CENTRALIZERS OF ELEMENTARY ABELIAN P - SUBGROUPS, THE BOREL CONSTRUCTION OF THE SINGULAR LOCUS

AND APPLICATIONS TO THE COHOMOLOGY OF DISCRETE GROUPS

by

Hans–Werner Henn

0. Statement of results

Let G be either a compact Lie group or a discrete group and X a space with a continuous G - action. Assume the action has finite stabilizers if G is discrete. For a fixed prime p let X_{s} be the p - singular locus of X, i.e. the set of all points in X which are fixed by some element of order p. In this paper we study the Borel construction EG ×_{G} X_{s} (EG denoting as usual the total space of the universal principal G - bundle) by relating it to the Borel constructions EG ×_{C_{G}(E)} X^{E} where E runs through the non-trivial elementary abelian p - subgroups of G, C_{G}(E) denotes the centralizer of E in G and X^{E} is the E - fixed point set of X.

Our starting point is the following recent result of Jackowski and McClure [JM]. For a compact Lie group G and a fixed prime p let A(G) be the category whose objects are all elementary abelian p - subgroups E of G, i.e. subgroups of the form E ≅ ( /p)^{k}; if E_{1} and E_{2} are elementary abelian p - subgroups of G, then the set of morphisms from E_{1} to E_{2} in A(G) consists precisely of those homomorphisms α : E_{1} ⟶ E_{2} of abelian groups for which there exists an element g ∈ G with α(e) = geg^{−1} ∀e ∈ E_{1}. Jackowski and McClure considered the full subcategory A_{∗}(G) of A(G) whose objects are all elementary abelian p - subgroups except the trivial subgroup. The assignment E ↦ EG ×_{G} G/C_{G}(E) extends in a canonical way to a functor from the opposite category A_{∗}(G)^{op} to spaces and the projections π(E) : EG×_{G} G/C_{G}(E) ⟶ EG×_{G} pt = BG , pt denoting a one point space and BG the classifying space of G, give a natural transformation from the functor EG ×_{G} G/C_{G}(−) to the constant functor with value BG. The main result of Jackowski and McClure claims that the space BG can be recovered from the spaces EG ×_{G} G/C_{G}(−) by taking the homotopy colimit of the maps π(E) (the definition of a homotopy colimit will be recalled in section 1).

THEOREM 0.0 [JM]. Let G be a compact Lie group. Then the maps π(E) induce an isomorphism in mod p cohomology

π : hocolim_{A_{∗}(G)^{op}} EG ×_{G} G/C_{G}(E) ⟶ BG .

Of course, in Theorem 0.0 we can replace mod p cohomology by mod p homology. In fact, as the integral homology of all spaces involved is finitely generated in each degree, we can use homology with coefficients in the p - local integers _{(p)}. A map inducing an isomorphism in homology with coefficients in _{(p)} will be called a _{(p)} - homology isomorphism. The space EG×_{G} G/C_{G}(E) is homotopy equivalent to BC_{G}(E) and the theorem gives therefore a tool to study homological and homotopy theoretic aspects of BG in terms of those of classifying spaces of subgroups of G. In fact, this tool has turned out to be extremely useful in recent developments in the homotopy theory of classifying spaces and of finite loop spaces (see [DW], [JMO]). In this paper we will prove extensions of this theorem for Borel constructions EG ×_{G} X of G - CW - complexes X; Theorem 0.0 will then correspond to the case X = pt.

So let X be a G - space and p be a fixed prime. The assignment E ↦ G ×_{C_{G}(E)} X^{E} extends to a functor from A_{∗}(G)^{op} to G - spaces and the G - action gives G - maps μ_{X}(E) : G×_{C_{G}(E)} X^{E} ⟶ X_{s} which define a natural transformation from the functor E ↦ G ×_{C_{G}(E)} X^{E} to the constant functor with value X_{s}. Here is a first extension of Theorem 0.0.

THEOREM 0.1. Let G be a compact Lie group and X be any G - CW - complex. Then the maps μ_{X}(E) induce a _{(p)} - homology isomorphism

μ_{X} : hocolim_{A_{∗}(G)^{op}} EG ×_{G} (G ×_{C_{G}(E)} X^{E}) ⟶ EG ×_{G} X_{s} .

We note that as in [JM] it is important in this theorem to pass to the Borel construction, i.e. the obvious map hocolim_{A_{∗}(G)^{op}}(G ×_{C_{G}(E)} X^{E}) ⟶ X_{s} is not a mod p equivalence in general. However, when G is abelian, it is not difficult to see that this map is even a homotopy equivalence (cf. section 2). At this point we should point out that the statement of Theorem 0.0 needs to be stated with a little more care in order to be correct. One has to throw in the further assumption that the category A_{∗}(G) is not empty, i.e. that G contains non-trivial elements of order p (which, of course, is automatic for any positive dimensional Lie group); otherwise the homotopy colimit gives the empty space which cannot be homology equivalent to a non empty space. This remark may seem a little pedantic but it essentially gives the explanation why X_{s} appears in Theorem 0.1 rather than X. Note that in case that G does not contain any non-trivial element of order p, then for any G - space X (in particular for X = pt) the subspace X_{s} is empty and therefore both the source and the target of μ_{X} are empty.

All the notions in Theorem 0.1 make sense for more general groups, in particular for discrete groups. In fact, Theorem 0.1 can be extended to the case of suitable actions of discrete groups as follows.

THEOREM 0.2. Let G be a discrete group and let X be any G - CW - complex such that the isotropy group of each cell of X is finite. Then the maps μ_{X}(E) induce a _{(p)} - homology isomorphism

μ_{X} : hocolim_{A_{∗}(G)^{op}} EG ×_{G} (G ×_{C_{G}(E)} X^{E}) ⟶ EG ×_{G} X_{s} .

Theorem 0.1 and Theorem 0.2 are both deduced from Theorem 0.0. In fact, a common generalization of Theorem 0.1 and Theorem 0.2 (Theorem 2.6) will be proved in section 2 after some recollections on homotopy colimits have been given in section 1. In section 3 we will discuss the homotopy colimit spectral sequences which arise from Theorem 0.1 resp. Theorem 0.2 and converge to the cohomology of the space EG ×_{G} X_{s}. In the situation of Theorem 0.0 and in case of mod p cohomology the E_{2} - term of this spectral sequence is concentrated on the vertical edge and the edge homomorphism is an isomorphism [JM]. Results of Oliver [O] and the author [H1] show that even in the situation of Theorem 0.1 and Theorem 0.2 there are often very strong vanishing results for the E_{2} - terms of these spectral sequences, and at least in the case of ordinary mod p cohomology H^{∗}(−; _{p}) the behaviour of the spectral sequences is almost as in the case of Theorem 0.0. For a generalized cohomology theory h^{∗} we denote the h^{∗} cohomology of the Borel construction EG ×_{G} X by h_{G}^{∗}X. We call a generalized cohomology theory p - local if its coefficients are _{(p)} - modules. In the result below lim^{s} is the s - th derived functor of the inverse limit functor and the p - rank r_{p}(G) is defined as the supremum of all natural numbers k such that G contains an elementary abelian p - subgroup E of rank k, i.e. E ≅ ( /p)^{k}.

