The nonabelian tensor square and Schur multiplicator of nilpotent groups of class 2

by Manfred Hartl

Institut de Recherche Mathématique Avancée, ULP et CNRS, 7 rue René-Descartes, 67084 Strasbourg Cedex, FRANCE

Abstract

The nonabelian tensor square and the Schur multiplicator are determined for arbitrary groups of class 2 in a closed form. A functorial description is given in terms of a polynomial quotient of the integral group ring, as well as a more explicit formula for finite groups which can be evaluated by matrix calculus. This is carried out for 2-generator groups. Also some homotopical data of the suspended classifying space of class 2 groups are derived. The approach is based on the use of polynomial maps and constructions in the sense of Passi.

1 Introduction

The nonabelian tensor square G ⊗ G of an arbitrary group G is a special case of a construction introduced by Lue in [21], and was also discussed by Dennis in [10]. It came into more general notice through the topological application of the more general construction G ⊗ H introduced by [8]. Nevertheless, a computation of the tensor square itself is desirable because of its close relation to certain homotopy groups as well as to group homology, in particular by the isomorphisms

π_{3}ΣK(G, 1) ≅ Ker(G ⊗ G → G) , [9]  (1)
H_{2}(G) ≅ Ker(G ∧ G → G) , [23], cf. also [11]  (2)

where ΣK(G, 1) denotes the suspension of the classifying space of G, and where the nonabelian exterior square G ∧ G is defined by reducing G ⊗ G modulo the subgroup generated by the diagonal, and both maps on the right-hand side are induced by the commutator pairing. Notably for nilpotent groups the Schur multiplicator H_{2}(G) is known to be a very rich invariant, encoding key information about extensions, presentations, homomorphisms and dimension subgroups; nevertheless, as a functor on nilpotent groups of a given class, H_{2}(G) is known in a closed form only in the case of abelian groups (by Miller’s theorem (2) above, since the nonabelian tensor square of abelian groups then coincides with the ordinary tensor square).

Research supported by successive research fellowships of the Sonderforschungsbereich 343, Bielefeld, and of the Human Capital and Mobility Programme of the European Union.

But also computation of G ⊗ G turns out to be difficult. It was calculated, for example, by computer aid for all groups up to order 48 (see [7]); moreover, it was determined for dihedral and quaternionic groups in [loc.cit], for metacyclic groups in [loc.cit] and [20], for many linear groups over finite fields in [13], and for free products of groups in [12].

For those nilpotent groups of class 2 which are not covered by the results cited above G⊗G is known only in some special cases: in the free case it was determined for two generators in [1] and subsequently in [3] in general, and for 2-generator p-groups for odd p in [4]. In this paper we determine the structure of the groups G ⊗ G, G ∧ G, H_{2}(G) and π_{3}ΣK(G, 1) for all groups of class 2: in a functorial way in terms of the ring P_{2}(G) = I(G)/I(G)^{3}, where I(G) = augmentation ideal of ZZ(G), and also by an explicit abelian presentation read off directly from a free presentation of G or from the extension γ_{2}(G) ↪ G → → G^{ab} (for finite groups G). Actually, the latter results are derived from the first, which underlines the meaning of the polynomial construction P_{n}(G) as a sensible way of abelianization, encoding also nonabelian information into abelian data from which certain invariants can be computed by a calculation involving only abelian groups.

For concrete finite p-groups of class 2 an explicit calculation of G ⊗ G, H_{2}(G) etc. as above is thus reduced to matrix calculus; this is used to complete the result of Bacon and Kappe for p = 2. Our result also yields a description of the third filtration quotient of the Schur multiplicator H_{2}(G) for all groups G; this result is used in [18] for a systematic construction of new counterexamples to the dimension subgroup conjecture, and a proof of completeness of the list is obtained under certain restrictions.

We point out that our description of H_{2}(G) generalizes nicely Miller’s old result for abelian groups: here H_{2}(G) equals G ∧G ≅ P_{1}(G) ∧P_{1}(G), the exterior square of Passi’s polynomial construction of degree 1 (which is just abelianization). Now for class 2 groups we identify H_{2}(G) as a natural subquotient of P_{2}(G) ∧ P_{2}(G), the exterior square of the polynomial construction of degree 2, and also the corresponding quotient has a nice structural interpretation. This raises the interesting question of whether analogous phenomena occur for nilpotent groups of arbitrary class.

In the following three sections we restrict ourselves to a description of the results obtained, while all proofs are postponed to the last section.

2 The nonabelian tensor square

Let G be a group. Recall from [9] that G ⊗ G is the group generated by the symbols a ⊗ b, a, b ∈ G, and defined by the relations

ab ⊗ c = (^{a}b ⊗^{a}c)(a ⊗ c)  (3)
a ⊗ bc = (a ⊗ b)(^{b}a ⊗^{b}c) ,  (4)

with ^{a}b = aba^{−1}. Recall from [9] the central homomorphism

ψ: Γ(G^{ab}) → G ⊗ G  (5)

where G^{ab} = G/G′, where Γ is Whitehead’s universal quadratic functor [26] and where ψγ(aG′) = a ⊗ a. One defines G ∧ G = coker(ψ) and writes a ∧ b for the coset of a ⊗ b.

Now let G be nilpotent of class 2 . Then the group G⊗G is abelian by the first of the following two relations in G ⊗ G which we recall from [7]: Let a, b, c, d ∈ G. Then defining

[a, b] =^{def} aba^{−1}b^{−1}

one has

(a ⊗ b)(c ⊗ d)(a ⊗ b)^{−1} = ^{[a,b]}c ⊗^{[a,b]} d  (6)
ab ⊗^{a}c = (a ⊗ [b, c])(b ⊗ c)  (7)

To compute G ⊗ G we consider a couple of other abelian groups and homomorphisms related to G.

First of all we need various constructions defined for abelian groups A (in particular, for A = G^{ab}). The symbol A^{⊗n} denotes the n-fold tensor power. The group L_{n}(A) is the subgroup of the free Lie ring over A which is generated by n-fold commutators. For example, L_{2}(A) = A^{∧ 2} is the quotient group of A^{⊗2} modulo the subgroup generated by the elements x ⊗ x, x ∈ A; we write x ∧ y for the coset of x ⊗ y. If G is a p-group, A = ∑ ZZ/p^{r(i)} · x_{i}, r(i) ≤ r(j) if i ≤ j, then we have

L_{3}(A) = ∑_{i≤j>k} ZZ/p^{r(min{i,k})} · [x_{i}, [x_{j}, x_{k}]] .

These constructions are interrelated by the following natural homomorphisms (with x, y, z ∈ A):

\begin{matrix}l_{2} : A^{∧ 2} → A^{⊗2} , & l_{2}(x ∧ y) = x ⊗ y − y ⊗ x \\ ν_{2} : A^{⊗2} → →A^{∧ 2} , & ν_{2}(x ⊗ y) = x ∧ y \\ ν_{3} : A^{⊗3} → →L_{3}(A) , & ν_{3}(x ⊗ y ⊗ z) = [x, [y, z]] \\ [ , ] : A ⊗ (A^{∧ 2})→ →L_{3}(A) , & [x, y ∧ z] = [x, [y, z]] \\ T_{12} : A^{⊗3} → A^{⊗3} , & T_{12}(x ⊗ y ⊗ z) = y ⊗ x ⊗ z . \end{matrix}

Now the nonabelian structure of G is reflected in the following two abelian constructions. The commutator map of G gives rise to the homomorphism

c_{2} : (G^{ab})^{∧ 2} → →G′ = [G, G] , c_{2}(x ∧ y) = [x′, y′]

where for x ∈ G^{ab} the symbol x′ denotes some arbitrarily chosen element in G such that x′G′ = x. This convention will be used throughout in this paper.

