Homology of groups

Homological Finiteness Properties of Groups

8.1 Homology of groups

8

For a not-necessarily-commutative ring Λ with 1 = 0 the tensor product B⊗^ΛAof a rightΛ-moduleB and a leftΛ-moduleAhas the structure of an abelian group; it is generated by elements of the formb⊗awhereb∈B and a∈Asubject to bilinearity relations of the formbλ⊗a=b⊗λawhereλ∈Λ.

IfΛis commutative, the left action ofΛonB⊗ΛAdefined byλ(b⊗a) =bλ⊗a makesB⊗ΛAinto aΛ-module.

When dealing with the caseΛ=RGwe must elaborate on this. In general, RGis not a commutative ring; butRis commutative, soB⊗RGAhas a natural R-module structure, and this will always be understood in what follows.² It is often convenient to abbreviate⊗RG to⊗G.

The left action of GonR defined byg.1 = 1 for allg ∈GmakesR into anRG-module. Unless we say otherwise, it is thistrivial action which is to be understood when we regardRas anRG-module.

Afree RG-resolution ofR is an exact sequence

· · · →F2

∂₂

−→F1

∂₁

−→F0

−→ R→0

of leftRG-modules in which eachFiis free. Associated with this free resolution is the chain complex · · · −→^∂2 F1

∂₁

−→ F0 → 0, whose homology modules in positive dimensions are trivial. Denoting this by (F, ∂), we see thatinduces an isomorphismH0(F)→R. In the spirit of Sect. 2.9, one thinks of the free resolution as the augmentation of the chain complex (F, ∂).

The basic fact about free resolutions, sometimes called the “Fundamental Lemma of Homological Algebra,” is that they are unique up to chain homotopy equivalence. Versions of this purely algebraic theorem can be found in standard textbooks (e.g., [29, Chap. 1, Sect. 7] or [83, Chap. 4, Sect. 4]). We will state one such version without proof:

Theorem 8.1.1.Let

· · · →F1→F0

−→ R→0 and

· · · →F₁→F₀ −→ R→0

be free RG-resolutions of R. For every isomorphism φ : H0(F) → H0(F) there is a chain homotopy equivalence{φi:Fi→F_i}inducingφ, and{φi}is

unique up to chain homotopy.

By convention there is a “canonical” choice for φin applications of 8.1.1, namelyH0(F)−→^∗ R

−1∗

−→H0(F).

LetM be a rightRG-module. Thehomology R-modules of Gwith coeffi- cients in M are computed from theR-chain complex

2 The point is that whenR =Q, for example, we want to think ofB⊗QGA as a Q-vector space, not just as an abelian group.

8.1 Homology of groups 183

· · · →M ⊗GF2 id⊗∂₂

−→ M ⊗GF1 id⊗∂₁

−→ M⊗GF0→0.

They are denotedH_∗(G, M). By 8.1.1 (and the convention which follows it) there is no ambiguity in this definition.

These definitions and Theorem 8.1.1 are motivated by topological ideas.

Let (X, v) be an oriented pointed CW complex whose universal cover is ( ˜X,v).˜ Let G =π₁(X, v). By 3.2.9 (see also 3.3.3) we may regard ˜X as a free left G-CW complex. Letp: ( ˜X,v)˜ →(X, v) be the covering projection. For each n-celleⁿ_αofX, pick ann-cell ˜eⁿ_αof ˜X such thatp(˜eⁿ_α) =eⁿ_α; in particular, pick

˜

v forv. Orient ˜eⁿ_α so that p|: ˜eⁿ_α →eⁿ_α is orientation preserving. Eachn-cell of ˜X overeⁿ_α has the form g˜eⁿ_α, whereg ∈Gis unique. The CW complex ˜X is to be oriented so that the covering transformationsx→g.xare orientation preserving on cells.

For eachg∈Gand eachn, the covering transformationx→g.xinduces an isomorphismg#:Cn( ˜X;R)→Cn( ˜X;R). Thus we have a leftRG-module structure onC_n( ˜X;R) given byg.c=g_#(c).