THEOREM 0.3. Let G be a compact Lie group and X be any G - CW complex, or let G be a discrete group and X a G - CW complex with the property that the isotropy group of each cell is finite. Let h^{∗} be any p - local generalized cohomology theory.

a) Then there is a spectral sequence

E_{2}^{s,t} = lim_{A_{∗}(G)}^{s} h_{C_{G}(E)}^{t}(X^{E}) ⟹ h_{G}^{s+t}(X_{s}) ,

with E_{2}^{s,t} = 0 if s ≥ r_{p}(G) and t is arbitrary.

b) Furthermore, if X has finitely many G - cells and h^{∗} = H^{∗}(−; _{p}), then E_{2}^{s,t} = 0 whenever s is positive and t is sufficiently large, and the map

ρ : H_{G}^{∗}(X_{s}; _{p}) ⟶ lim_{A_{∗}(G)} H_{C_{G}(E)}^{∗}(X^{E}; _{p})

induced by the obvious inclusions on the space and group level is an isomorphism in sufficiently large cohomological degrees.

\underline{Remarks:}

a) Theorem 0.3 can be used to get information on the ordinary cohomology H_{G}^{∗}(X; M) in large cohomological degrees for any _{(p)} - module M. If X is a G - CW - complex then the projection map EG ×_{G} X ⟶ G\X induces an isomorphism between the relative groups H_{G}^{∗}(X, X_{s}; M) and the relative groups H^{∗}(G\X, G\X_{s}; M). In particular, the natural map H_{G}^{∗}(X; M) ⟶ H_{G}^{∗}(X_{s}; M) is an isomorphism if ∗ > dim X, the dimension of X.

b) Theorem 0.3 gives a very tight relation between H_{G}^{∗}(X; _{p}) and the various H_{C_{G}(E)}^{∗}(X^{E}; _{p}) for the fixed point sets X^{E}, E ⊂ G, E ≅ ( /p)^{k} with k > 0; it describes H_{G}^{∗}(X; _{p}) by the H_{C_{G}(E)}^{∗}(X^{E}; _{p}) up to a finite error. The result may thus be regarded as an extension of classical Smith theory to actions of arbitrary compact Lie groups. Smith theory works well only for elementary abelian p - groups (or for tori if one works with rational cohomology); in its modern form it says that for an elementary abelian p - group V the restriction map H_{V}^{∗} (X; _{p}) ⟶ H_{V}^{∗} (X^{V}; _{p}) becomes an isomorphism after inverting the action of a suitable element in H_{V}^{∗} (pt; _{p}). Of course, the approximation to H_{V}^{∗} (X; _{p}) provided by such a localization is much cruder than that provided by the inverse limit in Theorem 0.3.b. From that point of view Theorem 0.3 provides also a strenghtening of classical Smith theory in the case of the action of an elementary abelian p - group.

c) We do not know whether the vanishing result in part b) of the Theorem holds for more general coefficients than _{p}. The proof of the vanishing result [H1] takes advantage of the action of the Steenrod algebra on mod p cohomology and in particular of the theory of Lannes’ T - functor. A naturally arising question is now whether something intelligible can be said about the higher limits, in particular those in part b). In the case of mod p cohomology and the action of an elementary abelian p - group V this question will be addressed in [HLO]. In particular we will show there that these higher limits are related to new invariants of V - CW - complexes X which refine the invariants d_{0}H_{V}^{∗} (X; _{p}) and d_{1}H_{V}^{∗} (X; _{p}) introduced in [HLS, Chap II.1]. These new invariants give lower bounds for the equivariant embedding dimension of X (i.e. the mimimal dimension of a smooth V - manifold into which X can be V - equivariantly embedded).

d) Note that Theorem 0.2 and Theorem 0.3 do not allow us to take X to be a point if G is discrete (unless G is finite) and, in fact, both theorems need not hold in this case (cf. [H2] for explicit examples). So in general we cannot use these results directly to get information on the classifying space BG of a discrete group. However, for many discrete groups G there exists an X as in Theorem 0.3 which is also finite dimensional and contractible so that EG ×_{G} X ≃ BG. So if M is a _{(p)} - module we obtain in this case a spectral sequence converging to H_{G}^{∗}(X_{s}; M) which is isomorphic to H_{G}^{∗}(X; M) ≅ H_{G}^{∗}(pt; M) if ∗ > dim X. If the spaces X^{E} are also contractible then we get EG ×_{C_{G}(E)} X^{E} ≃ BC_{G}(E) so the input in the spectral sequence is the cohomology of the groups C_{G}(E). In fact, we do not need contractibility; it is enough that the spaces X and X^{E} be acyclic with respect to cohomology with coefficients in M to get H_{G}^{∗}(X; M) ≅ H_{G}^{∗}(pt; M) and H_{C_{G}(E)}^{∗}(X^{E}; M) ≅ H_{C_{G}(E)}^{∗}(pt; M). For M ≅ _{p} classical Smith theory implies that the spaces X^{E} are mod p acyclic if X is. In particular we obtain the following result.

COROLLARY 0.4. Let G be a discrete group and assume there exists a finite dimensional mod p acyclic G - CW complex X such that the isotropy group of each cell is finite. a) Then there is a spectral sequence

E_{2}^{s,t} = lim_{A_{∗}(G)}^{s} H_{C_{G}(E)}^{t}(pt; _{p}) ⟹ H_{G}^{s+t}(X_{s}; _{p}) ,

with E_{2}^{s,t} = 0 if s ≥ r_{p}(G) and t is arbitrary.

b) Assume furthermore X has only finitely many G - cells. Then E_{2}^{s,t} = 0 whenever s is positive and t is sufficiently large. Furthermore the natural map

ρ : H_{G}^{∗}(pt; _{p}) ⟶ lim_{A_{∗}(G)} H_{C_{G}(E)}^{∗}(pt; _{p})

induced by the restriction homomorphisms is an isomorphism in sufficiently large cohomological degrees.

\underline{Remarks:}

a) Important classes of such groups are arithmetic or S - arithmetic groups [S2], mapping class groups of orientable surfaces [Ha], outer automorphism groups of free groups [CV] and hyperbolic groups in the sense of Gromov [GH]. In all these cases X can actually be chosen to be contractible with finitely many G - cells. If G has a finite index subgroup of finite cohomological dimension (such a G is said to be of finite virtual cohomological dimension, f.v.c.d. for short) then X can be chosen to be finite dimensional and contractible [S1].

b) The case of rational coefficients might also be of some interest. In this case we have H_{G}^{∗}(Y; ) ≅ H^{∗}(G\Y; ) for any G - CW complex Y for which the stabilizers of all cells are finite. Therefore, if X and X^{E} are rationally acyclic (in addition to the assumptions in Theorem 0.3), we obtain a spectral sequence

E_{2}^{s,t} = lim_{A_{∗}(G)}^{s} H_{C_{G}(E)}^{t}(pt; ) ⟹ H_{G}^{s+t}(X_{s}; ) ≅ H^{s+t}(G\X_{s}; ) ,

i.e. we get information about the part of H^{∗}(BG; ) ≅ H^{∗}(G\X; ) coming from the p - singular locus G\X_{s}.

c) Corollary 0.4 is somewhat reminiscent of a spectral sequence for the Farrell cohomology Ĥ^{∗}(Γ; _{p}) of a f.v.c.d. group Γ. Its E_{1} - term is given by the Farrell cohomology of intersections of normalizers of non-trivial elementary abelian p - subgroups of Γ and it converges to Ĥ^{∗}(Γ; _{p}) (see [B]). We will call this spectral sequence the “normalizer spectral sequence” as opposed to the “centralizer spectral sequence” of Theorem 0.3. Vanishing results as spelled out in Corollary 0.4.a) hold for trivial reasons in this normalizer spectral sequence but there are no known vanishing results for this spectral sequence which would correspond to the vanishing results in 0.4.b). However, most recently Chun-Nip Lee [L] has used Corollary 0.4.b) to show that the E_{2} - terms of the two spectral sequences agree on the vertical edge in large degrees (i.e. for s = 0 and t large) if the coefficients are in _{p}. From this he concludes that in the normalizer spectral sequence one has E_{2}^{0,t} = E_{∞}^{0,t} for t large, and at least E_{∞}^{s,t} = 0 if s > 0 and s + t is sufficiently large. We refer the reader to section 3.3 and section 4.5 for some further comments on the relation between the normalizer spectral sequence and the centralizer spectral sequence.