Furthermore, we consider Passi’s polynomial construction

P_{n}(G) = I(G)/I(G)^{n+1}

where I(G) denotes the augmentation ideal of the group ring ZZ(G). As proved in [2] (cf. also [24, Theorem 8.7], or [17] for a very simple proof) there is a short exact sequence

(G^{ab})^{⊗2}/l_{2}Ker(c_{2}) ⟶^{>}^{μ} P_{2}(G) ⟶^{ρ} G^{ab}  (8)

of abelian groups; here μ{x⊗y} = {(x′ −1)(y′ −1)}, ρ{a−1} = aG′, where {−} denotes the corresponding coset. Applying the functors (− ⊗ G^{ab}) and (G^{ab} ⊗ −) to sequence (8) one obtains connecting homomorphisms

(G^{ab})^{⊗3}/l_{2}Ker(c_{2}) ⊗ G^{ab} ⟵^{τ_{1}} G^{ab} ∗ G^{ab} ⟶^{τ_{2}} (G^{ab})^{⊗3}/G^{ab} ⊗ l_{2}Ker(c_{2})

using the notation G^{ab} ∗ G^{ab} =^{def} Tor_{1}^{Z Z}(G^{ab}, G^{ab}) of the book of Mac Lane [22].

It is natural to compare the nonabelian tensor square G ⊗ G with the ordinary tensor square which is (G^{ab})^{⊗2}, via the canonical quotient map

π : G ⊗ G→ →(G^{ab})^{⊗2} , π(a ⊗ b) = (aG′) ⊗ (bG′) .

Theorem 2.1 Let G be a group of class 2. Then there is a natural exact sequence of abelian groups

G^{ab} ∗ G^{ab} ⟶^{ν_{3}(τ_{1}−T_{12}τ_{2})} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} ⟶^{i} G ⊗ G ⟶^{π} (G^{ab})^{⊗2} → 1

with i{[x, [y, z]]} = x′ ⊗ [y′, z′] . More explicitely, the kernel of i is generated by the elements

[x_{1}, f_{k}x_{2}] + [x_{2}, f_{k}x_{1}] + \binom{k}{2}[x_{1} + x_{2}, [x_{1}, x_{2}]] ,

where k ∈ ZZ, x_{1}, x_{2} ∈ G^{ab} such that x_{1}^{k} = 1G′ = x_{2}^{k} and where f_{k}x_{i} ∈ (G^{ab})^{∧ 2} such that c_{2}f_{k}x_{i} = (x′_{i})^{k}. □

The corresponding extension problem is solved in the complete descriptions of G ⊗ G in 2.2, 2.3 and 2.4 below.

Note that the maps ν_{3}τ_{1} and ν_{3}T_{12}τ_{2} : G^{ab} ∗ G^{ab} → \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} are well

defined since

ν_{3}(l_{2}Ker(c_{2}) ⊗ G^{ab}) = [G^{ab}, Ker(c_{2})] = ν_{3}T_{12}(G^{ab} ⊗ l_{2}Ker(c_{2})) ,  (9)

by the relations

ν_{3}(l_{2} ⊗ G^{ab})(x ∧ y ⊗ z) = −[z, [x, y]] ,  (10)
ν_{3}T_{12}(G^{ab} ⊗ l_{2}) = [ , ]  (11)

which are immediate consequences of the Jacobi identity.

The following result of Bacon (previously obtained by Aboughazi in the case of two generators) is an immediate consequence of the theorem, since here Ker(c_{2}) = 0 = G^{ab} ∗ G^{ab}.

Example [3], [1]: Let G be the free nilpotent group of class 2 with basis {x_{i}} where i runs through a wellordered index set I. Then G⊗G ≅ L_{3}(G^{ab})⊕G^{ab}⊗G^{ab} is the free abelian group with basis {x_{i} ⊗ x_{j} | i, j ∈ I} ∪ {x_{i} ⊗ [x_{j}, x_{k}] | i ≤ j > k}.

In order to determine G ⊗ G completely and functorially we shall prove that a natural group epimorphism

ζ : P_{2}(G)^{⊗2} → →G ⊗ G  (12)

is well defined by sending {a−1}⊗{b−1} to a⊗b, a, b ∈ G. Using this we obtain the following description of G ⊗ G where we use the abbreviation X^{⊕2} = X ⊕ X.

Theorem 2.2 Let G be a group of class 2. Then there is a natural exact sequence of abelian groups

G^{ab} ∗ G^{ab} ⊕ ((G^{ab})^{⊗2} ⊗ P_{2}(G))^{⊕2} ⟶^{κ} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} ⊕ P_{2}(G)^{⊗2} ⟶^{λ} G ⊗ G → → 1

with λ = (i, ζ) and

κ = ((1, 0)^{t}ν_{3}(τ_{1} − T_{12}τ_{2}) , κ_{2}, κ_{3}) where
κ_{2}(x ⊗ y ⊗ {a − 1}) = ( {[x, [y, aG′]]} , − {(x′ − 1)(y′ − 1)} ⊗ {a − 1})
κ_{3}(u ⊗ w ⊗ {b − 1}) = ( {[u, [bG′, w]]} , − {b − 1} ⊗ {(u′ − 1)(w′ − 1)})

for x, y, u, w ∈ G^{ab}, a, b ∈ G. □

As an application, G ⊗ G can be calculated directly from a free presentation of G by a formula involving only abelian groups, as follows.

Let G be a group defined by a free presentation G = ⟨X|R⟩, where X = {x_{i}} is some set and R = {r_{j}} is some subset of the free group F generated by X. Write each r_{j} as a word

r_{j} = x_{j1}^{ϵ(j1)} ⋯ x_{js_{j}}^{ϵ(js_{j})}

in F where x_{jt} ∈ X, ϵ(jt) = ±1 for 1 ≤ t ≤ s_{j}. Let D, S be the free abelian groups with basis X, R, respectively, and define homomorphisms

A^{⊗2} ⟵^{e_{2}} S ⟶^{e_{1}} D ⟶^{q} A =^{def} coker(e_{1}) ,

by

e_{1}(r_{j}) = ∑_{1≤t≤s_{j}} ϵ(jt)x̄_{jt} with x̄ = qx ,

e_{2}(r_{j}) = ∑_{\begin{matrix}1≤t≤s_{j} \\ ϵ(jt)<0 \end{matrix}} x̄_{jt} ⊗ x̄_{jt} + ∑_{1≤p<q≤s_{j}} ϵ(jp)ϵ(jq)x̄_{jp} ⊗ x̄_{jq} .

If G is nilpotent of class 2 then G ⊗ G and H_{2}(G) can be computed in terms of the ‘abelian data’ e_{1}, e_{2}, as follows. Consider the following diagram of abelian groups.

(A^{⊗2}/e_{2}Ker(e_{1})) ⊗ A ⟵^{d} D ⊗ S ⟶^{q⊗e_{2}} A ⊗ A^{⊗2}
↑^{ē_{2}⊗q}  ↓^{D⊗e_{1}}
S ⊗ D  ⟶^{e_{1}⊗D} D ⊗ D ⟶^{q⊗D} A ⊗ D

Here ē_{2} denotes e_{2} followed by the canonical quotient map. By exactness of the bottom row a homomorphism

d: Ker(q ⊗ e_{1}) → (A^{⊗2}/e_{2}Ker(e_{1})) ⊗ A

is well defined by

d = (ē_{2} ⊗ q)(e_{1} ⊗ D)^{−1}(D ⊗ e_{1}) .

Thus we obtain a homomorphism

σ = ν_{3}(d − T_{12}(q ⊗ e_{2})) : Ker(q ⊗ e_{1}) → \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A)} .

Let T : B ⊗ C → C ⊗ B denote the interchange map, for any abelian groups B, C.

Corollary 2.3 Let G = ⟨X|R⟩ be a group of class 2. Then the following sequence is exact:

Ker(q ⊗ e_{1}) ⟶^{σ} \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A)} ⟶^{i} G ⊗ G ⟶^{π} A^{⊗2} → 1 .

