention to higher 2 - torsion. For this consider the mod - 2 Bockstein spectral sequence for SL(3, ℤ[1/2]): We know from Theorem 1.4 that H^{∗}(SL(3, ℤ[1/2]); 𝔽_{2}) maps injectively onto the subalgebra of H^{∗}(SD_{3}; 𝔽_{2}) ≅ 𝔽_{2}[x, y] ⊗ E(f, g) generated by v_{2} = x^{2} + xy+y^{2}, v_{3} = x^{2}y+xy^{2}, d_{3} = x^{2}g+y^{2}f and d_{5} = x^{4}g+y^{4}f (cf. [H1]). Therefore we have Sq^{1}v_{2} = v_{3} while Sq^{1} is zero on the other algebra generators and thus we see that the E_{2} - term of this spectral sequence is isomorphic to 𝔽_{2}[v_{2}^{2}] ⊗ E(d_{3}, d_{5}). The crucial point is now which order Bockstein of d_{3} kills v_{2}^{2}. To settle this we consider the mod - 2 Bockstein spectral sequence for GL(2, ℤ[1/2]): In this case H^{∗}(GL(2, ℤ[1/2]); 𝔽_{2}) maps injectively onto the subalgebra of H^{∗}(D_{2}; 𝔽_{2}) ≅ 𝔽_{2}[x, y] ⊗ E(f, g) generated by w_{1} = x + y, w_{2} = xy, e_{1} = e + f and e_{3} = x^{2}g + y^{2}f [H1] and the E_{2} - term identifies with 𝔽_{2}[w_{2}^{2}] ⊗ E(e_{1}, e_{3}). The restriction map from H^{∗}(SL(3, ℤ[1/2]); 𝔽_{2}) to H^{∗}(GL(2, ℤ[1/2]); 𝔽_{2}) maps d_{3} to e_{3} and v_{2} to w_{2} + w_{1}^{2}, hence v_{3}^{2} to w_{2}^{2} + w_{1}^{4} which in the E_{2} - term is identified with w_{2}^{2}. Therefore it suffices to determine which higher order Bockstein of e_{3} kills w_{2}^{2}. Now we compare the mod - 2 Bockstein spectral sequence for GL(2, ℤ[1/2]) with that of SL(2, ℤ[1/2]); we recall that H^{∗}(SL(3, ℤ[1/2]); 𝔽_{2}) ≅ 𝔽_{2}[w_{2}] ⊗ E(e_{3}) (cf. [Mi]). The notation suggests the behaviour of the restriction map, i.e. the elements w_{2} and e_{3} of H^{∗}(GL(2, ℤ[1/2]); 𝔽_{2}) map to the elements in H^{∗}(SL(2, ℤ[1/2]); 𝔽_{2}) with the same name. Furthermore, the element w_{2} comes from an integral class in H^{∗}(SL(2, ℤ[1/2]); ℤ) (namely the first Chern class c_{1}), hence Sq^{1} acts trivially on it. Therefore, the E_{2} - term in the case of SL(2, ℤ[1/2]) is isomorphic to 𝔽_{2}[w_{2}] ⊗ E(e_{3}) and hence it is enough to determine which higher order Bockstein of e_{3} kills w_{2}^{2} in the Bockstein spectral sequence for SL(2, ℤ[1/2]), or equivalently which is the additive order of the second power of the integral lift c_{1} of w_{2}. This can be checked to be of order 8, e.g. by playing off the mod - 2 cohomology computation [Mi] against an integral cohomology computation based on the amalgam description SL(2, ℤ[1/2]) ≅ SL(2, ℤ) ∗_{Δ} SL(2, ℤ) [Se]. Here Δ is the subgroup of SL(2, ℤ) consisting of all matrices which are upper triangular modulo 2. We summarize our discussion in the following result.

Proposition 4.15 The higher 2 - torsion in H^{∗}(SL(3, ℤ[1/2]); ℤ) is all of order 8 and is represented in the mod - 2 Bockstein spectral sequence by the classes v_{2}^{2n} and d_{5}v_{2}^{2n} (n > 0); the classes 1 and d_{5} represent classes of infinite order. □

The integral cohomology of SL(3, ℤ[1/2]) can now be easily written down explicitly. We leave the details to the interested reader.

5 The cohomology of GL(3, ℤ[1/2])

Let GL_{±}(3, ℤ[1/2]) be the preimage of the subgroup {±1} of (ℤ[1/2])^{×} under the determinant GL(3, ℤ[1/2]) ⟶ (ℤ[1/2])^{×}. The group GL_{±}(3, ℤ[1/2]) splits as SL(3, ℤ[1/2]) × ℤ/2, so we understand its mod p - cohomology by Theorem 1.4, Corollary 1.7 and Theorem 1.10. We will work out the mod - p cohomology spectral sequences of the group extension

1 ⟶ GL_{±}(3, ℤ[1/2]) ⟶ GL(3, ℤ[1/2]) ⟶ ℤ ⟶ 1  (5.1)

where the homomorphism from GL(3, ℤ[1/2]) to ℤ is the determinant followed by the quotient map (ℤ[1/2])^{×} ⟶ (ℤ[1/2])^{×}/{±1} ≅ ℤ. Note that the matrix 2·id is central in GL(3, ℤ[1/2]), hence it acts trivially on GL_{±}(3, ℤ[1/2]) by conjugation. Its determinant is 8 = 2^{3} which corresponds to the element 3 in ℤ under the determinant map. It follows that the conjugation action of ℤ on H^{∗}(GL_{±}(3, ℤ[1/2]); 𝔽_{p}) factors through an action of ℤ/3.