In section 4 we apply Corollary 0.4 to compute the mod 2 cohomology of the special linear group SL(3, [1/2]) (i.e. of its classifying space) in large dimensions. For earlier work on this case using the “normalizer spectral sequence” we refer to [B] (see also section 4 for further comments on the relation of the two approaches). In the following theorem which summarizes our computation E is an exterior algebra and Σ^{n} denotes n -fold suspension, e.g. Σ^{3} _{2} is a graded one dimensional _{2} - vector space concentrated in dimension 3. The indices of the elements give their cohomological dimensions. For a definition of the elements the reader is refered to section 4.

THEOREM 0.5. Let p = 2 and Γ = SL(3, [1/2]).

a) Then we obtain

lim_{A_{∗}(Γ)} H^{∗}(BC_{Γ}(E); _{2}) ≅ _{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5}),
lim_{A_{∗}(Γ)}^{1} H^{∗}(BC_{Γ}(E); _{2}) ≅ Σ^{3} _{2},
lim_{A_{∗}(Γ)}^{s} H^{∗}(BC_{Γ}(E); _{2}) ≅ 0 for all s > 1.

Furthermore, if X is 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} _{2} ⟶ H_{Γ}^{∗}(X_{s}; _{2}) ⟶^{ρ} _{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5}) ⟶ 0

in which ρ is an algebra homomorphism.

b) The canonical map

H^{∗}(BΓ; _{2}) ⟶ lim_{A_{∗}(Γ)} H^{∗}(BC_{Γ}(E); _{2}) ≅ _{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5})

is an isomorphism in dimensions larger than 5.

In [H2] we will finish the computation of the mod 2 cohomology H^{∗}(BSL(3, [1/2]); _{2}) by combining Theorem 0.5 with a computation of the relative cohomology H_{Γ}^{∗}(X, X_{s}; _{2}) ≅ H^{∗}(Γ\X, Γ\X_{s}; _{2}) (with Γ = SL(3, [1/2])) for a suitable space X as in Theorem 0.5. In particular we will prove in [H2].

THEOREM 0.6. The canonical map

H^{∗}(BΓ; _{2}) ⟶ lim_{A_{∗}(Γ)} H^{∗}(BC_{Γ}(E); _{2}) ≅ _{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5})

is an isomorphism in all dimensions.

\underline{Acknowledgements:} While this research was carried out, the author was supported by a “Heisenberg Stipendium der Deutschen Forschungsgemeinschaft”. It is a pleasure to thank Jean Lannes for numerous inspiring discussions which stimulated my interest in the cohomology of SL(3, [1/2]). Finally I would like to thank Alejandro Adem for his interest and comments on a first draft of this paper.

1. Recollections on homotopy colimits

In this section we recall the definition and collect some basic properties of the homotopy colimit of a functor F from a small category C to the category T of compactly generated topological spaces (cf. [BK], [JMO] or [HV]). All spaces in this paper will be assumed to be compactly generated.

\underline{1.1.} Let C be a small discrete category and F be a functor from C to T . Then hocolim_{C} F is the classifying space BC_{F} of the following topological category C_{F} whose objects are all pairs (c, x) with c ∈ ob C and x ∈ F c, and whose morphisms are given as mor_{C_{F}} ((c, x), (c′, x′)) = {α ∈ mor_{C}(c, c′)|(F α)x = x′}. The topology on C_{F} is induced in the obvious way from the topology on the spaces F c.

\underline{1.2.} Alternatively, hocolim_{C} F may be desribed as follows: Let Δ^{n} denote the standard geometric n - simplex. Then the homotopy colimit of F is given as the following identification space:

(1.2)  hocolim_{C} F := (∐_{n≥0} ∐_{c_{0}→···→c_{n}} F (c_{0}) × Δ^{n})/ ∼ .

Here the second coproduct is indexed by the n - simplices of the nerve of C and the identifications are induced in the obvious way by the face and degeneracy maps. The reader can easily verify the following properties of the homotopy colimit construction.

\underline{1.3.} Let G be a topological group. If F is a functor from C to G - spaces then hocolim commutes with the Borel construction:

(1.3.1)  EG ×_{G} hocolim_{C} F = hocolim_{C} EG ×_{G} F .

Similarly, if H is a closed subgroup of G and F is a functor from C to H - spaces then hocolim commutes with the process of induction:

(1.3.2)  G ×_{H} hocolim_{C} F = hocolim_{C} G ×_{H} F .

\underline{1.4.} Let C and C′ be small categories, i : C ⟶ C′, F : C ⟶ T , F′ : C′ ⟶ T functors and θ a natural transformation from F to F′i. From these data we obtain a continuous functor θ̃: C_{F} ⟶ C′_{F′} given on objects by sending a pair (c, x) to the pair (ic, (θc)x). This functor induces in turn a map of classifying spaces, i.e. a map hocolim_{C} F ⟶ hocolim_{C′} F′ which by abuse of notation will be called θ again. In the second description of hocolim in (1.2) this map is given by sending the summand F (c_{0}) × Δ^{n} corresponding to the n - simplex c_{0} → ⋯ → c_{n} of the nerve of C via the map θc_{0} × id_{Δ^{n}} to the summand F′(ic_{0}) × Δ^{n} corresponding to the n - simplex ic_{0} → ⋯ → ic_{n} of the nerve of C′.

\underline{1.5.1.} If C has a terminal object c_{0} than the natural map hocolim_{C} F ⟶ F (c_{0}) is a homotopy equivalence. \underline{1.5.2.} Let C and C′ be small categories and i : C ⟶ C′ a functor. For a functor F : C ⟶ T let the “homotopy push down” i_{∗}F : C′ ⟶ T be given by (i_{∗}F)(c′) = hocolim_{i↓c′} F where F on the right hand side denotes, by abuse of notation, the obvious functor on i ↓ c′ determined by F. (Here i ↓ c′ denotes the “over category” with objects consisting of pairs (c, g) with c ∈ ob C and g : ic ⟶ c′ a morphism in C′; morphisms from (c_{1}, g_{1}) to (c_{2}, g_{2}) are morphisms f : c_{1} ⟶ c_{2} in C such that g_{1} = g_{2}(if).) Then the obvious map hocolim_{C′} i_{∗}F ⟶ hocolim_{C} F is a homotopy equivalence (cf. [DK], or [HV] for an elegant and more detailed discussion.)