Moreover, G ⊗ G can be explicitly computed by the following exact sequence

Ker(q ⊗ e_{1}) ⊕ (S ⊗ D)^{⊕2} ⟶^{e} \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A)} ⊕ D^{⊗2} ⟶^{(i′,ζ′)} G ⊗ G → 1 (13)

where

e = ( \begin{matrix}σ & ν_{3}(e_{2} ⊗ q) & 0 \\ 0 & e_{1} ⊗ D & (1 + T)(e_{1} ⊗ D) \end{matrix} )

and ζ′(x ⊗ y) = {x} ⊗ {y} for x, y ∈ X. □

For finite class 2 groups G the explicit calculation of G ⊗ G is now a matter of matrix calculus. In fact, one uses the formula

(G × H) ⊗ (G × H) ≅ G ⊗ G × H ⊗ H × G^{ab} ⊗ H^{ab} × H^{ab} ⊗ G^{ab}

in [7] to reduce to p-groups; here applying 2.3 to a standard presentation gives the following result.

Corollary 2.4 Let G be a finite p-group of class 2, G^{ab} = ∑_{i=1}^{d} ZZ/p^{r(i)} · x̄_{i}, r(i) ≤ r(j) if i ≤ j. Let D = ∑_{i=1}^{d} ZZ · x_{i}. For all i choose x′_{i} ∈ G such that x′_{i}G′ = x̄_{i} and fx_{i} ∈ (G^{ab})^{∧ 2} with c_{2}fx_{i} = (x′_{i})^{p^{r(i)}} . Let U be the subgroup of L_{3}(G^{ab}) generated by the elements

p^{r(j)−r(i)}[x̄_{j}, fx_{i}] + [x̄_{i}, fx_{j}] + \binom{p^{r(j)}}{2}[x̄_{i} + x̄_{j}, [x̄_{i}, x̄_{j}]] ,

i ≤ j. Then the following sequence of abelian groups is exact:

D^{⊗2} ⟶^{κ′} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + U} ⊕ D^{⊗2} ⟶^{λ′} G ⊗ G → 1 ,

where

κ′(x_{i} ⊗ x_{j}) = \{ \begin{matrix}( [x̄_{j}, fx_{i}] + \binom{p^{r(i)}}{2}[x̄_{i}, [x̄_{i}, x̄_{j}]] , p^{r(i)}x_{i} ⊗ x_{j}) & if i ≤ j \\ (0 , p^{r(i)}(x_{i} ⊗ x_{j} + x_{j} ⊗ x_{i}) & else \end{matrix} ,

λ′(k{[x, [y, z]]} , l(x_{i} ⊗ x_{j})) = (x′ ⊗ [y′, z′])^{k}(x′_{i} ⊗ x′_{j})^{l}

for k, l ∈ ZZ, x, y, z ∈ G^{ab}. □

Example: As an immediate application one computes the tensor square of twogenerator 2-groups of class 2; the corresponding case of odd order is treated in [4] and can be easily rederived from corollary 2.4.

Let G be a group of class 2 with G^{ab} = ZZ/2^{a} · x_{1} ⊕ ZZ/2^{b} · x_{2}, a ≤ b, Ker(c_{2}) = 2^{c}[x_{1}, x_{2}], 0 ≤ c ≤ a, and (x_{1})^{′a} = 2^{r}u_{1}[x_{1}, x_{2}], (x_{2})^{′b} = 2^{s}u_{2}[x_{1}, x_{2}] with 0 ≤ r, s ≤ c and u_{1}, u_{2} being odd integers. Depending on the parameters, a number of cases has to be distinguished. We omit the special cases a = b = c = s and s = b − 1 which split themselves into numerous subcases. Otherwise one gets

r < s < min{b − 1, r + b − a} ⟹

G ⊗ G = ZZ/2^{a+1} · ((x′_{1} ⊗ x′_{1})) ⊕ ZZ/2^{r} · (u_{1}(x′_{1} ⊗ [x′_{1}, x′_{2}]) + 2^{a−r}(x′_{1} ⊗ x′_{1})) ⊕
ZZ/2^{b+1} · ((x′_{2} ⊗ x′_{2})) ⊕ ZZ/2^{a} · ((x′_{2} ⊗ x′_{1}) + (x′_{1} ⊗ x′_{2})) ⊕
ZZ/2^{a+s−r} · (u_{2} (x′_{1} ⊗ x′_{2}) + 2^{b−a+r−s}u_{1}(x′_{2} ⊗ x′_{2})) ⊕
ZZ/2^{r} · (u_{1}(x′_{2} ⊗ [x′_{1}, x′_{2}]) + 2^{a−r}(x′_{1} ⊗ x′_{2}))

max{r, r + b − a − 1} < s < b − 1 ⟹

G ⊗ G = ZZ/2^{a+1} · ((x′_{1} ⊗ x′_{1})) ⊕ ZZ/2^{r} · (u_{1}(x′_{1} ⊗ [x′_{1}, x′_{2}]) + 2^{a−r}(x′_{1} ⊗ x′_{1})) ⊕
ZZ/2^{b} · ((x′_{2} ⊗ x′_{2})) ⊕ ZZ/2^{b} · ((x′_{1} ⊗ x′_{2})) ⊕
ZZ/2^{a} · ((x′_{2} ⊗ x′_{1}) + (x′_{1} ⊗ x′_{2})) ⊕
ZZ/2^{r} · (u_{1}(x′_{2} ⊗ [x′_{1}, x′_{2}]) + 2^{a−r}(x′_{1} ⊗ x′_{2}))

r ≥ s < b − 1 ⟹ (with k = 2^{b−s−1} + w_{2} and l = 2^{b−a+r−s}u_{1} + 2^{b−s−1})

G ⊗ G = ZZ/2^{b+1} · ((x′_{2} ⊗ x′_{2})) ⊕ ZZ/2^{a} · (ku_{2} (x′_{1} ⊗ x′_{1}) − 4^{b−a+r−s}u_{1}^{2} (x′_{2} ⊗ x′_{2})) ⊕
ZZ/2^{a} · (u_{2}((x′_{1} ⊗ x′_{2})) + 2^{b−a+r−s}u_{1}((x′_{2} ⊗ x′_{2}))) ⊕
ZZ/2^{a} · ((x′_{2} ⊗ x′_{1}) + (x′_{1} ⊗ x′_{2})) ⊕
ZZ/2^{s} · (k(x′_{1} ⊗ [x′_{1}, x′_{2}]) + l(x′_{2} ⊗ [x′_{1}, x′_{2}])) ⊕
ZZ/2^{s} · (u_{2}(x′_{2} ⊗ [x′_{1}, x′_{2}]) + 2^{b−s}((x′_{2} ⊗ x′_{2}))) .
□

3 Applications to the low dimensional homotopy of a suspended classifying space

Theorem 2.2 allows us to compute some homotopical data of the classifying space BG = K(G, 1) of groups of class 2. By the natural isomorphism

π_{2}(ΩΣBG, BG) ≅ G ⊗ G

obtained in [9] (see also [5] for another approach) the former homotopy group is determined by theorem 2.2. To describe the group π_{3}(ΣBG) together with some related homotopy operations recall the natural isomorphism

π_{3}(ΣBG) ≅ Ker([ , ] : G ⊗ G → G)  (14)

obtained in [9], and call its inverse Φ. Note that the following diagram commutes:

\begin{matrix}P_{2}(G)^{⊗2} & ⟶^{ρ^{⊗2}} & (G^{ab})^{⊗2} & ⟶^{ν_{2}} & (G^{ab})^{∧ 2} & ⟶^{c_{2}} & G′ \\ ↓^{ζ} & & & & ↓^{ι} \\ G ⊗ G & & & ^{[ , ]  \overline{}} & G \end{matrix}  (15)

Corollary 3.1 Let G be a group of class 2. Then there are natural exact sequences of abelian groups

G^{ab} ∗ G^{ab} ⟶^{ν_{3}(τ_{1}−T_{12}τ_{2})} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} ⟶^{Φi} π_{3}(ΣBG) ⟶^{πΦ^{−1}} Ker(c_{2}ν_{2}) → 0 ,

G^{ab}∗G^{ab}⊕((G^{ab})^{⊗2}⊗P_{2}(G))^{⊕2} ⟶^{κ} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]}⊕Ker(c_{2}ν_{2}ρ^{⊗2}) ⟶^{Φλ}π_{3}(ΣBG) → 0.

