Finite free resolutions and homology of finitely generated torsion-free nilpotent groups

Yu. Semenov

Abstract

For any f.g. nilpotent group G such that Lie ring G associated to the lower central series of G is Z-torsion-free we construct a finite free resolution of minimal length of the kind ZG ⊗ Λ^{∗}G → Z → 0. Moreover, for any f.g. torsion-free nilpotent group G there exists a resolution of the kind ZG ⊗ Λ^{∗}G → Z → 0 where G is the Lie ring associated to the central series of G consisting of isolators of γ_{i}G. We prove also that H_{n}G ≅ H_{n}G for any free 2-step nilpotent group G.

Introduction

Let G be a free abelian group on r generators. It is well known how to construct a quite simple resolution of the trivial G-module Z over ZG. One can take the exterior algebra of G over ZG, i.e. ZG ⊗ Λ^{∗}G, supplied with a suitable differential d (see (3) below).

The general case of a finitely generated torsion-free (f.g.t.-f.) nilpotent group G is obviously more complicated. In 1989 J. Huebschmann [Hu] constructed small resolutions for 2-step nilpotent groups G (his perturbation theory construction admits a generalisation to groups of higher class of nilpotency). But the question of computing the integral homology groups for any f.g.t.-f. nilpotent group still remains unsolved.

In this paper we develop the approach to the construction of finite free resolutions for f.g.t.-f. nilpotent groups that appeared in [K-S] for the case of 2-step free nilpotent groups. Our main result is the following Theorem 4. Let G be a f.g.t.-f. nilpotent group and G be the Lie ring associated to the central series of G consisting of the isolators of the lower central series terms γ_{i}G. Then there exists a finite free resolution of Z over ZG of the kind ZG⊗Λ^{∗}G where Λ^{∗}G is the exterior algebra of G. In particular, it is of minimal length r = cd(G) (= Hirsch number of G).

The proof of this theorem provides an inductive procedure for constructing such resolutions. Moreover, it is valid not only for nilpotent groups but also for nilpotent Lie rings (in this case it yields well-known resolutions of the kind U(G) ⊗ Λ^{∗}G). The parallels are depicted in the notion of an E-triple and in the theorem 3.

In [K-S] we used similar construction to prove that the integral homology groups H_{n}G and H_{n}G are isomorphic for free 2-step nilpotent groups G (see also sections 3,4). In addition to this we should mention a paper by St.Sigg [Si] where he computes the groups H_{n}(G, C). In general, there are torsion elements in these integral homology groups ([L]). Question. Is it true that H_{n}G ≅ H_{n}G for f.g.t.-f. nilpotent groups G? One knows that H_{n}(G, Q) ≅ H_{n}(G, Q) (see [Ku]). In view of this result the eventual difference between H_{n}G and H_{n}G must be of torsion.

The paper is organised as follows. In the section 1 we present the main construction of a resolution for the case of a f.g.t.-f. nilpotent group G such that the Lie ring G associated to the lower central series of G is Z-torsion-free. In the section 2 we generalise this construction for any f.g.t.-f. nilpotent group G. We consider the case of a free 2step nilpotent group G as an example of how our construction works and we prove in sections 3,4 that there exists an isomorphism between H_{n}G and H_{n}G (the proof differs from that of [K-S] in some details).

Acknoledgements. The author is very grateful to the Max-Planck-Institut für Mathematik for its hospitality. This work was also partially supported by Russian Foundation for Fundamental Research (# 97-01-00290).

1 Main construction

Let G be a finitely generated nilpotent group of class c. We’ll suppose that Lie ring G associated to the lower central series of G is torsion-free as a Z-module (then G is called Magnus nilpotent group). Consider the central extension

1 → γ_{c}G → G → \overline{G} → 1,  (0)

where γ_{c}G = A is the last term of the lower central series of the group G. We also consider A as an abelian Lie ring; then there is the extension

0 → A → G → \overline{G} → 0,

where Lie ring \overline{G} is associated to quotient group \overline{G}. As a Z-module G ≅ A ⊕ \overline{G}. We fix Z-bases a_{1}, …, a_{k} in A and x_{1}, …, x_{m} in \overline{G}. In the sequel the usual notation Λ^{p}M for the p-th exterior power of a Z-module M is used (Λ^{0}M ≅ Z).

Theorem 1. Let G be a f.g. Magnus nilpotent group of class c. Given the extension (0) let’s suppose that there is a free resolution of the trivial \overline{G}-module Z over Z\overline{G} of the kind

… ⟶^{\overline{d}} Z\overline{G} ⊗ Λ^{p}\overline{G} ⟶^{\overline{d}} … ⟶^{\overline{d}} Z\overline{G} ⟶^{\overline{π}} Z → 0  (1)

Here \overline{d} is a differential and \overline{π} is the canonical augmentation epimorphism.

Then there exists a free resolution of the trivial G-module Z over ZG of the similar kind

… ⟶^{d} ZG ⊗ Λ^{n}G ⟶^{d} … ⟶^{d} ZG ⟶^{π} Z → 0  (2)

for some differential d depending on \overline{d}. Proof. It is well known that for a free abelian group A there is a resolution

… ⟶^{d_{0}} ZA ⊗ Λ^{q}A ⟶^{d_{0}} … ⟶^{d_{0}} ZA ⟶^{π} Z → 0,  (3)

where A-homomorphism d_{0} carries the basic element 1 ⊗ a_{j_{1}} ∧ … ∧ a_{j_{q}} to

∑_{t=1}^{q} (−1)^{t−1}(a_{j_{t}} − 1) ⊗ a_{j_{1}} ∧ … â_{j_{t}} … ∧ a_{j_{q}}.

Applying to (3) the functor ZG⊗_{ZA} we get the exact sequence

… ⟶^{d_{0}} ZG ⊗ Λ^{q}A ⟶^{d_{0}} … ⟶^{d_{0}} ZG ⟶^{μ} Z\overline{G} → 0,  (4)

where μ is induced by the canonical epimorphism G → \overline{G} and d_{0} is a G-homorphism. Note that Ker μ is generated by a_{j} − 1, j = 1, …, k as a left G-module.

Tensoring (4) by Λ^{p}\overline{G} we obtain the following family of exact sequences (p = 0, …, m = rk_{Z}\overline{G}):

… ⟶^{d_{0}⊗1} ZG ⊗ Λ^{q}A ⊗ Λ^{p}\overline{G} ⟶^{d_{0}⊗1} … ⟶^{d_{0}⊗1} ZG ⊗ Λ^{p}\overline{G} ⟶^{μ⊗1} Z\overline{G} ⊗ Λ^{p}\overline{G} → 0.

(4p)

Now let E_{pq} = ZG ⊗ Λ^{q}A ⊗ Λ^{p}\overline{G}. This notation agrees with that used for spectral sequences: p is the horizontal and q is the vertical coordinate (thus, every sequence (4p) becomes a column with the ”underground floor” term Z\overline{G} ⊗ Λ^{p}\overline{G} which is not included in E but should be always kept in mind in what follows). Having chosen an additive basis in \overline{G} x_{1}, …, x_{m} we order it so that x_{1} < … < x_{m}. The typical basic element of E_{po} is 1 ⊗ x_{i_{1}} ∧ … ∧ x_{i_{p}}, i_{1} < … < i_{p} . To simplify the notation it will be usually denoted by 1 ⊗ X (if there is no need to specify it).