2. Proof of Theorem 0.1 and Theorem 0.2

Let p be a fixed prime, G be a topological group and X be any G - space. Furthermore let E_{1}, E_{2} be any elementary abelian p - subgroups of G and α a homomorphism from E_{1} to E_{2} of the form α(e) = h^{−1}eh for some h ∈ G, independent of e ∈ E_{1}. Note that α determines h up to left multiplication by an element in C_{G}(E_{1}). For the sake of completeness we record the following lemmata whose easy verification is left to the reader.

LEMMA 2.1. The map x ↦ hx induces a map from X^{E_{2}} to X^{E_{1}}.

LEMMA 2.2. The map g ↦ hgh^{−1} induces a homomorphism α_{h} : C_{G}(E_{2}) ⟶ C_{G}(E_{1}).

LEMMA 2.3. The action of G on X restricts to an action of C_{G}(E) on X^{E}.

LEMMA 2.4. The map G × X^{E_{2}} ⟶ G ×_{C_{G}(E_{1})} X^{E_{1}}, (g, x) ↦ (gh^{−1}, hx) induces a G - equivariant map α^{∗} : G ×_{C_{G}(E_{2})} X^{E_{2}} ⟶ G ×_{C_{G}(E_{1})} X^{E_{1}} which only depends on α, not on h.

As an immediate consequence we get

COROLLARY 2.5. The assignment E ↦ G ×_{C_{G}(E)} X^{E}, α ↦ α^{∗} defines a functor F_{X} : A_{∗}(G)^{op} to G - spaces.

Next we note that the action of G on X induces maps μ_{X}(E) : G ×_{C_{G}(E)} X^{E} ⟶ X_{s} which combine to give a natural transformation μ = μ_{X} from F_{X} to the constant functor with value X_{s}. If we want to emphasize the role of the group that is acting we will write F_{X}^{G} resp. μ_{X}^{G} instead of F_{X} resp. μ_{X}. Theorem 0.1 and Theorem 0.2 are concerned with the map

hocolim_{A_{∗}(G)^{op}} EG ×_{G} F_{X}^{G} ⟶ EG ×_{G} X_{s}

which is induced by the natural transformation μ_{X} and which we will also call μ_{X}. The following result includes Theorem 0.1 and Theorem 0.2.

THEOREM 2.6. Let G be a topological group. Let C be the smallest class of G - spaces which is closed with respect to G - homotopy equivalences, arbitrary coproducts and homotopy pushouts (i.e. double mapping cylinders), and which contains all orbits G/H with H a subgroup which is a compact Lie group (maybe finite). Then the map

μ_{X} : hocolim_{A_{∗}(G)^{op}} EG ×_{G} F_{X}^{G} ⟶ EG ×_{G} X_{s}

is a _{(p)} - equivalence for all X in C.

\underline{Proof of 2.6:} We consider the following two functors from G - CW - complexes to graded _{(p)} - modules:

• F_{1}(X) = H_{∗}(hocolim_{A_{∗}(G)^{op}} EG ×_{G} F_{X}^{E}; _{(p)})

• F_{2}(X) = H_{∗}(EG ×_{G} X_{s}; _{(p)}).

The maps μ_{X} induce a natural transformation between these functors which we will also call μ. The functors F_{1} and F_{2} are both homotopy invariant, they preserve arbitrary coproducts and come with Mayer-Vietoris sequences for each homotopy pushout. Therefore it suffices to show the assertion for the orbits X = G/H where H is a compact Lie group. For this we consider two cases. Case 1: H does not contain a nontrivial element of order p. In this case we have F_{i}(X) = 0 for i = 1, 2 because of (G/H)^{E} = ∅ for each non - trivial E. Case 2: H does contain elements of order p. In this case we find X_{s} = X = G/H, i.e. F_{2}(X) = H_{∗}(BH; _{(p)}). By [JM] we know that μ_{pt}^{H} is a _{(p)} - homology isomorphism, and hence the following proposition and 1.3.1 imply that μ_{G/H}^{G} is one, too.

PROPOSITION 2.7. There is a canonical map

G ×_{H} hocolim_{D∈A_{∗}(H)^{op}}(H ×_{C_{H}(D)} pt) ⟶ hocolim_{E∈A_{∗}(G)^{op}}(G ×_{C_{G}(E)} (G/H)^{E})

which is a homotopy equivalence and commutes with the two maps to G/H induced by the natural transformations G ×_{H} μ_{pt}^{H} resp. μ_{G/H}^{G}.

\underline{Proof:} First we construct the map. We denote the inclusion functor of A_{∗}(H)^{op} into A_{∗}(G)^{op} by i. By 1.3.2 and 1.4 it suffices to consider the functors

G ×_{H} F_{H/H}^{H} : A_{∗}(H)^{op} → G − spaces , D ↦ G ×_{C_{H}(D)} pt

and

F_{G/H}^{G} : A_{∗}(G)^{op} → G − spaces , E ↦ G ×_{C_{G}(E)} (G/H)^{E}

and to construct a natural transformation θ : G ×_{H} F_{H/H}^{H} ⟶ F_{G/H}^{G} ∘ i.

For D ∈ A_{∗}(H) the map θ(D) : G ×_{C_{H}(D)} pt ⟶ G ×_{C_{G}(D)} (G/H)^{D} is given by sending the equivalence clas of (g, pt) to the equivalence class of (g, eH) where eH denotes the coset of the neutral element in G/H. It is easy to see that these maps are well defined and determine a natural transformation θ as required. Furthermore it is clear that the resulting map

θ : G ×_{H} hocolim_{D∈A_{∗}(H)^{op}}(H ×_{C_{H}(D)} pt) ⟶ hocolim_{E∈A_{∗}(G)^{op}}(G ×_{C_{G}(E)} (G/H)^{E})

commutes with the two obvious maps to G/H. The following two lemmata together with 1.5.1 will allow us to identify the “homotopy push down” i_{∗}(G ×_{H} F_{H/H}^{H}) with G ×_{H} F_{H/H}^{H} and then 1.5.2 will finish the proof of 2.7.

LEMMA 2.8. Let E be an elementary abelian p - subgroup of G. Consider pairs (D, α) where D is an elementary abelian p - subgroup of H and α : E ⟶ D is an isomorphism in A(G). Declare (D, α) and (D′, α′) to be equivalent iff there is an isomorphism β : D ⟶ D′ in A(H) with α′ = (iβ)α. Choose one representative (D_{i}, α_{i}) from each equivalence class. Then the compositions (α_{i})^{∗}θ(D_{i}) (with (α_{i})^{∗} the map introduced in 2.4) combine to give a homeomorphism

∐_{(D_{i},α_{i})} G ×_{C_{H}(D_{i})} pt ⟶ G ×_{C_{G}(E)} (G/H)^{E} .

LEMMA 2.9. Let E be an object in A_{∗}(G)^{op} and i : A_{∗}(H)^{op} ⟶ A_{∗}(G)^{op} be as above. Then the over category i ↓ E is a disjoint union of categories indexed by equivalence classes of pairs (D, α) as in 2.8 and with terminal objects given by chosen representative (D_{i}, α_{i}) from each equivalence class.