Let η: S^{3} → S^{2} be the Hopf map. Then the homotopy operations η^{∗} and Whitehead product [ , ],

π_{2}(ΣBG) ⟶^{η^{∗}} π_{3}(ΣBG) ⟵^{[ , ]} π_{2}(ΣBG) ⊗ π_{2}(ΣBG) ,

are given as follows: Using the canonical isomorphisms π_{2}(ΣBG) ≅^{h} H_{2}(ΣBG) ≅^{Σ^{−1}} H_{1}(BG) ≅ G^{ab} given by the Hurewicz and suspension isomorphism one has

η^{∗}(h^{−1}Σ(aG′)) = Φζ{a − 1} ⊗ {a − 1} ,
[h^{−1}Σ(aG′), h^{−1}Σ(bG′)] = {a − 1} ⊗ {b − 1} + {b − 1} ⊗ {a − 1} .

□

The assertion on π_{3}(ΣBG) follows from the foregoing theorems by using (14) and (15); the statements on η^{∗} and [ , ] are then obtained by combining 2.4 and 3.4 in [6].

4 The Schur multiplicator of groups of class 2

Now we describe analogous results for G ∧ G and H_{2}(G). We shall prove that a natural homomorphism

ζ̄ : P_{2}(G)^{∧ 2} → →G ∧ G  (16)

is well defined by sending {a − 1} ∧ {b − 1} to a ∧ b, a, b ∈ G.

Let the subgroup V ⊂ \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} be generated by the elements [x, c_{2}^{−1}((x′)^{o(x)})]

for x ∈ A of even finite order o(x). Write q_{1} : \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} → → \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + V}

and q_{2} : P_{2}(G)^{⊗2} → →P_{2}(G)^{∧ 2} for the canonical quotient maps.

Theorem 4.1 Let G be a group of class 2. Then there is a natural exact sequence of abelian groups

G^{ab} ∗ G^{ab} ⟶^{q_{1}ν_{3}(τ_{1}−T_{12}τ_{2})} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + V} ⟶^{ī} G ∧ G ⟶^{π̄} (G^{ab})^{∧ 2} → 1 ,

with ī{[x, [y, z]]} = x′ ∧ [y′, z′] and π̄(a ∧ b) = (aG′) ∧ (bG′).

Moreover, G ∧ G is completely and functorially determined by the following natural exact sequence of abelian groups:

G^{ab} ∗ G^{ab} ⊕ ((G^{ab})^{⊗2} ⊗ P_{2}(G))^{⊕2} ⟶^{κ̄} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} ⊕ P_{2}(G)^{∧ 2} ⟶^{λ̄} G ∧ G → → 1 ,

with λ̄ = (ī, ζ̄) and κ̄ = (id ⊕ q_{2})κ, where κ is defined in 2.2.

If G = ⟨X|R⟩ is a free presentation of G then G ∧ G can be computed by the following exact sequence, where we adopt the notation of 2.3.

Ker(q ⊗ e_{1}) ⊕ (S ⊗ D)^{⊕2} ⟶^{(id⊕ν_{2})e} \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A)} ⊕ D^{∧ 2} ⟶^{(i′,ζ′)} G ∧ G → 1 .

□

We remark that the term V can be put into the denominator occuring in the second sequence but is not necessary there.

To compute the Schur multiplicator first recall the formula of Hopf, i.e.

H_{2}(G) = \frac{R ∩ γ_{2}(F)}{[R, F]}  (17)

for any presentation R ↪ F ⟶^{β} G, R = Ker(β), and where F = γ_{1}(F) ⊃ γ_{2}(F) ⊃ … denotes the lower central series. If G is nilpotent of class 2 then natural homomorphisms

L_{3}(G^{ab}) ⟶^{d_{3}} H_{2}(G) ⟶^{j} G ∧ G

are induced by the maps

L_{3}(G^{ab}) \overleftarrow{⟵}^{L_{3}(β^{ab})} L_{3}(F^{ab}) ≅ \frac{γ_{3}(F)}{γ_{4}(F)} ↪ \frac{R ∩ γ_{2}(F)}{γ_{4}(F)} → → \frac{R ∩ γ_{2}(F)}{[R, F]} ≅ H_{2}(G) ,

H_{2}(G) \overleftarrow{⟵} R ∩ γ_{2}(F) ↪ γ_{2}(F) ≅^{(⋆)} F ∧ F ⟶^{β ∧ β} G ∧ G ,

respectively, and where relation (⋆) is proved in [11].

Now the following result is easily derived from (2) and theorem 4.1 by using an obvious modification of (15).

Corollary 4.2 Let G be a group of class 2. Then there is a natural exact sequence of abelian groups

G^{ab} ∗ G^{ab} ⟶^{q_{1}ν_{3}(τ_{1}−T_{12}τ_{2})} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + V} ⟶^{d̄_{3}} H_{2}(G) ⟶^{π̄j} Ker(c_{2}) → 0

where d̄_{3} is induced by d_{3}.

Moreover, H_{2}(G) is completely and functorially determined by the following natural exact sequence of abelian groups.

G^{ab} ∗ G^{ab} ⊕ ((G^{ab})^{⊗2} ⊗ P_{2}(G))^{⊕2} ⟶^{κ̄} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})]} ⊕ Ker(c_{2}ρ^{∧ 2}) ⟶^{λ′} H_{2}(G) → 0

with λ′ = (d̄_{3}, j^{−1}ζ̄| Ker(c_{2}ρ^{∧ 2})). □

From this result we derive a formula for H_{2}(G) in terms of a free presentation of G which, unlike the formula of Hopf, involves only abelian groups.

Let G = ⟨X|R⟩ as considered before 2.3. Adopting the notation there, let S^{2}(D) =^{def} D^{⊗2}/l_{2}(D^{∧ 2}), and consider the following diagram.

Ker(q^{⊗2}σ_{2})  (S ⊗ D)^{⊕2} ⟶^{(e_{2}⊗q,−e_{2}⊗q)} A^{⊗2} ⊗ A
↓^{ι}  ↓^{(e_{1}⊗D, T (e_{1}⊗D))}
S^{2}(D)  ⟶^{σ_{2}}  D^{⊗2}  ⟶^{q^{⊗2}}  A^{⊗2}

Here ι denotes the inclusion and σ_{2}({x}⊗{y}) = x⊗y +y ⊗x. Since Ker(q^{⊗2}) = Im((e_{1} ⊗ D, T (e_{1} ⊗ D))) , the additive relation

δ = ν_{3}(e_{2} ⊗ q, −e_{2} ⊗ q)(e_{1} ⊗ D, T (e_{1} ⊗ D))^{−1}σ_{2}ι:
Ker(q^{⊗2}σ_{2}) ⟶ \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A) + Im(σ)}

is well defined. Actually, δ is a homomorphism, as is easily derived from 2.3. Moreover, we need the composite map

l: D^{∧ 2} ⟶^{l_{2}} D^{⊗2} ⟶^{q^{⊗2}} A^{⊗2}/e_{2}Ker(e_{1}) .