We consider E = ⊕_{p,q≥0}E_{pq} as a bi-graded G-module. Now we have a differential (G-homorphism) d_{0} : E_{pq} → E_{p,q−1} which is the same as d_{0} ⊗ 1 in (4p) and equal to 0 when q = 0 (the vertical differential). It also may be viewed as an element of Homgr_{G}^{0,−1}(E, E) ⊂ Homgr_{G}(E, E). For a fixed basic element a_{j} ∈ A consider a Ghomorphism R_{j} ∈ Homgr_{G}^{01}(E, E) such that

R_{j}(1⊗a_{j_{1}} ∧… a_{j_{q}} ∧x_{i_{1}} ∧…∧x_{i_{p}}) = 1⊗a_{j} ∧a_{j_{1}} ∧… a_{j_{q}} ∧x_{i_{1}} ∧…∧x_{i_{p}}.

It is not hard to see that in Homgr_{G}(E, E) one has

d_{0}R_{j} + R_{j}d_{0} = (a_{j} − 1)Id  (5)

since all the a_{j} belong to the centre of G.

Given d_{0} we are interested in the following problem: how to find the elements d_{s} ∈ Homgr_{G}^{−s,s−1}(E, E), s = 1, …, m so that one has the equality

d^{2} = (d_{0} + d_{1} + … + d_{m})^{2} = 0.  (6)

[figure: •; •; •; •; •; •; •; p; q; d_{0}; d_{1}; d_{2}; d_{3}; …; …; …; E_{pq}; E_{00}; E_{10}; …; Pic.1]

Let’s consider the equation (6) as a system of equations

d_{0}d_{s} + d_{s}d_{0} = − ∑_{i=1}^{s−1} d_{i}d_{s−i} , s = 1, …, m.  (7)

First of all, we’ll assume that all the desired d_{s} commute with all R_{j} (in the graded sense): d_{s}R_{j} + R_{j}d_{s} = 0. This assumption enables to define d_{s} only on the elements of the type 1⊗x_{i_{1}} ∧…∧x_{i_{p}} = 1⊗X . The correctness is based on the following lemma. Lemma 1. Suppose that the endomorphism F_{s} = d_{0}d_{s} + d_{s}d_{0} coin-

cides with K_{s} = − ∑_{i=1}^{s−1} d_{i}d_{s−i} on E_{p0}. Then the system (7) holds in

Homgr_{G}(E, E). Proof. It is enough to prove that both F_{s} and K_{s} commute with any R_{j}. The equality K_{s}R_{j} = R_{j}K_{s} follows immediately from the above assumption and F_{s}R_{j} = R_{j}F_{s} from (5).

In order to solve (7) we’ll proceed inductively. Since d_{1} should be related with the differential \overline{d} in (1) we define d_{1} in the following manner. Let’s construct a commutative diagram

… → ZG ⊗ Λ^{p}\overline{G} ⟶^{d_{1}} ZG ⊗ Λ^{p−1}\overline{G} → …
↓  ↓
… → Z\overline{G} ⊗ Λ^{p}\overline{G} ⟶^{\overline{d}} Z\overline{G} ⊗ Λ^{p−1}\overline{G} → …,

where the vertical arrows are homomorphisms μ ⊗ 1. This diagram is a diagram of G-modules (via μ). Thus we can define d_{1} since all the G-modules ZG ⊗ Λ^{p}\overline{G} are free and the vertical arrows are surjective.

Obviously, d_{0}d_{1}+d_{1}d_{0} = 0 on E_{p0}. By the lemma 1 d_{0}d_{1}+d_{1}d_{0} ≡ 0.

A G-homorphism d_{2} should be defined from the equality (X is fixed)

(d_{0}d_{2} + d_{2}d_{0})(1 ⊗ X) = −d_{1}^{2}(1 ⊗ X).

Since \overline{d}^{2} = 0 then z = −d_{1}^{2}(1 ⊗ X) belongs to Ker μ ⊗ Λ^{p−2}\overline{G}. It follows that z can be written in the form

z = ∑_{i=1}^{k} ∑_{t} w_{it}(a_{i} − 1) ⊗ X_{it}

for some w_{it} ∈ ZG and basic elements X_{it} of Λ^{p−2}\overline{G}.

The differential d_{0} = 0 on E_{p0}. Consequently, it remains only to prove that z is a boundary. In fact,

z = d_{0}(∑_{i=1}^{k} ∑_{t} w_{it} ⊗ a_{i} ∧ X_{it}),

and we can define

d_{2}(1 ⊗ X) = ∑_{i=1}^{k} ∑_{t} w_{it} ⊗ a_{i} ∧ X_{it}.

Suppose that we have already defined d_{1}, …, d_{s−1}, s ≥ 3, so that the first s − 1 equations of the system (7) hold. Consider K_{s} =

− ∑_{i=1}^{s−1} d_{i}d_{s−i}.

Lemma 2. The equality K_{s}d_{0} = d_{0}K_{s} holds. Proof. Using (7) we have on the one hand

\begin{matrix}K_{s}d_{0} = − ∑_{i=1}^{s−1} d_{i}d_{s−i}d_{0} = − ∑_{i=1}^{s−1} d_{i}(−d_{0}d_{s−i} − ∑_{j=1}^{s−i−1} d_{j}d_{s−i−j}) = \\ = ∑_{i=1}^{s−1} d_{i}d_{0}d_{s−i} + ∑_{i=1}^{s−1} ∑_{j=1}^{s−i−1} d_{i}d_{j}d_{s−i−j} = ∑_{i=1}^{s−1} d_{i}d_{0}d_{s−i} + ∑_{{p+q+r=s \atop p,q,r>0}} d_{p}d_{q}d_{r}. \end{matrix}

On the other hand,

\begin{matrix}d_{0}K_{s} = − ∑_{i=1}^{s−1} d_{0}d_{i}d_{s−i} = − ∑_{i=1}^{s−1}(−d_{i}d_{0} − ∑_{j=1}^{i−1} d_{j}d_{i−j}d_{s−i−j}) = \\ = ∑_{i=1}^{s−1} d_{i}d_{0}d_{s−i} + ∑_{i=1}^{s−1} ∑_{j=1}^{i−1} d_{j}d_{i−j}d_{s−i} = ∑_{i=1}^{s−1} d_{i}d_{0}d_{s−i} + ∑_{{p+q+r=s \atop p,q,r>0}} d_{p}d_{q}d_{r}. \end{matrix}

In other words, this lemma asserts that K_{s} is a cycle relative to a differential D in Homgr_{G}(E, E) such that Df = d_{0}f − (−1)f̃fd_{0} where f̃ is the total degree of f. To find a solution of the system (7) is equivalent to prove that all K_{s} are boundaries.

By the lemma 2

0 = (d_{0}K_{s} − K_{s}d_{0})(1 ⊗ X) = d_{0}K_{s}(1 ⊗ X).

It follows that K_{s}(1⊗X) is a cycle of E_{p−s,s−2}. Since all the sequences (4p) are exact we deduce that there is an element Y ∈ E_{p−s,s−1} such that d_{0}(Y) = K_{s}(1 ⊗ X) (recall that s ≥ 3). Set d_{s}(1 ⊗ X) = Y. Thus we have

(d_{0}d_{s} + d_{s}d_{0})(1 ⊗ X) = d_{0}(Y) = K_{s}(1 ⊗ X).