\underline{Proof of 2.8:} We begin by analyzing the spaces (G/H)^{E} as C_{G}(E) - spaces. We remark that there are only finiteley many conjugacy classes of elementary abelian p subgroups of H (cf. [Q, Lemma 6.3]) and therefore the disjoint sum is indexed by a finite set. Consider an E - fixed point gH ∈ G/H. The element g gives an isomorphism α : E → g^{−1}Eg, e ↦ g^{−1}eg in A_{∗}(G) such that g^{−1}Eg ⊂ H, i.e. it gives a pair (D, α) = (g^{−1}Eg, e ↦ g^{−1}eg). By definition of our equivalence relation the equivalence class of this pair (D, α) is independent of the coset representative in gH. If the coset g′H gives rise to the same equivalence class as gH then gH and g′H differ by left multiplication by an element in C_{G}(E). Furthermore, the isotropy of gH with respect to the C_{G}(E) action is given by C_{G}(E) ∩ gHg^{−1}. In other words, picking a set of elements g_{i} such that the corresponding pairs (D_{i}, α_{i}) run through a set of representatives of the equivalence classes gives us first a homeomorphism of G(E) - spaces

∐_{(D,α)} C_{G}(E)/C_{G}(E) ∩ g_{i}Hg_{i}^{−1} ⟶^{≅} (G/H)^{E}

and then one of G - spaces

∐_{(D_{i},α_{i})} G ×_{C_{G}(E)∩g_{i}Hg_{i}^{−1}} pt ⟶^{≅} G ×_{C_{G}(E)} (G/H)^{E}

which on the i -th summand is given by [g, pt] ↦ [g, g_{i}H] if [g, pt] resp. [g, g_{i}H] denote the equivalence class of the pair (g, pt) resp. (g, g_{i}H). Combining this with the homeomorphism

G ×_{C_{H}(D_{i})} pt ⟶^{≅} G ×_{C_{G}(E)∩g_{i}Hg_{i}^{−1}} pt

given by [g, pt] ↦ [gg_{i}^{−1}, pt] we get a homeomorphism of G - spaces

∐_{(D_{i},α_{i})} G ×_{C_{H}(D_{i})} pt ⟶^{≅} G ×_{C_{G}(E)} (G/H)^{E}

which on the i -th summand is given by [g, pt] ↦ [gg_{i}^{−1}, g_{i}H]. By its definition the map (α_{i})^{∗}θ(D_{i}) is given by the very same formula.

\underline{Proof of 2.9:} Translating back to the categories A_{∗}(H) and A_{∗}(G) we have to decompose the “under category” E ↓ i and exhibit appropriate initial objects for each “component”. An object in E ↓ i is a pair (D, β) with D ∈ ob A_{∗}(H) and β : E ⟶ iD a morphism in A_{∗}(G); a morphism from (D, β) to (D′, β′) is a morphism α : D ⟶ D′ in A_{∗}(H) with β = (iα)β′. The morphism β has a unique factorization E ⟶ Im β ⊂ D in A_{∗}(G) (with Im β a subgroup of H); we write β = jβ̃. This factorization gives a morphism j̃ : (Im β, β̃) ⟶ (D, β) in E ↓ i. The decomposition of E ↓ i into components is given by the equivalence relation

(D, β) ∼ (D′, β′) ⟺ (Im β, β̃) ≅ (Im β′, β̃′) (≅ in A_{∗}(H))

and a choice of representatives gives initial objects for the “components” of E ↓ i.

3. The spectral sequences

Let G and X be as in Theorem 0.1 resp. Theorem 0.2. Assume h^{∗} is a p - local generalized cohomology theory. Because of Theorem 0.1 and Theorem 0.2 we obtain a spectral sequence [BK, XII, 4.5]

E_{2}^{s,t} = lim_{A_{∗}(G)}^{s} h_{C_{G}(E)}^{t}(X^{E}) ⟹ h_{G}^{s+t}(X_{s}) .

It is remarked in [JMO] that this spectral sequence is, in fact, the Leray spectral sequence of the projection map hocolim_{A_{∗}(G)} EG ×_{C_{G}(E)} X^{E} ⟶ hocolim_{A_{∗}(G)} pt. In the case of Theorem 0.0 and mod p cohomology this spectral sequence degenerates completely, i.e. lim_{A_{∗}(G)}^{s} H^{∗}(BC_{G}(E); _{p}) = 0 for all s > o and the edge homomorphism H^{∗}(BG; _{p}) ⟶ lim H^{∗}(BC_{G}(E); _{p}) is an isomorphism [JM]. Such a strong result is too much to ask for in the general situation even in the case of mod p cohomology (see Theorem 0.5 for an explicit example). However, there are still interesting vanishing results which will imply Theorem 0.3. The first result is due to Oliver.

THEOREM 3.1 [O]. Let G be any group and denote by r_{p}(G) the p - rank of G, i.e. the supremum of all natural numbers k such that G contains an elementary abelian p - subgroup E of rank k. Let F be any functor from A_{∗}(G) to _{(p)} - modules. Then lim_{A_{∗}(G)}^{s} F = 0 for all s ≥ r_{p}(G).

This result is stated in [O] only for compact Lie groups but its proof shows that it actually holds in the generality stated here. Oliver describes also an explicit small chain complex whose homology gives the higher limits.

THEOREM 3.2 [H1]. Assume that either

a) G is a compact Lie group and X a G - CW - complex with finitely many G - cells, or b) G is a discrete group and X is a G - CW - complex with finitely many G - cells such that the isotropy group of each cell is finite. Then lim_{A_{∗}(G)}^{s} H_{C_{G}(E)}^{∗}(X^{E}; _{p}) is a finite dimensional graded _{p} - vector space for each s > 0, in particular it is trivial if ∗ is sufficiently large.

\underline{Proof of Theorem 0.3:} Part (a) follows immediately from Theorem 0.1 resp. Theorem 0.2 and Theorem 3.1. Theorem 3.2 implies the further vanishing result of part (b). Combining both vanishing results gives that in sufficiently large cohomological degrees the spectral sequence collapses at E_{2}, and hence the edge homomorphism is an isomorphism in large degrees. Finally consider the map ρ. It is given as composition of the map H_{G}^{∗}(X; _{p}) ⟶ H_{G}^{∗}(X_{s}; _{p}) followed by the edge homomorphism. The first map is an isomorphism above the dimension of X (see Remark a) after Theorem 0.3), hence the composition is an isomorphism in sufficiently large degree.

We would like to point out that the assertion in Theorem 0.3 concerning ρ appear here as a consequence of the vanishing results for the E_{2} - term. However, this is somewhat misleading because the vanishing results depend on 3.2, and the proof of Theorem 3.2 given in [H1] derives the vanishing result for the higher limits from a purely algebraic theorem on certain unstable modules M over the Steenrod algebra, which in case M = H_{G}^{∗}(X; _{p}), X as in Theorem 0.3, specializes to the assertion concerning ρ.