Corollary 4.3 Let G = ⟨X|R⟩ be a nilpotent group of class 2. Then the following sequences are exact:

Ker(q^{⊗2}σ_{2}) ⟶^{δ} \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A) + Im(σ)} ⟶^{d̄_{3}} H_{2}(G) ⟶^{π̄j} Ker(c_{2}) → 0

S ⊗ D ⟶^{(ν_{3}(e_{2}⊗q),ν_{2}(e_{1}⊗D))} \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A)} ⊕ Ker(l) ⟶^{λ̄′} H_{2}(G) → 0

Here λ̄′ = (d̄_{3}, j^{−1}ζ̄′|Ker(l)), andζ̄′ sends x ∧ y to {x} ∧ {y} for x, y ∈ X.

For finite groups we obtain the following formula.

Corollary 4.4 Let G be a finite p-group of class 2. Adopting the notation of 2.4 let the subgroup W ⊂ L_{3}(G^{ab}) be generated by the elements [x̄_{i}, fx_{i}], 1 ≤ i ≤ d. Then the sequence

0 → \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + U + W} ⟶^{d̄_{3}} H_{2}(G) ⟶^{π̄j} Ker(c_{2}) → 0

is exact, and H_{2}(G) is isomorphic to the homology of the sequence

D^{∧ 2} ⟶^{κ′′} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + U + W} ⊕ D^{∧ 2} ⟶^{(0,c_{2}q^{∧ 2})} G′ ,

where κ′′(x_{i} ∧ x_{j}) = ( [x̄_{j}, fx_{i}] + \binom{p^{r(i)}}{2}[x̄_{i}, [x̄_{i}, x̄_{j}]] , p^{r(i)}x_{i} ∧ x_{j}). □

Remarks. The method in [19] principally allows to calculate the whole integral cohomology ring of groups of class 2, in particular, for finite groups G, H^{3}(G, ZZ) ≅ Ext_{Z Z}^{1}(H_{2}(G), ZZ) ≅ H_{2}(G). It seems, however, not to produce a closed formula depending directly on data which are easily accessible from the group structure of G, as is obtained in 4.3 and 4.4 above.

Moreover, the results above allow to controll also central extensions of groups of class 2, via the universal coefficient sequence

0 → Ext_{Z Z}^{1}(G^{ab}, B) → H^{2}(G, B) → Hom_{Z Z}(H_{2}(G), B) → 0

for any abelian group B. In particular, groups of class 3 with certain interesting properties may be constructed in this way. As an example, an enormous variety of new counterexamples to the dimension subgroup conjecture is exhibited in this way [18], and also completeness of the list obtained under certain restrictions can be shown.

We point out that the key term ν_{3}(τ_{1} − T_{12}τ_{2}) arising throughout in this paper essentially measures the difference between ‘left’ and ‘right’ connecting homomorphism, a fundamental phenomenon which seems not to have appeared so far and which also turns out to explain nicely the ‘irregular’ torsion phenomenon in the third filtration quotient of the integral group ring, see [17].

Note that by the results above, the groups G ⊗ G, G ∧ G, π_{3}ΣK(G, 1) and H_{2}(G) depend only on the truncated group ring ZZ(G)/I(G)^{3} for groups of class 2 (which actually determines G under rather mild assumptions, in particular if G is finitely generated, see [25] and [14]).

Finally we give an application of the first sequence in 4.2 to the natural filtration Γ_{n}H_{2}(G) of H_{2}(G) for arbitrary groups G. This is defined via the formula of Hopf (17) by

Γ_{n}H_{2}(G) = Im (R ∩ γ_{n}(F) ↪ R ∩ γ_{2}(F) → → H_{2}(G)) ⊂ H_{2}(G)

for n ≥ 2.

We obtain the following functorial description of the third filtration quotient Q_{3}(G) = Γ_{3}H_{2}(G)/Γ_{4}H_{2}(G).

Corollary 4.5 For all groups G there is a natural isomorphism

Q_{3}H_{2}(G) ≅ \frac{Ker( c_{3} : L_{3}(G^{ab})→ →γ_{3}(G)/γ_{4}(G) )}{[G^{ab}, Ker(c_{2})] + Im(ν_{3}(τ_{1} − T_{12}τ_{2})) + V}

where c_{3}[x, [y, z]] = [x′, [y′, z′]]. □

For finitely generated groups G this result was obtained in [16].

5 Proofs

First of all, we need Passi’s notion of polynomial maps on groups.

Recall [24] that a function f : G → B from a group G to some abelian group B is called polynomial of degree ≤ n if its linear extension to ZZ(G) annihilates I(G)^{n+1}. Clearly, the map

p_{n}: G → P_{n}(G) = I(G)/I(G)^{n+1} , p_{n}(a) = a − 1 + I(G)^{n+1} ,

is universal among polynomial maps of degree ≤ n from G to abelian groups which annihilate the unit e ∈ G.

We need the following inductive characterization of polynomial maps [17].

Lemma: A function f : G → B as above with f(e) = 0 is polynomial of degree ≤ n if and only if the deviation map

w_{f} : G × G → B , w_{f}(a, b) = f(ab) − f(a) − f(b)

is polynomial of degree ≤ n − 1 in the first (or, equivalently, the second) variable for all b ∈ G (a ∈ G). □

Proof of theorem 2.1: Consider the canonical map

G × G → G ⊗ G , (a, b) ↦ a ⊗ b .

By (3), (4) and (7) its deviation from linearity is given by the terms

w_{(−⊗c)}(a, b) = (ab ⊗ c)(a ⊗ c)^{−1}(b ⊗ c)^{−1}
= a ⊗ [b, c]  (18)

w_{(c⊗−)}(a, b) = (c ⊗ ab)(c ⊗ a)^{−1}(c ⊗ b)^{−1}
= a ⊗ [c, b]  (19)

Both terms are linear in b since [b, c] is, and by relation (4). So by the lemma above the maps (− ⊗ c), (c ⊗ −) : G → G ⊗ G are polynomial of degree ≤ 2. Thus the surjective homomorphism

ζ : P_{2}(G) ⊗ P_{2}(G)→ →G ⊗ G

in (12) is well defined by linear extension of the map p_{2}(a) ⊗ p_{2}(b) ↦ a ⊗ b. It satisfies the following equations:

ζ((p_{2}(a)p_{2}(b)) ⊗ p_{2}(c)) = ζ((p_{2}(ab) − p_{2}(a) − p_{2}(b)) ⊗ p_{2}(c))
= w_{(−⊗c)}(a, b) = ζ(p_{2}(a) ⊗ p_{2}[b, c])
= ζ(p_{2}(a) ⊗ [p_{2}(b), p_{2}(c)])  (20)
ζ(p_{2}(c) ⊗ (p_{2}(a)p_{2}(b))) = ζ(p_{2}(a) ⊗ [p_{2}(c), p_{2}(b)])  (21)

with [p_{2}(b), p_{2}(c)] = p_{2}(b)p_{2}(c) − p_{2}(c)p_{2}(b).

By linearity in c of the right hand term in (21) one has

ζ (P_{2}(G)^{2} ⊗ P_{2}(G)^{2}) = 0 .  (22)

From now on, we shall abbreviate A =^{def} G^{ab}. Homomorphisms

(A^{⊗2}) ⊗ A ⟶^{j_{1}} \frac{P_{2}(G) ⊗ P_{2}(G)}{P_{2}(G)^{2} ⊗ P_{2}(G)^{2}} ⟵^{j_{2}} A ⊗ (A^{⊗2})

are well defined by j_{1}(x_{1}⊗x_{2}⊗x_{3}) = (p_{2}(x′_{1})p_{2}(x′_{2}))⊗p_{2}(x′_{3}) and j_{2}((x_{1}⊗x_{2}⊗x_{3}) = p_{2}(x′_{1}) ⊗ (p_{2}(x′_{2})p_{2}(x′_{3})) . Consider the sequence of homomorphisms

(A^{⊗2}) ⊗ A ⊕ A ⊗ (A^{⊗2}) ⟶^{(R_{1},R_{2})} \frac{P_{2}(G) ⊗ P_{2}(G)}{P_{2}(G)^{2} ⊗ P_{2}(G)^{2}} ⟶^{ζ̄} G ⊗ G  (23)

where ζ̄ is the factorization of ζ according to (22) and where

R_{1}(x ⊗ y ⊗ z) = j_{1}(x ⊗ y ⊗ z) − j_{2}(x ⊗ [y, z])  (24)
R_{2}(x ⊗ y ⊗ z) = j_{2}(x ⊗ y ⊗ z − y ⊗ [x, z])  (25)

Sequence (23) is exact since annihilating Im(R_{1}) + Im(R_{2}) makes the relations (20), (21) hold which themselves just correspond to the distributive laws (3), (4) defining G ⊗ G.