By the lemma 1 we obtain the equality d_{0}d_{s} + d_{s}d_{0} = K_{s}.

Let

E_{n} = ⨁_{p+q=n}E_{pq} = ⨁_{p+q=n}ZG⊗Λ^{q}A⊗Λ^{p}\overline{G} ≅ ZG⊗Λ^{n}(A⊕\overline{G}) ≅ ZG⊗Λ^{n}G.

The differential d = d_{0} + … + d_{m} maps E_{n} to E_{n−1}, i.e. we have the augmented complex

… ⟶^{d} E_{n} ⟶^{d} … ⟶^{d} E_{0} ⟶^{π} Z ⟶ 0.  (8)

In order to prove the theorem we have to show that (8) is exact. Step 1. Exactness in dimension 0. Since E_{0} = ZG and π is the augmentation epimorphism let’s show that Im d : E_{1} → E_{0} coincides with the augmentation ideal Ker π ⊂ ZG.

Consider a commutative diagram

ZG ⊗ \overline{G} ⟶^{d_{1}} ZG ⟶^{π} Z → 0
↓  μ ↓  ||
Z\overline{G} ⊗ \overline{G} ⟶^{\overline{d}} Z\overline{G} ⟶^{\overline{π}} Z → 0.

It is not hard to see that Ker π = Ker μ+Im d_{1}. But Ker μ ≅ Im d_{0} : ZG ⊗ A → ZG, i.e.

Ker π = Im d_{0} + Im d_{1} = Im d.

Step 2. Exactness in dimensions ≥ 1. Consider the following finite filtration on E:

F_{p}E_{n} = ⨁_{{s+q=n \atop s≤p}} E_{sq}.

The differential d respects this filtration. Let (E_{pq}^{r}, d_{pq}^{r}) be the associated spectral sequence converging to homology of E.

Since there is the decomposition d = d_{0} + … + d_{m} then E_{pq}^{0} ≅ E_{pq} and d_{pq}^{0} = d_{0}. From the exact sequences (4p) we deduce that E_{pq}^{1} = 0 if q > 0, and E_{p0}^{1} ≅ Z\overline{G}⊗Λ^{p}\overline{G}. Due to these isomorphisms d_{p0}^{1} becomes \overline{d}. It follows from (1) that E_{pq}^{2} = 0 if p + q > 0, i.e. (8) is in fact a resolution of Z over ZG. Q.E.D.

As a direct consequence of the theorem 1 we obtain Theorem 2. Let G be finitely generated nilpotent Magnus group of class c (e.g. a free nilpotent group). Then there is a resolution of the kind ZG ⊗ Λ^{∗}G → Z → 0. Proof. It is well known for c = 1 (abelian case). Step of induction is provided by the theorem 1. Question. One can prove for a f.g. free nilpotent group G of class 2 that there is an isomorphism between integral homology groups H_{n}G and H_{n}G (sections 3,4). Is it true for arbitrary f.g. Magnus nilpotent groups?

2 Resolutions for torsion-free nilpotent groups.

Given a free Z-module A of finite rank k with a basis a_{1}, …, a_{k}, let S(A), Λ(A) be symmetric and exterior algebras of A over Z and ε : S(A) → Z be the canonical augmentation epimorphism.

Let’s suppose that we’re also given an associative ring R with identity and a monomorphism φ : S(A) → Z(R) where Z(R) is the centre of R. By \overline{R} we denote the quotient ring R ⊗_{S(A)} Z (R is a right S(A)-module via φ and Z is a left S(A)-module via ε). Let μ be the canonical epimorphism R → \overline{R}. In the sequel we’ll suppose that both R, \overline{R} are free Z-modules.

One knows ([Bo]) that there is an exact sequence of S(A)-modules (Koszul resolution)

0 ⟶ S(A) ⊗ Λ^{k}A ⟶^{δ} … ⟶^{δ} S(A) ⟶^{ε} Z → 0,  (9)

where δ is a S(A)-differential such that

δ(1 ⊗ a_{j_{1}} ∧ … ∧ a_{j_{q}}) = ∑_{s=1}^{q}(−1)^{s−1}a_{j_{s}} ⊗ a_{j_{1}} ∧ … â_{j_{s}} … ∧ a_{j_{q}}.

Definition. The triple (A, R, φ) is said to have E-property (or to be an E-triple) if the sequence

0 ⟶ R ⊗ Λ^{k}A ⟶^{d} … ⟶^{d} R ⟶^{μ} \overline{R} → 0  (10)

obtained from (9) by applying the functor R ⊗_{S(A)} − is exact. Examples. Let R be a flat S(A)-module. Then (A, R, φ) has Eproperty. We are mostly interested in the following case: G is a Z-torsion-free finite dimensional nilpotent Lie ring, A is its centre (supposed to be a direct summand of G as a Z-module), R = U(G) (universal enveloping ring), φ : S(A) → U(G) is induced by the inclusion A → G. It follows from Poincarè-Birkhoff-Witt theorem that U(G) is a free S(A)-module, i.e. E-property holds.

Second example deals with torsion-free nilpotent groups. Let A be a central subgroup of a f.g. torsion-free nilpotent group G. If a_{1}, …, a_{k} is a free basis of A (as a Z-module) then we take R = ZG and φ : S(A) → ZG is induced by the mapping which carries each a_{i} ∈ A to a_{i} − 1 ∈ ZG. The E-property easily follows (cf. (4)). (Note that the flatness of R over S(A) also can be proved).

Theorem 3. Let (A, R, φ) be an E-triple. Suppose that there is a finite free resolution of an \overline{R}-module L of the kind

0 ⟶ \overline{R} ⊗ B_{m} ⟶^{\overline{d}} … ⟶^{\overline{d}} \overline{R} ⊗ B_{0} ⟶^{\overline{π}} L → 0  (11)

Here \overline{d} is a differential and B_{0}, …, B_{m} are finitely generated free Zmodules.

Then there exists a finite free resolution of L over R of the kind

0 ⟶ R ⊗ C_{n} ⟶^{d} … ⟶^{d} R ⊗ C_{0} ⟶^{π} L → 0  (12)

where C_{n} = ⨁_{p+q=n} Λ^{q}A ⊗ B_{p} and π is the composition

R ⊗ C_{0} = R ⊗ B_{0} ⟶^{μ⊗1} \overline{R} ⊗ B_{0} ⟶^{\overline{π}} L

(for a suitable differential d depending on \overline{d}). Proof. The proof of this theorem is quite analogous to that of theorem 1 ( we used only the properties of an E-triple).

Consequently, we obtain Theorem 4. Let G be a f.g. torsion-free nilpotent group and r = cdG be the Hirsch number of G. Then there exists a free Z-module M of rank r and a finite free resolution of Z over ZG of the kind

0 ⟶ ZG ⊗ Λ^{r}M ⟶^{d} … ⟶^{d} ZG ⟶^{π} Z → 0.  (13)

Moreover, we can choose M to be G – the Lie ring associated to the central series of G consisting of the isolators of γ_{i}G in G.

Proof. It is known that G admits a central series

1 = G_{l+1} ⊂ G_{l} ⊂ … ⊂ G_{1} = G

with torsion-free abelian factors G_{i}/G_{i+1}, and

r = ∑_{i=1}^{l} rk_{Z}G_{i}/G_{i+1}

is independent of the choice of such a central series.