Proposition 8.1.2.The oriented n-cells ˜eⁿ_α of X˜ freely generate C_n( ˜X;R) as anRG-module. The boundary∂:Cn( ˜X;R)→Cn−1( ˜X;R)is a homomor- phism of left RG-modules. If X is aspherical, this gives a freeRG-resolution ofR:

· · · −−−−−→^∂3 C₂( ˜X;R) −−−−−→^∂2 C₁( ˜X;R) −−−−−→^∂1 C₀( ˜X;R) −−−−−→ R −−−−−→ 0

whereis defined by

α,g

m_α,gg˜e⁰_α

=

α,g

m_α,g.

Note that the underlying R-chain complex of this free resolution is the augmented cellular chain complex C_∗( ˜X;R) and is the augmentation; so freeRG-resolutions ofRarise in topology as the augmented chain complexes of the universal covers ofK(G,1)-complexes. The uniqueness up toRG-chain homotopy of such resolutions follows from the algebraic Proposition 8.1.1, but this can also be seen topologically:

Proposition 8.1.3.Let(X, x)and(Y, y)beK(G,1)-complexes, with pointed universal covers ( ˜X,x)˜ and ( ˜Y ,y). Let the groups˜ π₁(X, x) and π₁(Y, y) be identified withGvia given isomorphisms so that C_∗( ˜X;R)andC_∗( ˜Y;R)are RG-chain complexes. Then these RG-chain complexes are canonically chain homotopy equivalent.

Proof. By hypothesis, there is a given isomorphismφ: π1(X, x) →π1(Y, y) inducing id : G → G. By 7.1.7, there is a cellular homotopy equivalence f : (X, x)→(Y, y) inducingφ, and f is unique up to pointed homotopy. Let k: (Y, y)→(X, x) be a cellular homotopy inverse forf. LetF :k◦f id_X

andK:f◦kidY be cellular homotopies relative to the base pointsxand y. By repeated use of 3.3.4, one finds cellular homotopies ˜F : ˜k◦f˜idX˜ and K˜ : ˜f ◦k˜ idY˜ relative to the base points ˜xand ˜y; here, ˜F : ˜X ×I →X˜ and ˜K : ˜Y ×I→Y˜ are lifts ofF and K. Hence the chain maps ˜f# and ˜k#

induced by ˜f and ˜k, and the chain homotopies ˜DF and ˜DK induced by ˜F and K˜ (as described in 2.7.14) are homomorphisms ofRG-modules. Thus ˜f# is a chain homotopy equivalence ofRG-chain complexes.³ The chain homotopy equivalence constructed in the last proof induces f˜_∗:H₀( ˜X;R)→H₀( ˜Y;R). By 8.1.1, it is unique up to chain homotopy.

Proposition 8.1.4.Let (X, v) be a K(G,1)-complex. Then H_∗(G, R) ∼= H_∗(X;R).

Proof. The chain complex (R⊗GC_∗( ˜X;R),id⊗∂) is isomorphic⁴to the chain

complex (C_∗(X;R), ∂).

Even when one knows that a free resolution “comes from” a K(G,1)- complex in the sense of 8.1.2, it is sometimes easier to describe the free reso- lution than the complex. For example, letG=Zn=t|tⁿ, the cyclic group of ordern. LetN = 1 +t+· · ·+tⁿ⁻¹∈RG. Consider:

· · · −−−−→^t−1 RG −−−−→^N RG −−−−→^t−1 RG −−−−→ R −−−−→ 0.

Here,t−1 andNdenote multiplication by those elements, and _n−1

i=0

mitⁱ

=

n−1

i=0

mi. Obviously this sequence is exact, hence it is a free RG-resolution of R. Applying the functor R⊗G·we find thatH_∗(G, R) is calculated from

· · · −−−−→⁰ R −−−−→^×n R −−−−→⁰ R −−−−→ 0.

Thus Hk(G, R)∼=R/(n) when k is odd, and Hk(G, R) ∼={r∈ R |nr = 0} whenkis even.

By taking R=Z, and applying 8.1.4, we conclude:

Proposition 8.1.5.For n≥2,Zn has infinite geometric dimension.

This plugs the gap in the proof of 7.2.12.

Indeed, there is a K(Zn,1)-complex (X, v) such that C_∗( ˜X;R) R is a freeRZn-resolution ofR, but some careful work is needed to describe the attaching maps. The skeleta of suchK(Zn,1)-complexes are calledgeneralized lens spaces. For the details, see [42].

3 The word “canonical” indicates that the chain homotopy equivalence is deter- mined up to chain homotopy by the hypotheses.

4 Compare with 13.2.1.