\underline{3.3.} We finish this section with some comments on the relation between the “centralizer spectral sequence” considered here and the “normalizer spectral sequence” [B] converging to the mod - p Farell cohomology of a discrete group Γ of finite virtual cohomological dimension. For the convenience of the reader we briefly recall how this spectral sequence arises. The group Γ acts by conjugation on the the poset _{p}(Γ) of non-trivial elementary abelian p - subgroups of Γ and then also on the simplicial complex | _{p}Γ)| associated to it. We denote Farrell cohomology of a Γ - space X with coefficients in _{p} by Ĥ_{Γ}^{∗}(X; _{p}) and abbreviate Ĥ_{Γ}^{∗}(pt; _{p}) by Ĥ^{∗}(Γ; _{p}) and similarly H_{Γ}^{∗}(pt; _{p}) by H^{∗}(Γ; _{p}). Then Ĥ^{∗}(Γ; _{p}) agrees with Ĥ_{Γ}^{∗}(| _{p}(Γ)|; _{p}) and the spectral sequence of the map from EΓ ×_{Γ} | _{p}(Γ)| to Γ\| _{p}(Γ)| has E_{1} - term given by the cohomology of the stabilizers of simplices in | _{p}(Γ)|, i.e. by

E_{1}^{s,∗} = ∏_{E_{0}⊂···⊂E_{s}} Ĥ^{∗}(⋂_{i=0}^{s} N_{Γ}(E_{i}); _{p})

where the product is indexed by all the conjugacy classes of proper chains E_{0} ⊂ ⋯ ⊂ E_{s} in the poset | _{p}(Γ)| of length s + 1 and N_{Γ}(−) denotes the normalizer in Γ. In particular, E_{1}^{s,∗} = 0 if s > r_{p}(Γ).

\underline{3.3.1.} If r_{p}(Γ) = 1 then both spectral sequences collapse. The “normalizer spectral sequence” gives

Ĥ^{∗}(Γ; _{p}) ≅ ∏_{(E)} Ĥ^{∗}(N_{Γ}(E); _{p}) ,

the product being indexed by conjugacy classes of non-trivial elementary abelian p - subgroups of Γ. On the other hand, the category A_{∗}(Γ) has as a skeleton a discrete category; objects are in one to one correspondence with the conjugacy classes of elementary abelian p - subgroups of Γ, all morphisms are automorphisms and Aut_{A_{∗}(Γ)}(E) identifies with N_{Γ}(E)/C_{Γ}(E). Therefore the centralizer approach gives an isomorphism

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

the right hand side denoting the invariants with respect of the action of N_{Γ}(E)/C_{Γ}(E). In this case the two approaches are essentially equivalent because H^{∗}(Γ; _{p}) ≅ H_{Γ}^{∗}(X_{s}; _{p}) above the virtual cohomological dimension and because the invariants H^{∗}(C_{Γ}(E); _{p})^{N_{Γ}(E)/C_{Γ}(E)} are isomorphic to H^{∗}(N_{Γ}(E); _{p}) (Note that the groups N_{Γ}(E)/C_{Γ}(E) are of order prime to p, being subgroups of Aut(E) ≅ /(p − 1)). We remark that the “normalizer approach” has been used in the p - rank 1 case to determine the Farrell cohomology of certain arithmetic groups and of certain mapping class groups (see [A], [X1, X2]).

\underline{3.3.2.} It would be interesting to understand the relation between the two approaches in cases of higher p - rank. The case of p - rank 1 and the example of SL(3, [1/2]) (see section 4.5 below) suggest that there ought to be some connection. \underline{3.3.3.} In the case of general p - rank the “normalizer spectral sequence” requires the knowledge of the cohomology of as many subgroups of Γ as there are simplices in Γ\| _{p}(Γ)|, while in the “centralizer approach” the number of subgroups involved is equal to the number of conjugacy classes of elementary abelian p - subgroups of Γ, i.e. the number of 0 - simplices in Γ\| _{p}(Γ)|. Because the computation of the cohomology of each normalizer (and centralizer) is usually non-trivial this seems to represent a significant computational advantage of our approach over the “normalizer approach”. However, one serious limitation of the “normalizer approach” is still present in our approach: the cohomology of the subgroups remains often inaccessible.

In those cases where the subgroups are accessible, our approach should be more effective. Unfortunately this is not easily demonstrated; each interesting example still seems to require a major effort. In the following section we illustrate our method by the particularly favourable example of the special linear group SL(3, [1/2]).

4. The example of the group Γ = SL(3, [1/2])

In this section we will prove Theorem 0.5. We will abbreviate SL(3, [1/2]) by Γ and write H^{∗}G instead of H^{∗}(BG; _{2}) if G is a discrete group.

\underline{4.1.} We begin by describing a skeleton of the category A_{∗}(Γ) which we will denote by A. The category A has two objects, say E_{1} and E_{2} of rank 1 and 2 respectively. We take E_{1} to be the subgroup generated by the diagonal matrix whose first two diagonal entries are −1 and whose third diagonal entry is 1, and E_{2} to be the subgroup of all diagonal matrices with diagonal entries 1 or −1 and determinant 1. The automorphism group of E_{1} is trivial, of course, while Aut_{A}(E_{2}) is isomorphic to the group of all abstract automorphisms of E_{2} which we can identify with 𝔖_{3}, the symmetric group on three elements. There are three morphisms from E_{1} to E_{2} which are permuted by Aut_{A}(E_{2}).

\underline{4.2.} The next step is to descibe the functor on A which sends E to H^{∗}C_{Γ}(E). For the centralizers themselves we find C_{Γ}(E_{1}) ≅ GL(2, [1/2]) and C_{Γ}(E_{2}) ≅ D_{3}, the diagonal matrices inside Γ which can be identified with GL(1, [1/2]) × GL(1, [1/2]). The mod 2 cohomology of these groups is known. In the case of E_{2} this is trivial, because GL(1, [1/2]) is equal to the units in [1/2] which are isomorphic to /2 × . Hence H^{∗}(GL(1, [1/2]) × GL(1, [1/2])) is isomorphic to the tensor product of a polynomial algebra on two generators x, y of degree 1 with an exterior algebra on two classes f, g, also of degree 1, i.e.

H^{∗}C_{Γ}(E_{2}) ≅ _{2}[x, y] ⊗ E(f, g) .

We agree to chose the generators so that x and f come from the first factor GL(1, [1/2]) and y and g from the second factor. The case of E_{1} is less obvious but has been computed by Mitchell [M]. He proved that H^{∗}GL(2, [1/2]) is isomorphic to a tensor product _{2}[w_{1}, w_{2}] ⊗ E(e_{1}, e_{3}) where the indices give the degrees of the generators, i.e.

H^{∗}C_{Γ}(E_{1}) ≅ _{2}[w_{1}, w_{2}] ⊗ E(e_{1}, e_{3}) .

Furthermore Mitchell shows that H^{∗}GL(2, [1/2]) is detected on the subgroup of diagonal matrices in GL(2, [1/2]) which is, of course, again isomorphic to GL(1, [1/2]) × GL(1, [1/2]). The elements w_{i}, i = 1, 2 map to the i − th elementary symmetric polynomial in x and y, e_{1} maps to the symmetrization (with respect to the natural action of 𝔖_{2}) of f and e_{3} maps to the symmetrization of x^{2}g. Now the inclusion of E_{1} into E_{2} induces an inclusion of C_{Γ}(E_{2}) into C_{Γ}(E_{1}) which corresponds via our identifications of the centralizers to the inclusion of the subgroup of diagonal matrices into GL(2, [1/2]). If we denote the inclusion be α we can therefore write for the induced map on the cohomology of the centralizers:

α_{∗}(w_{1}) = x + y, α_{∗}(w_{2}) = xy
α_{∗}(e_{1}) = f + g, α_{∗}(e_{3}) = x^{2}g + y^{2}f .