Let A = G_{l}. Then (A, ZG, φ) is an E-triple as it was described in the second example before the theorem 3. We may assume by induction hypothesis that there is a resolution of desired type for the quotient group \overline{G} = G/A. Thus, theorem 3 provides such a resolution for a group G. Note that we can choose a module M to be A ⊕ \overline{M} where \overline{M} had been chosen for \overline{G}.

Consider now a special kind of a the central series

1 = \sqrt{γ_{c+1}G} ⊂ \sqrt{γ_{c}G} ⊂ … ⊂ \sqrt{γ_{1}G} = G.  (14)

Here \sqrt{γ_{i}G} = \{g ∈ G|g^{m} ∈ γ_{i}G for some m ≥ 1\} is the isolator of γ_{i}G in G. In fact, (14) is an N-series, i.e. [\sqrt{γ_{i}G}, \sqrt{γ_{j}G}] ⊂ \sqrt{γ_{i+j}G} for all i, j. The associated Lie ring

G = ⨁_{i≥1} \sqrt{γ_{i}G}/\sqrt{γ_{i+1}G}

is a free Z-module of rank r.

Let A be \sqrt{γ_{c}G} and \overline{G} be G/A. Since

\sqrt{γ_{i}\overline{G}} = \sqrt{γ_{i}G}/A

we obtain the isomorphism of Z-modules G ≅ A ⊕ \overline{G}. By induction hypothesis we may choose \overline{M} to be \overline{G} and the theorem is proved.

Remark. Since Z-rank of M is equal to Hirsch number of G that is cdG ([Bi]), the resolution (13) has minimal possible length.

3 Example: a resolution for a free twostep nilpotent group

In this section we give an explicit construction of a free finite resolution for a free 2-step nilpotent group G.

3.1. Presentations of G and G . Let G = G(r) be a free nilpotent group on r generators x_{1}, …, x_{r}. Consider the following presentation of G:

G =< x_{i}, c_{ij}; 1 ≤ i < j ≤ r | c_{ij} = [x_{i}, x_{j}], c_{ij} are central >,

where [a, b] = a^{−1}b^{−1}ab. Corresponding Lie ring G = G(r) has the similar presentation

G =< x_{i}, c_{ij}; 1 ≤ i < j ≤ r | c_{ij} = [x_{i}, x_{j}], c_{ij} are central > .

Remark. In fact, it is more convenient to use the same notation for generators and basic commutators of G and G. In few doubtful cases we’ll always mention what object is considered.

In what follows we will use a geometric interpretation of generators x_{k} and commutators c_{ij}. Let Δ = Δ(r) be a standard simplex on r vertices embedded in the euclidian space R^{r}, i.e.

Δ = {(a_{1}, …, a_{r}) ∈ R^{r}| ∑_{i=1}^{r} a_{i} = 1, a_{i} ≥ 0}.

Let the generator x_{i} be represented by the vertex v_{i} = (0…. , 1, …, 0) (1 is standing on the i-th position) and the commutator c_{ij} - by the directed edge e_{ij}, i < j, issuing from v_{i} and terminating in v_{j} (in coordinates e_{ij} = (0, …, −1, 0, …, 1, 0, …) ). If v ∈ Δ is a vertex (or e is a directed edge) then we denote by x(v) (c(e)) the generator (commutator) corresponding to v (resp. e).

Next we order the set of x_{i} according to their indices, i.e. x_{1} < … < x_{r}. We’ll say that c_{ij} < c_{mk} if and only if either j < k or j = k and i < m. Also c_{ij} < x_{k} for all i, j, k. Due to the above correspondence the sets of vertices and edges of Δ are ordered, too.

3.2. Trees in Δ and their labels. First of all, let T be a maximal tree in the 1-skeleton of Δ. Evidently, T has r vertices and r − 1 edges. Let e_{1}(T) < … < e_{r−1}(T) be vectors corresponding to the directed edges of the tree T. It’s not hard to see that they form a basis of the subspace of R^{r} consisting of all vectors with the zero sum of coordinates. If we add to the above vectors some vector e_{0} = (a_{1}, …, a_{r}) such that a_{1} + … + a_{r} ≠ 0 we obtain a basis for R^{r}. Let δ = det(e_{0}, e_{1}(T), …, e_{r−1}(T)) be determinant composed from the coordinates of vectors e_{0}, e_{1}(T), …, e_{r−1}(T). It can be proved by induction on r that δ = (−1)^{ε(T)}(a_{1} + … + a_{r}) where ε(T) = 0 or 1 and depends only on the tree T. By definition, ε(T) is said to be the parity of the tree T.

Let V = {v_{s_{1}}, …, v_{s_{p}}} be an ordered subset of vertices. Consider a face Γ(V) of the simplex Δ spanned by V. Γ(V) and Δ(p) are isomorphic as simplicial sets: vertex v_{s_{j}} is in a correspondence with v_{j}. To a maximal tree T ∈ Γ(V) corresponds a tree T′ ∈ Δ(p). By definition, ε(T) = ε(T′). Let’s denote by max(V) the set of all maximal trees in Γ(V). One knows (due to Cayley and others) that the number of maximal trees in max(V) is equal to p^{p−2}.

Let T be, as earlier, a maximal tree in Γ(V). Since T is maximal then any complementary edge e in Γ(V) defines a circuit Z = Z(e) in the 1-skeleton of Γ(V). We orient Z according to the direction pointed out by the senior edge of Z (recall that we have ordered all commutators, i.e. edges of Δ). If the orientation of Z given by e is the same (resp. opposite) then we call e positive relative to T (resp. negative).

Let’s define now the label u(T) ∈ G of the tree T as

u(T) = ∏_{e>0} c(e),

where the product is taken over all positive edges relative to T. Note that u(T) belongs to the centre of G.

Examples. Let V = {v_{1}, …, v_{r}} and T be the star tree with edges e_{1j}, j = 2, …, r. Then all complementary edges are positive and u(T) = ∏_{1<i<j≤r} c_{ij}. On the contrary, if T is the non-branched tree

with edges e_{i,i+1}, i = 1, …, r − 1 then all complementary edges are negative and u(T) = 1.

[figure: •; v_{2}; v_{1}; v_{3}; …; v_{r−1}; v_{r}; •; •; •; •; …; v_{1}; v_{2}; v_{3}; v_{r−1}; v_{r}; Pic.2 Examples of trees.]

3.3. Parity of partition. Let V be a finite ordered set, e.g. V = {1, …, n}. Let’s suppose that V is decomposed in a disjoint union of its components V = V_{1} ⊔ … ⊔ V_{h} where the order of subsets is fixed. Consider a shuffle of elements of V corresponding to such a partition, i.e. a permutation of V such that first we write down the elements of V_{1} (in the increasing order), next the elements of V_{2} and so on. By definition, the parity of the partition V = V_{1} ⊔ … ⊔ V_{h} is the parity of this permutation. We denote it by ε(V_{1}, …, V_{h}). If P is a subset of V then we denote ε(P, V \ P) simply by ε(P) (when it is no doubt about V). As usual, |P | denotes the number of elements of P.