We obtain a commutative diagram, labelled (26),

A ⊗ (A^{⊗2}/l_{2}Ker(c_{2})) ⟶^{j′_{2}}  \frac{P_{2}(G) ⊗ P_{2}(G)}{P_{2}(G)^{2} ⊗ P_{2}(G)^{2}} /(Im R_{1} + Im R_{2}) ⟶^{ρ^{⊗2}} A^{⊗2}
↓^{ζ′}  ζ_{o}↓≅  ∥
Ker π  ↪  G ⊗ G  ⟶^{π} A^{⊗2}

with j′_{2}, ζ_{o} induced by j_{2}, ζ̄, respectively.

The top row is exact by right exactness of the ordinary tensor product and by definition of R_{1}. So ζ′ is onto, and in order to determine Ker(π) we have to calculate Ker(j′_{2}).

With τ denoting the appropriate connecting homomorphism, one obtains from (8) an exact sequence

P_{2}(G) ∗ A ⟶^{τ} P_{2}(G) ⊗ (A^{⊗2}/l_{2}Ker(c_{2})) ⟶^{id⊗μ} P_{2}(G) ⊗ P_{2}(G) ⟶^{id⊗ρ} P_{2}(G) ⊗ A . (27)

This induces another exact sequence

P_{2}(G) ∗ A ⟶^{(ρ⊗id)τ} A ⊗ (A^{⊗2}/l_{2}Ker(c_{2})) ⟶^{j′_{2}} \frac{P_{2}(G) ⊗ P_{2}(G)}{P_{2}(G)^{2} ⊗ P_{2}(G)^{2}}  (28)

To determine Im(j′_{2}) ∩ Im(R_{1}) consider the following commutative diagram with canonical quotient map q and exact row.

A^{⊗2} ⊗ A  ⟶^{q} (A^{⊗2}/l_{2}Ker(c_{2})) ⊗ A
↓^{R_{1}}  ↓^{μ⊗A}
A ⊗ (A^{⊗2}/l_{2}Ker(c_{2})) ⟶^{j′_{2}}  \frac{P_{2}(G) ⊗ P_{2}(G)}{P_{2}(G)^{2} ⊗ P_{2}(G)^{2}}  ⟶^{id⊗ρ}  P_{2}(G) ⊗ A

Since Ker(μ ⊗ A) = Im(τ_{1}) we have

Im(j′_{2}) ∩ Im(R_{1}) = R_{1}(l_{2}Ker(c_{2}) ⊗ A) + R_{1}q^{−1}Im(τ_{1})
= j_{2}(A ⊗ L)(l_{2}Ker(c_{2}) ⊗ A) + R_{1}q^{−1}Im(τ_{1}) ,  (29)

with L: A^{⊗2} → A^{⊗2}, L(x ⊗ y) = x ⊗ y − y ⊗ x.

Let ⟨x_{1}, k, x_{2}⟩ be a typical generator of A ∗ A, with x_{1}, x_{2} ∈ A and k ∈ ZZ such that kx_{1} = kx_{2} = 0, cf. [22] V.6. Choose y_{j}, z_{j} ∈ A such that

μ^{−1}(kp_{2}(x′_{1})) = ∑_{j} y_{j} ⊗ z_{j} + l_{2} Ker(c_{2}) .

Then

R_{1}q^{−1}τ_{1}⟨x_{1}, k, x_{2}⟩ = kp_{2}(x′_{1}) ⊗ p_{2}(x′_{2}) − j_{2} ∑_{j} y_{j} ⊗ [z_{j}, x_{2}]
= j′_{2}(x_{1} ⊗ μ^{−1}(kp_{2}(x′_{2})) − ∑_{j} y_{j} ⊗ [z_{j}, x_{2}])
= j′_{2}(τ_{2}⟨x_{1}, k, x_{2}⟩− (A ⊗ L)(∑_{j} y_{j} ⊗ z_{j} ⊗ x_{2}))  (30)

Taking together (26), (28), (29) and (30) we see that ζ′ induces an isomorphism of Ker(π) with the double quotient

\frac{A ⊗ A^{⊗2}}{(id − (A ⊗ L)T_{12})(A^{⊗3}) + A ⊗ l_{2}Ker(c_{2}) + (A ⊗ L)(l_{2}Ker(c_{2}) ⊗ A)}/ (31)

(Im(ρ ⊗ id)τ + Im(τ_{2} − (A ⊗ L)τ_{1})) .

Here the term (id − (A ⊗ L)T_{12})(A^{⊗3}) corresponds to Im(R_{2}). To simplify this expression we use the following

Lemma: The following sequence of homomorphisms is exact:

A^{⊗3} ⟶^{id−(A⊗L)T_{12}} A^{⊗3} ⟶^{ν_{3}T_{12}} L_{3}(A) → 0 .

Indeed, writing ω = T_{12}(id − (A ⊗ L)T_{12}) , we have

ω(x ⊗ y ⊗ z) = y ⊗ x ⊗ z − x ⊗ y ⊗ z + z ⊗ y ⊗ x .

In particular, ω(x ⊗ y ⊗ x) = y ⊗ x ⊗ x, so there is a factorization

A^{⊗3} ⟶^{A⊗ν_{2}} A ⊗ (A^{∧ 2}) ⟶ coker(ω), ν_{2}(y ⊗ z) = y ∧ z. But

− (A ⊗ ν_{2})ω(x ⊗ y ⊗ z) = y ⊗ (z ∧ x) + x ⊗ (y ∧ z) + z ⊗ (x ∧ y) .

Thus coker(ω) is isomorphic to A ⊗ (A^{∧ 2}) modulo the Jacobi identity which is L_{3}(A). This proves the lemma.

Using the identification

A ⊗ (A^{⊗2})/Im(id − (A ⊗ L)T_{12}) ≅^{ν_{3}T_{12}} L_{3}(A)  (32)

provided by the lemma the term in (31) simplifies as follows, where we also use the relations (9) and

ν_{3}T_{12}(A ⊗ L) = ν_{3} .  (33)

Ker(π) ≅^{ζ^{′′−1}} \frac{L_{3}(A)}{[A, Ker(c_{2})]}/(Im(ν_{3}T_{12}(ρ ⊗ id)τ) + Im(ν_{3}T_{12}(τ_{2} − (A ⊗ L)τ_{1}))) .