To finish the description of H^{∗}C_{Γ}(−) as a functor on A it remains to describe the action of Aut_{A}(E_{2}) ≅ 𝔖_{3} on _{2}[x, y] ⊗ E(f, g), and because of the multiplicative structure we need it only on the generators. If τ corresponds to the transposition in 𝔖_{3} which permutes the first and second diagonal entry then

τ_{∗}(x) = y, τ_{∗}(y) = x
τ_{∗}(f) = g, τ_{∗}(g) = f ,

and if σ corresponds to the cyclic permutation of the diagonal entries (in the appropriate order) then

σ_{∗}(x) = y, σ_{∗}(y) = x + y
σ_{∗}(f) = g, σ_{∗}(g) = f + g .

In other words, if we identify 𝔖_{3} with GL(2, _{2}) then the representation on the subspace generated by x and y respectively f and g is the mod 2 reduction of the Steinberg representation of this group which we will denote by St_{_{2}} .

\underline{4.3.} Before we get to the proof of Theorem 0.5 we explain how to compute higher limits for any functor F from A to modules over _{(2)}, the integers localized at 2. Of course, this calculation can be done using the general theory developed by Oliver [O] . However, in the case of the category A everything is quite simple so we do this from scratch. Let St_{}be the Steinberg representation of GL(2, _{2}), defined over the integers. For our purposes the following description is appropriate: identifying as before GL(2, _{2}) with 𝔖_{3}, St_{}is the kernel of the GL(2, _{2}) - linear homomorphism from the permutation module [𝔖_{3}/𝔖_{2}] to the trivial module which sends each coset to the element 1 ∈ . We can choose an integral basis of St_{}consisting of elements a and b such that in the notation introduced above we have:

τ_{∗}(a) = b, τ_{∗}(b) = a
σ_{∗}(a) = −b, σ_{∗}(b) = a − b .

In particular, the mod 2 reduction of St_{}is the representation St_{_{2}} that we discussed above.

With this notation we can now describe how to compute the (higher) limits on A.

PROPOSITION 4.3. Let F be a functor F from A to _{(2)}− modules. Then

a) lim^{i}_{A}F = 0 if i ≥ 2

b) There is a short exact sequence

0 → lim^{0}_{A}F ⟶ F (E_{1}) ⟶^{φ} Hom_{GL(2, _{2})} (St_{}, F (E_{2})) ⟶ lim^{1}_{A}F → 0

where φ is the homomorphism given by φ(x)(a) = α_{∗}(x) − (σ_{∗})^{2}α_{∗}(x), φ(x)(b) = α_{∗}(x) − σ_{∗}α_{∗}(x) for x ∈ F (E_{1}).

\underline{Proof:} lim^{i}_{A}F can be computed as H^{i}(Hom_{F}(P_{•}, F)) where F is the abelian category of functors from A to the category of _{(2)}− modules and P_{•} is a projective resolution of the constant functor in this category. The proposition follows from the following explicit resolution of the constant functor which we will denote by \underline{}_{(2)}. Let P_{0} be the functor given by P_{0}(E) = _{(2)}[Hom_{A}(E_{1}, E)], the free _{(2)} - module with basis given by the set Hom_{A}(E_{1}, E). Then there are natural isomorphisms Hom_{F}(P_{0}, F) ≅ F (E_{1}) for each F ∈ F which shows that P_{0} is projective. The kernel of the natural epimorphism from P_{0} to \underline{}_{(2)} is the functor P_{1} with P_{1}(E_{1}) = 0 and, by our very definition of St_{}, P_{1}(E_{1}) = St_{}⊗ _{(2)}. For the functor P_{1} we get natural isomorphism Hom_{F}(P_{1}, F) ≅ Hom_{_{(2)}GL(2, _{2})} (St_{_{(2)}} , F (E_{1})) which show that P_{1} is also a projective functor because it is “induced” from the _{(2)}GL(2, _{2})− module St_{}⊗ _{(2)} which is well known to be a projective module. This proves (a) and (b) except for the explicit description of the map φ which follows from the inspection of the inclusion from P_{1} to P_{0}.

Now we consider the case of the functor F : E ↦ H^{∗}C_{Γ}(E). The main result in this section is the following theorem . Together with Theorem 0.4 and the well known fact that the virtual cohomological dimension of Γ is 5 it implies Theorem 0.5.

THEOREM 4.4 a) lim^{0}_{A}F ≅ _{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5}) where the indices give the degrees of the elements. Furthermore, if we identify lim^{0}_{A}F with a subalgebra of H^{∗}GL(2, [1/2]) as in Proposition 4.3 then v_{2} = w_{1}^{2} + w_{2}, v_{3} = w_{1}w_{2}, d_{3} = e_{3} and d_{5} = w_{1}^{2}e_{3} + w_{2}^{2}e_{1}.

b) lim^{1}_{A}F ≅ Σ^{3} _{2}, a one dimensional vector space over _{2} concentrated in degree 3.

\underline{Proof of a):} First of all it is straightforward to check that the subalgebra of H^{∗}GL(2, [1/2]) ≅ _{2}[w_{1}, w_{2}] ⊗ E(e_{1}, e_{3}) generated by the elements w_{1}^{2} + w_{2}, w_{1}w_{2}, e_{3} and w_{1}^{2}e_{3} + w_{2}^{2}e_{1} is isomorphic to the tensor product of a polynomial algebra on two generators v_{2} and v_{3} of degree 2 and 3 and an exterior algebra on two generators d_{3} and d_{5} of degree 3 and 5. From now on we identify v_{2}, v_{3}, d_{3} and d_{5} with w_{1}^{2} + w_{2}, w_{1}w_{2}, e_{3} and w_{1}^{2}e_{3} + w_{2}^{2}e_{1} respectively. Now we use Proposition 4.3. In our case α_{∗} is injective and identifying F (E_{1}) via α_{∗} with its image in F (E_{2}) we see that lim^{0}_{A}F agrees with the intersection of _{2}[w_{1}, w_{2}] ⊗ E(e_{1}, e_{3}) with the invariants in _{2}[x, y] ⊗ E(f, g) under the action of the order 3 cyclic subgroup of 𝔖_{3} generated by σ. We leave it to the reader to check that the elements w_{1}^{2} + w_{2} = x^{2} + xy + y^{2}, w_{1}w_{2} = xy(x+y), e_{3} = x^{2}g +y^{2}f and w_{1}^{2}e_{3} +w_{2}^{2}e_{1} = x^{4}g +y^{4}f all belong to this intersection. Now consider an arbitrary element p ∈ F (E_{1}) and write it as sum p = q + re_{1} + se_{3} + te_{1}e_{3} with unique elements q, r, s, t ∈ _{2}[w_{1}, w_{2}]. Then σ_{∗}p = p is equivalent to the three equations

(1)  σ_{∗}q = q ,

(2)  σ_{∗}(re_{1} + se_{3}) = re_{1} + se_{3}

and

(3)  σ_{∗}(te_{1}e_{3}) = te_{1}e_{3} .