In the sequel we’ll use also the following notations. Let V be a subset of vertices of Δ. We denote by X(V) the exterior product in Λ(G) of the elements x_{s} corresponding to vertices of V (written in the increasing order). For example, if V = {v_{1}, v_{3}, v_{4}} then X(V) = x_{1} ∧ x_{3} ∧ x_{4}. Analogously, we denote by C(E) the exterior product of basic commutators corresponding to an ordered set E of directed edges. In particular, for any tree T ∈ Δ (rather for the set of edges of T) the element C(T) = c_{1}(T) ∧ … ∧ c_{s−1}(T) is defined. The typical basic element of Λ(G) is consequently C(E) ∧ X(V).

Finitely we denote by x(V) the product of generators x(v), v ∈ V, in the group G written in the decreasing order (for V in the above example, x(V) = x_{4}x_{3}x_{1}).

3.4. Let’s return to a resolution ZG ⊗ Λ^{∗}G with the differential d = d_{0} + … + d_{r} constructed in the theorem 1. In our notations the application of d_{0} can be written as follows:

d_{0}(1 ⊗ C(E) ∧ X(V)) = ∑_{e∈E}(−1)^{ε(e)}(c(e) − 1) ⊗ C(E \ e) ∧ X(V)

(elements of the type 1 ⊗ C(E) ∧ X(V) represent a basis of free Gmodule ZG ⊗ Λ^{∗}G.

Let’s describe d_{1}, …, d_{r}. Recall that it is enough to determine these G-morphisms only on the elements of the type 1 ⊗ X(V). Proposition 5. In the above notations

d_{1}(1 ⊗ X(V)) = ∑_{v∈V}(−1)^{ε(v)}(x(v) − 1) ⊗ X(V \ v).

If s ≥ 2 then

d_{s}(1⊗X(V)) = ∑_{{P ∈V, |V |=s \atop T ∈max(P)}}(−1)^{ε(P)+ε(T)+s−1}x(P)u(T)⊗C(T)∧X(V \P).

In particular, if V = {v_{s_{1}}, …, v_{s_{p}}}

d_{2}(1 ⊗ X(V)) = ∑_{P ={v_{s_{i}},v_{s_{j}} }}(−1)^{i+j}x_{s_{j}}x_{s_{i}} ⊗ c_{s_{i}s_{j}} ∧ X(V \ P).

Proof. Roughly speaking, d_{s} removes step by step all subsets P of s vertices from V and inserts instead s − 1 commutators ”spanning” a maximal tree in Γ(P) (each tree contributes a summand). We can also write

d_{s}(1 ⊗ X(V)) = ∑_{P ∈V, |P |=s}(−1)^{ε(P)}d_{s}(1 ⊗ X(P)) ∧ X(V \ P). (15)

Our aim is to check for s ≥ 2 the equality

d_{0}d_{s}(1 ⊗ X(V)) = − ∑_{i=1}^{s−1} d_{i}d_{s−i}(1 ⊗ X(V)).  (16)

In view of (15) it is not hard to see that we may restrict ourselves to the case |V | = s and moreover, V = {v_{1}, …, v_{s}} (it may be done by direct computations but one should be careful with signs).

Consider the left-hand side of (16). It is equal to

∑_{{T ∈max(V ) \atop e∈T^{1}}}(−1)^{ε(T)+ε(e,T^{1}\e)+s−1}x(V)u(T)(c(e) − 1) ⊗ C(T \ e).

What happens to a tree T when a fixed edge e is taken off? Evidently, T just splits into 2 disjoint subtrees T_{1} and T_{2} (in order to fix indices of trees we’ll always assume that the maximal vertex v_{s} ∈ T_{2}). Suppose that T_{1} has i vertices, then T_{2} has exactly s − i vertices and both T_{1}, T_{2} define unique subsets P_{1}, P_{2} of its vertices.

Consider now the right-hand side of (16). In the full expression of it we can find 2 summands corresponding to the pair T_{1}, T_{2}. One of them is equal to

\begin{matrix}(−1)^{ε(T_{2})+ε(T_{1})+ε(P_{2})+s−i}w(T_{2})w(T_{1}) ⊗ C(T_{2}) ∧ C(T_{1}) = \\ = (−1)^{ε(T_{2})+ε(T_{1})+ε(P_{2})+s−i+(s−i−1)(i−1)}w(T_{2})w(T_{1}) ⊗ C(T_{1}) ∧ C(T_{2}) \end{matrix}

( if a tree Q has only one vertex v then we denote by w(Q) the element x(v) − 1 ∈ ZG else w(Q) = x(P)u(Q) ∈ G), the second

(−1)^{ε(T_{1})+ε(T_{2})+ε(P_{1})+i}w(T_{1})w(T_{2}) ⊗ C(T_{1}) ∧ C(T_{2}).

Let

μ = μ(T_{1}, T_{2}) = ε(T_{1}) + ε(T_{2}) + ε(P_{1}) + i.

It follows from the definition of signs that

ε(T_{2}) + ε(T_{1}) + ε(P_{2}) + s − i + (s − i − 1)(i − 1) ≡ μ − 1 (mod 2).

Thus, the equality (16) turns into

\begin{matrix}∑_{{T ∈max(V ) \atop e∈T^{1}}}(−1)^{ε(T)+ε(e,T^{1}\e)+s−1}w(T)(c(e) − 1) ⊗ C(T \ e) = \\ = ∑_{T_{1}, T_{2}}(−1)^{μ}(w(T_{1})w(T_{2}) − w(T_{2})w(T_{1})) ⊗ C(T_{1}) ∧ C(T_{2}). (17) \end{matrix}

For a fixed pair of disjoint trees T_{1}, T_{2} we denote by E_{+}(T_{1}, T_{2}) (resp. E_{−}(T_{1}, T_{2})) the set of directed edges going from T_{1} to T_{2} (resp. from T_{2} to T_{1}) and

E(T_{1}, T_{2}) = E_{+}(T_{1}, T_{2}) ⊔ E_{−}(T_{1}, T_{2}).

If e ∈ E(T_{1}, T_{2}) we denote by T_{1}#_{e}T_{2} the tree obtained from T_{1} and T_{2} by joining the edge e to them. Now, the equality (17) is the consequence of the equalities (for each pair T_{1}, T_{2})

\begin{matrix}∑_{T =T_{1}#_{e}T_{2}}(−1)^{ε(T)+ε(e,T^{1}\e)+s−1}w(T_{1}#_{e}T_{2})(c(e) − 1) ⊗ C(T \ e) = \\ = (−1)^{μ}(w(T_{1})w(T_{2}) − w(T_{2})w(T_{1})) ⊗ C(T_{1}) ∧ C(T_{2}). (18) \end{matrix}

We are going to obtain a simpler form of (18). First of all, the element

(−1)^{ε(T)}C(T) = (−1)^{ε(T)}c_{1}(T) ∧ … ∧ c_{s−1}(T)

is independent from the order of the commutators in the exterior product provided the determinant δ = det(e_{0}, e_{1}(T), …, e_{r−1}(T)) which is responsible for the sign of the tree T = T_{1}#_{e}T_{2} is computed in accordance with this order (if you make the transposition of two commutators in the exterior product you have to make a transposition of corresponding lines in the determinant, and you have 2 changes of sign). Thus we may assume that c_{1}(T) = c(e) (i.e. the second line of δ is related to e) and that C(T \ e) = C(T_{1}) ∧ C(T_{2}).