(34)

To make the map ζ′′ induced by ν_{3}T_{12} and ζ′ explicit, let x_{i} ∈ A, i = 1, 2, 3. Then

ζ′′[x_{1}, [x_{2}, x_{3}]] = ζ′(x_{1} ⊗ [x_{2}, x_{3}]) by (33)
= ζ(p_{2}(x′_{1}) ⊗ [p_{2}(x′_{2}), p_{2}(x′_{3})])
= ζ(p_{2}(x′_{1}) ⊗ p_{2}[x′_{2}, x′_{3}])
= x′_{1} ⊗ [x′_{2}, x′_{3}]  (35)

Finally we evaluate the terms on the right-hand side of (34). By (33) one has Im(ν_{3}T_{12}(τ_{2} − (A ⊗ L)τ_{1})) = Im(ν_{3}(τ_{1} − T_{12}τ_{2})). For a typical generator ⟨x_{1}, k, x_{2}⟩ of A ∗ A we use the identities

k p_{2}(x′_{i}) = p_{2}(x_{i}^{′k}) − \binom{k}{2}μ(x_{i} ⊗ x_{i})
= μl_{2}f_{k}x_{i} − \binom{k}{2}μ(x_{i} ⊗ x_{i})  (36)

and (10), (11) to calculate

ν_{3}(τ_{1} − T_{12}τ_{2})⟨x_{1}, k, x_{2}⟩ = ν_{3}(μ^{−1}(kp_{2}(x′_{1})) ⊗ x_{2} − T_{12}(x_{1} ⊗ μ^{−1}(kp_{2}(x′_{2})))
= − [x_{2}, f_{k}x_{1}] − \binom{k}{2}[x_{1}, [x_{1}, x_{2}]] +  (37)

− [x_{1}, f_{k}x_{2}] + \binom{k}{2}[x_{2}, [x_{1}, x_{2}]]

Now let ⟨y, k, x⟩ be a typical generator of P_{2}(G) ∗ A, i.e. y ∈ P_{2}(G), k ∈ ZZ, x ∈ A such that ky = 0 = kx. Then

ν_{3}T_{12}(ρ ⊗ id)τ⟨y, k, x⟩ = ν_{3}T_{12}(ρy ⊗ (l_{2}f_{k}x − \binom{k}{2}x ⊗ x))
= [ρy, f_{k}x] − \binom{k}{2}[x, [ρy, x]] ,  (38)

by (36) and (11). Let y′ ∈ G such that y′γ_{2}(G) = ρy. By (8) there is z ∈ A^{⊗2}/l_{2}Ker(c_{2}) such that y = p_{2}(y′) + μ(z), whence kp_{2}(y′) = kμ(z). Thus

ν_{3}(τ_{1} − T_{12}τ_{2})⟨−ρy, k, x⟩ = − ν_{3}(kz ⊗ x) + [ρy, f_{k}x] − \binom{k}{2}[x, [ρy, x]]
= ν_{3}T_{12}(ρ ⊗ id)τ⟨y, k, x⟩

by the two identities in array (37), by (38), and since kx = 0.

This shows that the term Im(ν_{3}T_{12}(ρ ⊗ id)τ) in (34) is superfluous. Together with (35) and (37) this proves the theorem. □

Proof of theorem 2.2: Consider the following diagram:

((G^{ab})^{⊗2} ⊗ P_{2}(G))^{⊕2}  ⟶^{μ′} P_{2}(G)^{⊗2} ⟶^{ρ^{⊗2}} (G^{ab})^{⊗2}
↓^{(1,0)(κ_{2},κ_{3})}  ↓^{ζ}  ∥
\frac{L_{3}(A)}{[A, Ker(c_{2})] + Im(ν_{3}(τ_{1} − T_{12}τ_{2}))} ⟶^{i} G ⊗ G ⟶^{π} (G^{ab})^{⊗2}

(39)

with

μ′(x ⊗ y ⊗ {a − 1} , u ⊗ w ⊗ {b − 1}) = {(x′ − 1)(y′ − 1)} ⊗ {a − 1} +
+ {b − 1} ⊗ {(u′ − 1)(w′ − 1)}

The left-hand square commutes in the first summand (G^{ab})^{⊗2} ⊗P_{2}(G) by (20) and in the second by (26) and the identification (32) used to derive (34). The right hand square is clearly commutative.

Since the map i is injective by theorem 2.1 we may replace ((G^{ab})^{⊗2}⊗P_{2}(G))^{⊕2} by its quotient modulo Ker(μ′). One obtains a commutative diagram of abelian group extensions with identical cokernel. Hence the bottom extension is induced by the top one, which implies that the left hand square is a push out square of abelian groups. This implies the theorem. □

Proof of corollary 2.3: First we recall some elementary facts, see [15], for example. For any n ≥ 1, the ring homomorphism P_{n}(F) → P_{n}(G) induced by the quotient map F → → G is surjective, and its kernel is the both-sided ideal generated by the elements p_{n}(r_{j}), r_{j} ∈ R. The ring P_{n}(F) itself is isomorphic to the truncated augmentation ideal T̄(D)/T̄(D)^{n+1} of the tensor algebra T (D), where p_{n}(x) corresponds to the monomial x̄ for x ∈ X. Using these facts we obtain the following commutative diagram with exact rows, with g(x) = p_{2}{x} for x ∈ X.

S  ⟶^{e_{1}}  D  ⟶^{q} A ≅ G^{ab}
∥  ↑^{(1,0)}  ↑^{ρ}
S  ⟶^{(e_{1},e_{2})^{t}} D ⊕ A^{⊗2} ⟶^{(g,μ)} P_{2}(G)
↑  ↑  ↑
Ker(e_{1}) ⟶^{e_{2}}  A^{⊗2}  ⟶^{μ} P_{2}(G)^{2}

Indeed, a short calculation shows that (e_{1}r_{j}, e_{2}r_{j}) is the image under (D ⊕ q^{⊗2}) of the element in D ⊕ D^{⊗2} which corresponds to p_{2}(r_{j}) ∈ P_{2}(F) under the isomorphism D ⊕ D^{⊗2} ≅ P_{2}(F) described above. By (8) there is a factorization of e_{2},

e_{2} : Ker(e_{1}) ⟶^{ẽ_{2}} A^{∧ 2} ⟶^{>}^{l_{2}} A^{⊗2} , and ẽ_{2} Ker(e_{1}) = Ker(c_{2}) .  (40)

Using (10) we conclude that

ν_{3}(e_{2}Ker(e_{1}) ⊗ A) = [A, ẽ_{2} Ker(e_{1})] = [A, Ker(c_{2})] .

Using the inclusion ι_{1} : Ker(q ⊗ e_{1}) ↪ D⊗S and the factorization S ⟶^{ẽ_{1}} Im(e_{1}) ↪^{ι_{2}} D of e_{1}, we get a surjection

Ker(q ⊗ e_{1}) ⟶^{(q⊗ẽ_{1})ι_{1}} L_{1}(A ⊗ −)(A) = Ker(A ⊗ ι_{2} : A ⊗ Im(e_{1}) → A ⊗ D)

where L_{1}(A ⊗ −) denotes the first left derived functor of (A ⊗ −). Now observe that the homomorphism

(ẽ_{1} ⊗ q)(e_{1} ⊗ D)^{−1}(D ⊗ e_{1})ι_{1} : Ker(q ⊗ e_{1}) → Im(e_{1}) ⊗ A

maps onto Ker(ι_{2} ⊗ A) = L_{1}(− ⊗ A)(A) and induces the canonical isomorphism

b_{1} : L_{1}(A ⊗ −)(A) ≅ L_{1}(− ⊗ A)(A)

between the two possible descriptions of A ∗ A as a derived functor. Then it is a routine exercise in elementary homological algebra to verify that

τ_{2}(q ⊗ ẽ_{1})ι_{1} = (q ⊗ ē_{2})ι_{1} and τ_{1}b_{1}(q ⊗ ẽ_{1})ι_{1} = d .

It follows that

Im(σ) = Im(ν_{3}(τ_{1} − T_{12}τ_{2})) in \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A)} = \frac{L_{3}(A)}{[A, Ker(c_{2})]} . (41)

Finally, consider the following diagram with exact rows, see 2.2.