It suffices to show q ∈ _{2}[v_{2}, v_{3}], re_{1} + se_{3} = r̃d_{3} + s̃d_{5} and te_{1}e_{3} = t̃d_{3}d_{5} with suitable elements r̃, s̃, t̃∈ _{2}[v_{2}, v_{3}]. As q is already invariant under τ the first equation is equivalent to saying that q considered as polynomial in x and y is invariant under 𝔖_{3}. The 𝔖_{3} – invariants in _{2}[x, y] are well known to be given by _{2}[v_{2}, v_{3}]. If we express (3) in terms of x, y, f, g we get (σ_{∗}t)x^{2}fg = t(x^{2} + y^{2})fg. Now fg is invariant under σ and hence t(x^{2} + y^{2}) is invariant as well. Furthermore we see that t(x^{2} + y^{2}) is divisible by x^{2} and, because of invariance, also by y^{2} and x^{2} + y^{2}. Using that _{2}[x, y] is a unique factorization domain we conclude t(x^{2} + y^{2}) = t̃v_{3}^{2} with t̃ invariant, i.e. t̃∈ _{2}[v_{2}, v_{3}]. Finally d_{3}d_{5} = v_{3}^{2}fg and we get t(x^{2} + y^{2})fg = t̃d_{3}d_{5}. Expressing (2) in terms of x, y, f, g and comparing coefficients of f and g shows that (2) is equivalent to the two equations

(4)  σ_{∗}r + (σ_{∗}s)y^{2} = r + sy^{2},

(5)  (σ_{∗}s)x^{2} = r + sx^{2} .

Using that _{2}[w_{1}, w_{2}] is a free module over _{2}[v_{2}, v_{3}] with basis w_{1}, w_{1}^{2} we write s = s_{0} + s_{1}w_{1} + s_{2}w_{1}^{2} with unique elements s_{0}, s_{1}, s_{2} ∈ _{2}[v_{2}, v_{3}]. Substituting into (5) and solving for r yields r = s_{1}x^{2}y+s_{2}x^{2}y^{2}. However, r is invariant under τ and therefore s_{1} = 0. This means r = s_{2}w_{2}^{2} and s = s_{0} +s_{2}w_{1}^{2}, and finally we get re_{1} +se_{3} = s_{2}w_{2}^{2}e_{1} +(s_{0} +s_{2}w_{1}^{2})e_{3} = r̃d_{3} +s̃d_{5} with r̃ = s_{0} and s̃ = s_{2}.

\underline{Proof of b):} This is just a computation with Poincare series in the exact sequence of Proposition 4.3. Clearly,

χ_{0} := Σ_{n=0}^{∞} dim_{_{2}} ( _{2}[v_{2}, v_{3}] ⊗ E(d_{3}, d_{5}))^{n}t^{n} = \frac{(1 + t^{3})(1 + t^{5})}{(1 − t^{2})(1 − t^{3})} ,

χ_{1} := Σ_{n=0}^{∞} dim_{_{2}} ( _{2}[w_{1}, w_{2}] ⊗ E(e_{1}, e_{3}))^{n}t^{n} = \frac{(1 + t)(1 + t^{3})}{(1 − t)(1 − t^{2})} .

We claim

χ_{2} := Σ_{n=0}^{∞} dim_{_{2}} (Hom_{GL(2, _{2})} (St_{}, _{2}[x, y] ⊗ E(f, g)))^{n}t^{n}

= \frac{(t + t^{2} + t^{3})}{(1 − t)(1 − t^{3})} + \frac{(t + t^{4})}{(1 − t^{2})(1 − t^{3})}

and then a simple computation shows

χ_{3} := Σ_{n=0}^{∞} dim_{_{2}} lim^{1}_{A}F^{n}t^{n} = χ_{0} + χ_{2} − χ_{1} = t^{3}

which proves (b). To prove the claim we observe that there is an isomorphism of graded GL(2, _{2}) – modules

2[x, y] ⊗ E(f, g)) ≅ _{2}[x, y] ⊕ (Σ _{2}[x, y] ⊗ St_{_{2}} ) ⊕ Σ^{2} _{2}[x, y] .

This yields a corresponding decomposition of Hom_{GL(2, _{2})} (St_{}, _{2}[x, y] ⊗ E(f, g)). The Poincare series corresponding to the first and third summand are given by \frac{t}{(1−t)(1−t^{3})} and \frac{t^{3}}{(1−t)(1−t^{3})} respectively (cf. [MP, sect. 4]). For the second summand we write

Hom_{GL(2, _{2})} (St_{}, Σ _{2}[x, y] ⊗ St_{_{2}} ) ≅ Hom_{GL(2, _{2})} (St_{_{2}} ⊗ St _{2}^{∗} , Σ _{2}[x, y])

where (−)^{∗} denotes the dual representation. We leave it to the reader to verify that St_{_{2}} ⊗ St _{2}^{∗} is isomorphic as GL(2, _{2}) – module to the direct sum of St_{_{2}} and _{2}[GL(2, _{2})/( /3)], the permutation module on the cosets of the cyclic subgroup of order 3. Hence the remaining summand is isomorphic to Hom_{GL(2, _{2})} (St_{}, Σ _{2}[x, y]) ⊕ Σ( _{2}[x, y]^{/3} ). The Poincare series of the first piece is \frac{t^{2}}{(1−t)(1−t^{3})}. Finally _{2}[x, y]^{/3} is a free module over _{2}[v_{2}, v_{3}] on two generators of degree 1 and 3 (cf. [MP, sect. 4]) and therefore the second piece has Poincare series \frac{(t+t^{4})}{(1−t^{2})(1−t^{3})} .

\underline{4.5.} Using the “normalizer spectral sequence” K. Brown [B] obtained the following exact sequence for the mod 2 Farrell cohomology Ĥ^{∗}(−) := Ĥ^{∗}(−; _{2}):

⋯ ⟶ Ĥ^{∗}(Γ) ⟶^{ι} Ĥ^{∗}N_{Γ}(E_{1}) ⊕ Ĥ^{∗}N_{Γ}(E_{2}) ⟶ Ĥ^{∗}(N_{Γ}(E_{1}) ∩ N_{Γ}(E_{2})) ⟶^{δ} Ĥ^{∗+1}(Γ) ⟶ ⋯

where the components of first two maps are given by the appropriate restriction homomorphisms. In principle it is possible to compute the cohomologies of N_{Γ}(E_{1}), N_{Γ}(E_{2}) and of N_{Γ}(E_{1}) ∩ N_{Γ}(E_{2}) and to compute Ĥ^{∗}(Γ) from this sequence. But even without carrying out this laborious explicit computations one can see that this exact sequence leads to the same answer in the range where Farrell cohomology and ordinary cohomology agree: In fact, a transfer argument shows that the restriction homomorphism H^{∗}N_{Γ}(E_{2}) ⟶ H^{∗}(N_{Γ}(E_{1}) ∩ N_{Γ}(E_{2})) is a split injection and it is not difficult to see that after splitting off this direct summand from ι one ends up with the map φ of Proposition 4.3 (again in the range where Farrell cohomology and ordinary cohomology agree). In particular, in this same range the E_{2} - term of the “normalizer spectral sequence” agrees with that of the “centralizer spectral sequence”. We leave the details to the interested reader.

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Hans–Werner Henn Mathematisches Institut der Universität Im Neuenheimer Feld 288 D–69120 Heidelberg Fed. Rep. of Germany