Secondly, using relations in G we have

x(P_{1})x(P_{2}) = x(V) ∏_{e∈E_{+}(T_{1},T_{2})} c(e); x(P_{2})x(P_{1}) = x(V) ∏_{e∈E_{−}(T_{1},T_{2})} c(e).

Then (18) follows from the equality in ZG

\begin{matrix}∑_{T =T_{1}#_{e}T_{2}}(−1)^{ε(T)+s−1+μ}u(T_{1}#_{e}T_{2})(c(e) − 1) = \\ = u(T_{1})u(T_{2}) ∏_{e∈E_{+}(T_{1},T_{2})}c(e) − u(T_{1})u(T_{2}) ∏_{e∈E_{−}(T_{1},T_{2})}c(e).  (19) \end{matrix}

Lemma 3. If in the equality (19) the edge e belongs to the set E_{+}(T_{1}, T_{2}) then (−1)^{ε(T)+s−1+μ} = 1, otherwise (−1)^{ε(T)+s−1+μ} = −1. Proof. Recall that μ = ε(T_{1}) + ε(T_{2}) + ε(P_{1}) + i. Let e = e_{jk}. Then

det(v_{j}, e_{jk}, e_{1}(T_{1}), …, e_{1}(T_{2}), …) =
= det(v_{j}, v_{j} + e_{jk}, e_{1}(T_{1}), …, e_{1}(T_{2}), …) =
= det(v_{j}, v_{k}, e_{1}(T_{1}), …, e_{1}(T_{2}), …) = (−1)^{ε(T)}.

Applying the Laplace theorem for the decomposition of determinant along the fixed columns corresponding to vertices of T_{1} (there is only one non-trivial minor) we obtain that

(−1)^{ε(T)} = (−1)^{μ+s−1}, if v_{j} ∈ T_{1}, v_{k} ∈ T_{2}.

The remaining case is treated analogously.

Now we can rewrite (19) in the following manner

\begin{matrix}∑_{e∈E(T_{1},T_{2})}(−1)ẽu(T_{1}#_{e}T_{2})(c(e) − 1) = \\ = u(T_{1})u(T_{2}) ∏_{e∈E_{+}(T_{1},T_{2})} c(e) − u(T_{1})u(T_{2}) ∏_{e∈E_{−}(T_{1},T_{2})} c(e).  (20) \end{matrix}

where ẽ = 0 if e ∈ E_{+}(T_{1}, T_{2}) and ẽ = 1 otherwise.

Note that by virtue of the definition of u(T) each label u(T_{1}#_{e}T_{2}) may be represented as u(T_{1})u(T_{2})z(e) where z(e) is the product of commutators corresponding only to the edges f ∈ E(T_{1}, T_{2}). In other words, u(T_{1}#_{e}T_{2}) is always ”divisible” by u(T_{1})u(T_{2}). Cancelling in (20) the common factor u(T_{1})u(T_{2}) finitely we have to check the equality

∑_{e∈E(T_{1},T_{2})}(−1)ẽz(e)(c(e) − 1) = ∏_{e∈E_{+}(T_{1},T_{2})} c(e) − ∏_{e∈E_{−}(T_{1},T_{2})} c(e). (21)

Ordering of the set E(T_{1}, T_{2}). Let e, f ∈ E(T_{1}, T_{2}) and Z = Z(e, f) be the uniquely defined circuit obtained from T_{1} and T_{2} by joining e, f. We’ll say that e ≺ f if walking along positive orientation of Z we occur in the tree T_{2} after passing through the edge e. Otherwise we’ll say that e ≻ f.

Let’s show that this relation is an order relation on the set E(T_{1}, T_{2}), i.e. if e ≺ f, f ≺ h then e ≺ h. Suppose that h ≺ f. Consider the union of all edges from the circuits Z(e, f), Z(f, h) and Z(e, h). It is a graph Φ = Φ(e, f, h) whose vertices have degree 2 except two vertices of degree 3. Moreover, Φ contains no separating edges. Any edge b of Φ belongs exactly to 2 of the above 3 circuits. If e ≺ f, f ≺ h, h ≺ e, then on succesive walk in positive direction of 3 circuits, we pass through the edge b twice: one time in accordance with its direction and second time in opposite one. Let b be the senior edge of Φ (in the sense of the order for the edges of the simplex Δ). Obviously, b is the senior edge in both circuits to which it belongs. Since we choose positive orientation of a circuit according to the direction of its senior edge then we obtain a contradiction, i.e. e ≺ h.

Thus, we order all edges of E(T_{1}, T_{2}). Write them down in the increasing order: e_{1} ≺ … ≺ e_{t}. We introduce the following notations (e ∈ E(T_{1}, T_{2})):

z_{−}(e) = ∏_{{f≻e \atop f∈E_{−}(T_{1},T_{2})}} c(f); z_{+}(e) = ∏_{{f≺e \atop f∈E_{+}(T_{1},T_{2})}} c(f).

In the left-hand side of (21) we have z(e) = z_{−}(e)z_{+}(e). In fact, if e is fixed (say, e ∈ E_{+}(T_{1}, T_{2})), then c(f) is a factor of z(e) in two cases, namely, if either f ≻ e, f ∈ E_{−}(T_{1}, T_{2}) or f ≻ e, f ∈ E_{+}(T_{1}, T_{2}). In both cases f is positive relative to the tree T = T_{1}#_{e}T_{2}. We will call such a decomposition of z(e) into 2 factors polar.

The proof of the equality (21) is based on the following Lemma 4. Let

S(p) = ∑_{i=1}^{p}(−1)ẽ_{i}(z(e_{i})c(e_{i}) − z(e_{i})), p = 1, …, t.

If e_{p} ∈ E_{+}(T_{1}, T_{2}) then

S(p) = − ∏_{e∈E_{−}(T_{1},T_{2})} c(e) + z(e_{p})c(e_{p}),

else

S(p) = − ∏_{e∈E_{−}(T_{1},T_{2})} c(e) + z(e_{p}),

Proof. We proceed by induction on p. If p = 1 then we have z(e_{1}) = z_{−}(e_{1}). Suppose that we have proved the lemma for some p, and e_{p} ∈ E_{+}(T_{1}, T_{2}) (the remaining case can be treated in a similar way). If e_{p+1} ∈ E_{+}(T_{1}, T_{2}) then

S(p + 1) = − ∏_{e∈E_{−}(T_{1},T_{2})} c(e) + z(e_{p})c(e_{p}) − z(e_{p+1}) + z(e_{p+1})c(e_{p+1}),

The polar decomposition yields z(e_{p})c(e_{p}) = z(e_{p+1}), i.e. step of induction is done. The case e_{p+1} ∈ E_{−}(T_{1}, T_{2}) is left to the reader.

In any case, if p = t we get just the equality (21), i.e. the proposition 5 is proved. Q.E.D.

4 An isomorphism between H_{n}G and H_{n}G

We continue the previous notations. In the section 3 we have constructed an explicit resolution of Z over ZG. Applying to it the trivialisation functor Z ⊗_{ZG} − we obtain the chain complex (Λ^{∗}G, d) with the differential d = −d_{2} + d_{3} … + (−1)^{r−1}d_{r} (here we especially alternate the signs). The d_{s} is defined by the following properties:

\begin{matrix}1. d_{s}(C(E) ∧ X(V)) = (−1)^{|E|}C(E) ∧ d_{s}(X(V)); \\ 2. d_{s}(X(V)) = ∑_{{P ⊂V, |P |=s \atop T ∈max(P)}}(−1)^{ε(P)+ε(T)}C(T) ∧ X(V \ P). \end{matrix}

The n-th homology group of this complex is isomorphic to H_{n}G.