The case p > 3. The conjugation action of ℤ/3 on H^{5}(GL_{±}(3, ℤ[1/2]); 𝔽_{p}) ≅ 𝔽_{p} comes from one on integral cohomology, hence it is necessarily trivial. Furthermore the spectral sequence necessarily collapses at E_{2} and we obtain the following result.

Proposition 5.1 Assume p > 3. Then there is an isomorphism of algebras

H^{∗}(GL(3, ℤ[1/2]); 𝔽_{p}) ≅ H^{∗}(SL(3, ℤ[1/2]); 𝔽_{p}) ⊗ H^{∗}((ℤ[1/2])^{×}; 𝔽_{p}) .

We have chosen ℤ[1/2]^{×} as second factor in order to get “symmetric statements” for the different primes.

The case p = 2. Again we look at the spectral sequence of the group extension (5.1). As in the case of primes p > 3 we claim that the conjugation action of ℤ/3 on H^{∗}(GL_{±}(3, ℤ[1/2]); 𝔽_{2}) ≅ H^{∗}(SL(3, ℤ[1/2]); 𝔽_{2}) ⊗ H^{∗}(ℤ/2; 𝔽_{2}) is trivial. First we note that this action leaves the two factors H^{∗}(SL(3, ℤ[1/2]); 𝔽_{2}) and H^{∗}(ℤ/2; 𝔽_{2}) invariant and is clearly trivial on the second factor. By dimensional reasons it is clear that the action is trivial on v_{2}, and because of Sq^{1}v_{2} = v_{3} it is also trivial on v_{3}. Now we know that the action of ℤ/3 on H^{3}(SL(3, ℤ[1/2]; 𝔽_{2}) ≅ (𝔽_{2})^{2} has an invariant subspace (namely the subspace generated by v_{3}) and this forces it to be also trivial on d_{3}. Next the formula Sq^{2}d_{3} = d_{5} and multiplicativity of the action shows that ℤ/3 acts trivially as claimed. We obtain E_{2} ≅ H^{∗}(SL(3, ℤ[1/2]); 𝔽_{2}) ⊗ H^{∗}((ℤ[1/2])^{×}; 𝔽_{2}) as algebras. By Theorem 1.1 E_{2} consists of permanent cycles, i.e. the spectral sequence collapses and we have finally proved Theorem 1.3.

The case p = 3. Once more we look at the spectral sequence of the group extension (5.1). Using the restriction map to the cohomology of the centralizers C_{SL(3,ℤ[1/2])}(E_{i}) together with the description of these groups as provided by Section 4.2, it is easy to see that the action of ℤ/3 on H̃^{∗}(GL_{±}(3, ℤ[1/2]); 𝔽_{3}) is trivial. So as before we obtain an isomorphism E_{2} ≅ H^{∗}(SL(3, ℤ[1/2]); 𝔽_{3}) ⊗ H^{∗}((ℤ[1/2])^{×}; 𝔽_{3}) as algebras, i.e. there is no room for differentials and the spectral sequence collapses. By using the restriction maps to the centralizers C_{GL(3,ℤ[1/2])}(E_{i}) we see that the E_{2} - term gives also the algebra structure. We state the result of our discussion in the following result.

Proposition 5.2 There is an isomorphism of algebras

H^{∗}(GL(3, ℤ[1/2]); 𝔽_{3}) ≅ H^{∗}(SL(3, ℤ[1/2]); 𝔽_{3}) ⊗ H^{∗}((ℤ[1/2])^{×}; 𝔽_{3}) .

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[H1] H.-W. Henn, Centralizers of elementary abelian p - subgroups, the Borel construction of the singular locus and applications to the cohomology of discrete groups, Topology 36 (1997), 271-286

[H2] H.-W. Henn, Commutative algebra of unstable K - modules, Lannes’ T - functor and equivariant mod - p - cohomology, Journal für die reine und angewandte Mathematik 478 (1996), 189-215

[HLS] H.-W. Henn, J. Lannes and L. Schwartz. Localization of unstable A - modules and equivariant mod - p cohomology, Math. Ann. 301 (1995), 23-68

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Hans-Werner Henn Mathematisches Institut der Universität Im Neuenheimer Feld 288 D-69120 Heidelberg Germany

Current address: Département de Mathématique Université Louis Pasteur 7, rue René Descartes F-67084 Strasbourg France