(S ⊗ D)^{⊕2}  ⟶^{h}  \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A) + Im(σ)} ⊕ D^{⊗2}  ⟶^{u} coker(h)
↓^{(e_{2}⊗g)^{⊕2}}  ↓^{id⊕g^{⊗2}}  ↓^{m}
((A^{⊗2} ⊗ P_{2}(G))^{⊕2} ⟶^{(κ_{2},κ_{3})} \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + Im(ν_{3}(τ_{1} − T_{12}τ_{2}))} ⊕ P_{2}(G)^{⊗2} ⟶^{λ} G ⊗ G

Here

h = ( \begin{matrix}ν_{3}(e_{2} ⊗ q) & − ν_{3}(e_{2} ⊗ q) \\ e_{1} ⊗ D & T (e_{1} ⊗ D) \end{matrix} ) .

It is straightforward to check that the left-hand square commutes, so there is a homomorphism m: coker(h) → G ⊗ G which makes the right-hand square commutative. Writing u = (u_{1}, u_{2}) we obtain a commutative diagram with exact rows, see 2.1, where v is induced by (0, q^{⊗2}).

\frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A) + Im(σ)}  ⟶^{u_{1}} coker(h) ⟶^{v} A^{⊗2} → 1
∥  ↓^{m}  ∥
1 → \frac{L_{3}(G^{ab})}{[G^{ab}, Ker(c_{2})] + Im(ν_{3}(τ_{1} − T_{12}τ_{2}))} ⟶^{i} G ⊗ G ⟶^{π} (G^{ab})^{⊗2} → 1

It follows that u_{1} is injective and whence m is an isomorphism. Now the assertion follows by observing that the map e has the same image as the map h. □

Proof of theorem 4.1: From theorem 2.2 and (5) we obtain a short exact sequence

1 → Im(i)/i^{−1}ψKer(πψ) → G ∧ G ⟶^{π̄} (G^{ab})^{∧ 2} → 1 .  (42)

So in order to prove exactness of the first sequence in the theorem we have to verify the relation

V = i^{−1}ψKer(πψ) .  (43)

Lemma 5.1 Let A be an abelian group. Then the kernel of the natural homomorphism πψ: Γ(A) → A ⊗ A is generated by the elements o(x)γ(x) where x runs through the elements of A of even finite order o(x).

Proof: Let K(A) be the subgroup of Γ(A) generated by the elements o(x)γ(x) as above; this is clearly a natural subgroup.

Recall the following properties of Γ [26]:

Γ(ZZ/n · x) = \{ \begin{matrix}ZZ/n · γ(x) , & n odd \\ ZZ/2n · γ(x) , & n even \end{matrix}  (44)
Γ(A ⊕ B) = Γ(A) ⊕ Γ(B) ⊕ A ⊗ B .  (45)

By (44) the assertion holds for cyclic groups, whence for all finitely generated abelian groups since Ker(πψ) is an additive functor as is easily derived from (45). Now let A be an arbitrary abelian group. Identifying A with the colimit of its finitely generated subgroups A_{i} we obtain an exact sequence

colim K(A_{i}) ⟶^{j} Γ(A) ⟶^{πψ} A^{⊗2} ,

since the functors Γ and (−)^{⊗2} commute with colimits over direct systems. But j maps into K(A) while K(A) ⊂ Ker(πψ) = Im(j), whence K(A) = Ker(πψ), as asserted.

Now let x be periodic in G^{ab} and choose fx ∈ (G^{ab})^{∧ 2} such that c_{2}fx = x^{o(x)}. Then

ψ(o(x)γ(x)) = ζ(o(x)p_{2}(x′) ⊗ p_{2}(x′))
= ζ((μl_{2}fx − \binom{k}{2}p_{2}(x′)^{2}) ⊗ p_{2}(x′))
= iν_{3}(l_{2} ⊗ A)(fx ⊗ x) by (20) and commutativity of diagram (39)
= − i[x, fx] by (10) .

This proves (43). Next we have to show that the map ζ̄ in (16) is well defined. Since the map ζ in (12) is already proved to be so it remains to verify that the subgroup W of P_{2}(G)^{⊗2} generated by the elements w ⊗ w, w ∈ P_{2}(G), maps onto Im(ψ) under ζ, cf. (5). Together with 2.2 this also proves exactness of the second sequence in the theorem.

But the group P_{2}(G) is generated by the elements p_{2}(a), a ∈ G, so W is generated by the elements p_{2}(a) ⊗ p_{2}(a), a ∈ G, and p_{2}(a) ⊗ p_{2}(b) + p_{2}(b) ⊗ p_{2}(a), a, b ∈ G. Under ζ, these map to a ⊗ a and

(a ⊗ b)(b ⊗ a) = (ba ⊗ ba)(b ⊗ b)^{−1}(a ⊗ a)^{−1} = ψ(γ(ba) − γ(b) − γ(a)) , (46)

respectively, where the first equation is taken from the proof of 2.12 in [9]. This verifies the claim.

To prove exactness of the third sequence in the theorem consider the following diagram.

\begin{matrix}S^{2}(D) & ⟶^{w} & Γ(D) & ⟶^{ψ} & D ⊗ D & ⟶^{q^{⊗2}} & A^{⊗2} \\ ↓^{S^{2}(q)} & & ↓^{Γ(q)} & & ↓^{ζ′} & & ∥ \\ S^{2}(A) & ⟶^{w} & Γ(A) & ⟶^{ψ} & G ⊗ G & ⟶^{π} & A^{⊗2} \end{matrix}  (47)

where w is the natural homomorphism defined by w{a ⊗ b} = γ(ab) − γ(a) − γ(b). This diagram is commutative; to check this for the middle square one needs the fact that Γ(D) is generated by the elements γ(x), w{x ⊗ y}, x, y ∈ X, and relation (46). Then exactness of the third sequence in the theorem follows from (13), surjectivity of Γ(q) and commutativity of the middle square. □

Proof of corollary 4.3: Exactness of the second sequence is obtained by combining (2), 4.1, commutativity of the following diagram and injectivity of the composite map l_{2}c_{2}^{−1} by (40).

[figure: G′; ≅^{c_{2}^{−1}}; A^{∧ 2}/Ker(c_{2}); l_{2}; A^{⊗2}/e_{2}Ker(e_{1})]
↑^{[ ,]}  ↑^{l}
G ∧ G \overleftarrow{⟵}^{(ī′,ζ̄′)} \frac{L_{3}(A)}{ν_{3}(e_{2}Ker(e_{1}) ⊗ A) + Im(σ)} ⊕ D^{∧ 2} ⟶^{(0,1)}  D^{∧ 2}

To prove exactness of the first sequence recall diagram (47). By 2.3 it remains to show that

(i′)^{−1}ψKer(qψ) = Im(δ) .  (48)

By (5.1) we have Ker(qψ) ⊂ 2Γ(A) ⊂ Im(w) , where the latter inclusion follows from the relation 2γ(x) = γ(2x)−2γ(x) = w{x⊗x} in Γ(A) for all x ∈ A. Using diagram (47) and surjectivity of S^{2}(q) we find that

ψKer(qψ) = ζ′(ψw)Ker(q^{⊗2}(ψw)) .

But ψw = σ_{2}, so (48) follows from (13). □

Acknowledgments

I am grateful to R. Brown for explaining to me the topological background of the nonabelian tensor product and for some comments on this work.

Also I thank C.-L. Kappe for sending me preprints of her and M. R. Bacon’s work.

Moreover, I would like to express my gratitude to the Sonderforschungsbereich 343 in Bielefeld and the Human Capital and Mobility programme of the European Union for awarding successive researchs positions to me which enable me to persue freely my work programme. Moreover, I am very grateful to Professor H. Abels at Bielefeld and Professors J.-L. Loday and C. Kassel at Strasbourg for creating a friendly and stimulating atmosphere and for many interesting discussions.

Last not least I would like to thank the referees for numerous detailed suggestions on the presentation of the paper, which may have made it much more readable than it was in the original version. I am also indebted to them for proposing some more applications, like the homotopy operations and the formula for the Schur multiplicator in terms of a presentation.

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