The homology groups H_{n}G can be computed by means of the complex (Λ^{∗}G, ∂) where

∂(z_{1} ∧ … ∧ z_{n}) = ∑_{i<j}(−1)^{i+j}[z_{i}, z_{j}] ∧ z_{1} … ẑ_{i} … ẑ_{j} … ∧ z_{n}.

It is not hard to see that ∂ = −d_{2} in the above notations.

The main goal of this section is to prove the existence of an isomorphism between H_{n}G and H_{n}G. It is sufficient to construct an automorphism F : Λ^{∗}G → Λ^{∗}G of the graded Z-module Λ^{∗}G for which the equality dF = F ∂ holds.

Note that Λ(G) is actually a bi-graded algebra:

Λ(G) = ⨁_{q,p≥0} Λ^{qp}(G),

where the elements of the type C(E) ∧ X(V), |E| = q, |V | = p, form a Z-basis of Λ^{qp}(G). Evidently, the morphism d_{s} belongs to the group Homgr^{s−1,−s}(Λ(G), Λ(G)).

We’ll look for F of the type F = 1 − f_{1} + … + (−1)^{r}f_{r} where 1 stands for the identity map and f_{k} ∈ Homgr^{k,−k}(Λ(G), Λ(G)), i.e. f_{k} maps Λ^{qp}(G) into Λ^{q+k,p−k}(G). Such an F is always an automorphism since F = 1 + N and N is nilpotent (N^{r+1} = 0).

Now we have to solve the equation

(−d_{2}+…+(−1)^{r−1}d_{r})(1−f_{1}+…+(−1)^{r}f_{r}) = (1−f_{1}+…+(−1)^{r}f_{r})(−d_{2}).

Since it is graded it splits into the following system

\{ \begin{matrix}d_{3} & = & f_{1}d_{2} − d_{2}f_{1} \\ … \\ d_{k} + d_{k−1}f_{1} + … + d_{3}f_{k−3} & = & f_{k−2}d_{2} − d_{2}f_{k−2} \\ … & & & (22) \\ d_{r} + d_{r−1}f_{1} + … + d_{3}f_{r−3} & = & f_{r−2}d_{2} − d_{2}f_{r−2}. \end{matrix}

Two equations in which f_{r−1}, f_{r} appear are of the kind 0 = 0, i.e. for fixed r f_{r−1} and f_{r} are irrelevant.

Let

Homgr(Λ(G), Λ(G)) = ⨁_{q,p} Homgr^{q,p}(Λ(G), Λ(G)).

Considering the summands with q + p ≡ 0 (mod 2) and the summands with q + p ≡ 1 (mod 2) we obtain a decomposition

Homgr(Λ(G), Λ(G)) = Homgr_{0}(Λ(G), Λ(G)) ⊕ Homgr_{1}(Λ(G), Λ(G)),

i.e. Homgr(Λ(G), Λ(G)) is a superalgebra. As usual, we call the corresponding morphisms even or odd. For example, all d_{s} are odd and f_{k} are even. Moreover, we introduce a differential D on the graded Z-module Homgr(Λ(G), Λ(G)) of total degree (−1) by the formula Df = d_{2}f − (−1)f̃fd_{2}. Note that the left-hand sides of (22) are boundaries relative D.

To get a solution of (22) we impose some restrictions on f_{k}. Definition 1. A morphism f ∈ Homgr(Λ(G), Λ(G)) is said to be a morphism of the type (0, q, p) if

1. f ∈ Homgr^{p,−p}(Λ(G), Λ(G)) ⊂ Homgr_{0}(Λ(G), Λ(G)) ;

2. f(Λ^{st}(G)) = 0 if s < q or t < p;

3. for any E,V f(C(E) ∧ X(V)) =

= ∑_{{B⊂E \atop |B|=q}} ∑_{{R⊂V \atop |R|=p}}(−1)^{μ}C(E \ B) ∧ f(C(B) ∧ X(R)) ∧ X(V \ R).

where μ = ε(E \ B) + ε(R). Definition 2. A morphism f ∈ Homgr(Λ(G), Λ(G)) is said to be a morphism of the type (1, q, p) if

1. f ∈ Homgr^{p−1,−p}(Λ(G), Λ(G)) ⊂ Homgr_{1}(Λ(G), Λ(G)) ; 2. f(Λ^{st}(G)) = 0 if s < q or t < p;

3. for any E,V f(C(E) ∧ X(V)) =

= (−1)^{|E|−q} ∑_{{B⊂E \atop |B|=q}} ∑_{{R⊂V \atop |R|=p}}(−1)^{μ}C(E \ B) ∧ f(C(B) ∧ X(R)) ∧ X(V \ R).

( μ as above).

In other words, the morphisms of the type (0, q, p) (of the type (1, q, p)) are even (resp. odd ) and are completely determined by their values on Λ^{qp}(G). In the sequel this property will be called shortly the principle of localisation. For example, d_{s} is of the type (1, 0, s) (see the beginning of this section).

The connection between these two types of morphisms is depicted in the following lemma.

Lemma 5. If f is a morphism of the type (0, q, p) then −Df = fd_{2} − d_{2}f is a morphism of the type (1, q − 1, p + 2). Proof. It can be checked by direct computations.

Let’s go back to solving the system (22). Consider a simpler system

\{ \begin{matrix}d_{3} & = & f_{(1)}^{(1)}d_{2} − d_{2}f_{(1)}^{(1)} & = & −Df_{(1)}^{(1)} \\ … & & & & & (23) \\ d_{r} & = & f_{(r−2)}^{(1)}d_{2} − d_{2}f_{(r−2)}^{(1)} & = & −Df_{(r−2)}^{(1)} \end{matrix}

Since d_{k} is of the type (1, 0, k), by lemma 5 one should look for f_{(k−2)}^{(1)} among the morphisms of the type (0, 1, k − 2). Let |V | = k. Then

d_{k}(X(V)) = ∑_{T ∈max(V)}(−1)^{ε(T)}C(T).

On the other hand,

−Df_{(k−2)}^{(1)}(X(V)) = f_{(k−2)}^{(1)}d_{2}(X(V))

since f_{(k−2)}^{(1)}(Λ^{0k}(G)) ≡ 0 by localisation principle. Again in accordance

with it the following equality is sufficient to determine f_{(k−2)}^{(1)}:

\begin{matrix}f_{(k−2)}^{(1)}d_{2}(X(V)) = \\ = ∑_{{L⊂V \atop |L|=2}}(−1)^{ε(L)}f_{(k−2)}^{(1)}(C(L) ∧ X(V \ L)) = ∑_{T ∈max(V)}(−1)^{ε(T)}C(T). \end{matrix}

Elements of the type C(L)∧X(V) = c_{ij} ∧X(P), |P | = k−2 form a basis of Λ^{1,k−2}(G). We define f_{(k−2)}^{(1)} on these basic elements as follows.

If {v_{i}, v_{j}} ∩ P ≠ ∅ then f_{(k−2)}^{(1)}(c_{ij} ∧ X(P)) = 0. For the other basic elements (they are of the type C(L) ∧ X(V \ L)), where L = {v_{i}, v_{j}}, V = L ⊔ P) the value of f_{(k−2)}^{(1)}(C(L) ∧ X(V \ L) ∈ Λ^{k−1,0}(G) may be arbitrary provided the only condition is satisfied (one condition for every fixed V):

∑_{{L⊂V \atop |L|=2}}(−1)^{ε(L)}f_{(k−2)}^{(1)}(C(L) ∧ X(V \ L)) = ∑_{T ∈max(V)}(−1)^{ε(T)}C(T).

4.2. Solution of the system (22). Let’s fix a solution of the system (23), i.e. a collection of morphisms (f_{(1)}^{(1)}, …, f_{(r−2)}^{(1)}). It turns out that a solution of (22) i.e. a collection (f_{1}, …, f_{r−2}), may be obtained in the following manner: 1.

f_{p} = ∑_{{t_{1}>...>tn≥1 \atop q_{1}t_{1}+...+qntn=p, q_{i}≥1}} f_{(t_{1},...,t_{n})}^{(q_{1},...,q_{n})}, p = 1, …, r − 2.

2. Morphisms f_{(t_{1},...,t_{n})}^{(q_{1},...,q_{n})} are of the type (0, q, p) where q = ∑_{i=1}^{n} q_{i} and

are defined inductively according to the localisation principle (|E| = q − 1, |V | = p) by the formula:

\begin{matrix}f_{(t_{1},...,t_{n})}^{(q_{1},...,q_{n})}(C(E) ∧ c ∧ X(V)) = \\ ∑_{i=1}^{n} ∑_{{R⊂V \atop |V \R|=t_{i}}}(−1)^{ν_{i}}f_{(t_{1},...,t_{i},...,t_{n})}^{(q_{1},...,q_{i}−1,...,q_{n})}(C(E) ∧ X(R)) ∧ f_{(t_{i})}^{(1)}(c ∧ X(V \ R)). \end{matrix}

Here ν_{i} = p − t_{i} + ε(R). If q_{i} = 1 for some i then we omit in the right-hand sum both indices q_{i} − 1 = 0 and t_{i}.

Note that one can parametrise the summands of f_{p} by all Young diagrams on p cells (having q_{1} lines of length t_{1},…, q_{n} lines of length t_{n}). The special role of the last commutator in the inductive definition of f = f_{(t_{1},...,t_{n})}^{(q_{1},...,q_{n})} is quite conventional (like a decomposition of a determinant along the last column). In fact, f always possesses the following property:

f(c_{1} ∧ … ∧ c_{q} ∧ X(V)) = (−1)^{sign(π)}f(c_{π(1)} ∧ … ∧ c_{π(q)} ∧ X(V)), where π ∈ Sym_{q}.

Lemma 6. The following equality is satisfied in Homgr(Λ(G), (Λ(G));

−Df_{(t_{1},...,t_{n})}^{(q_{1},...,q_{n})} = ∑_{i=1}^{n} d_{t_{i}+2}f_{(t_{1},...,t_{i},...,t_{n})}^{(q_{1},...,q_{i}−1,...,q_{n})}.  (24)

Proof. It is not hard to show that d_{k} is a morphism of the type (1, q, p + k) if f is of the type (0, q, p). Consequently, both sides of (24) are of the type (1, q − 1, p + 2) due to lemma 5. According to the localisation principle one should verify (24) only on the basic elements of Λ^{q−1,p+2}(G).

Let C(E) ∧ X(V) be a basic element of Λ^{q−1,p+2}(G). Let’s denote by f the morphism f_{(t_{1},...,t_{n})}^{(q_{1},...,q_{n})}, by f^{(i)} - f_{(t_{1},...,t_{i},...,t_{n})}^{(q_{1},...,q_{i}−1,...,q_{n})}.

On the one hand,

−Df(C(E) ∧ X(V)) = fd_{2}(C(E) ∧ X(V)) =

= ∑_{L⊂V, |L|=2}(−1)^{|E|+ε(L)}f(C(E) ∧ C(L) ∧ X(V \ L)) =

∑_{i=1}^{n} ∑_{{L⊂V \atop |L|=2}} ∑_{{R⊂V \L \atop |R|=p−t_{i}}}(−1)^{λ_{i}}f^{(i)}(C(E) ∧ X(R)) ∧ f_{(t_{i})}^{(1)}(C(L) ∧ X(V \ L \ R))

= ∑_{i=1}^{n} ∑_{{R⊂V \atop |R|=p−t_{i}}} (−1)^{μ_{i}}f^{(i)}(C(E) ∧ X(R)) ∧ f_{(t_{i})}^{(1)}d_{2}(X(V \ R)) =

= ∑_{i=1}^{n} ∑_{{R⊂V \atop |R|=p−t_{i}}} (−1)^{μ_{i}}f^{(i)}(C(E) ∧ X(R)) ∧ d_{t_{i}+2}(X(V \ R))

where λ_{i} = |E| + p − t_{i} + ε(L) + ε(R, V \ L \ R), μ_{i} = |E| + p − t_{i} + ε(R, V \ R).

On the other hand,

∑_{i=1}^{n} d_{t_{i}+2}f^{(i)}(C(E) ∧ X(V)) =

= ∑_{i=1}^{n} ∑_{{R⊂V \atop |R|=p−t_{i}}} (−1)^{ε(R,V \R)}d_{t_{i}+2}f^{(i)}(C(E) ∧ X(R)) ∧ X(V \ R) =

= ∑_{i=1}^{n} ∑_{{R⊂V \atop |R|=p−t_{i}}} (−1)^{μ_{i}}f^{(i)}(C(E) ∧ X(R)) ∧ d_{t_{i}+2}(X(V \ R))

(μ_{i} is the same as above). Proof is completed.

If we put the expression for f_{p} into the right-hand side of the p-th equation of the system (22), apply the lemma 6 and then collect in the left-hand side of the same equation all summands with the fixed factor d_{k}, k = 3, …, p + 2, we just obtain the equality

d_{p+2} + d_{p+1}f_{1} + … + d_{3}f_{p−1} = −Df_{p}.

Thus, the system (22) is solved and we have an isomorphism H_{n}G ≅ H_{n}G.

References.

[Bi] R.Bieri.Cohomological dimension of discrete groups, Queen Mary College Maths.Notes, London, 1976

[Bo] N.Bourbaki.Algèbre homologique, chap.X, Masson, 1980

[Hu] J.Huebschmann. Perturbation theory and free resolutions for nilpotent groups of class 2. J. of Algebra, v.126(2), 1989, 348-399 [L] L.A.Lambe. Cohomology of principal G-bundles over a torus when H^{∗}(BG; R) is polynomial. Bull.Soc.Math.Belg., 38, 1986. 247-264.

[Ku] Yu.V. Kuz’min. On the connection between group cohomology and Lie algebras. Russ.Math.Surveys, 37(4),1982, 123-124. [K-S] Yu.V. Kuz’min, Yu.S.Semenov. On homology of a free nilpotent group of class 2. (to appear).

[Si] St.Sigg. Laplacian and homology of free two-step nilpotent Lie algebras. J. of Algebra, v.185(1), 1996, 144-161