Homological Wideness

In this short chapter, we introduce a new topological notion: the action of a finite group G on a manifold is called K-homologically wide if the first homology group with coefficients in K contains the regular representation of G; this is in some sense complementary to the notion of homological triviality that is well studied in algebraic topology. We study how homological wideness behaves under reduction modulo primes. We also relate homological wideness to the previous conditions (∗) and (∗∗) for realisability of certain wreath products as covering groups of manifolds.

1 The Notion of Homological Wideness

Definition 8.1.1

Suppose G is a finite group acting (freely or not) on a closed connected (topological) manifold M. Let K denote a field. We say the action of G is K-homologically wide if the K-homology representation h _K = h of G, given by the induced action on the first homology group

$$\displaystyle \begin{aligned} h_K \colon G \rightarrow \operatorname{Aut}(\operatorname{H}_1(M,K)) \end{aligned} $$

(8.1)

contains the regular representation of G.

Recall that, for a ring R, an R-module M is called cyclic if there exists a cyclic vector m ∈ M, i.e., a vector such that Rm = M. The regular representation is a cyclic G-module (in this case, any element g ∈ G is a cyclic vector, since the vector space span of the orbit G ⋅ g spans the entire representation space), hence another way to formulate homological wideness is as follows: the action of G on M is homologically wide if and only if there exists a class $\omega \in \operatorname {H}_1(M,K)$ such that the orbit G ⋅ ω spans a vector space of dimension |G| inside $\operatorname {H}_1(M,K)$. Indeed, if the regular representation is contained in the homology representation, just take a cyclic vector for that subrepresentation. Conversely, if such ω exists, then all g ⋅ ω for g ∈ G are linearly independent, and hence span a copy of the regular representation.

Lemma 8.1.2

If the action of G on M is F _ℓ -homologically wide, then condition (∗∗), and hence condition (∗) for ℓ coprime to |G|, holds for any subgroup H ₁ in G.

Proof

It suffices to show that for any H ₁ ≤ G, $\operatorname {Ind}_{H_1}^G \mathbf {1}$ is a subrepresentation of the regular representation. For that, it suffices to prove that the multiplicity of any irreducible G-representation ρ in $\operatorname {Ind}_{H_1}^G \mathbf {1}$ is less than or equal to $\dim \rho$, the multiplicity of ρ in the regular representation. By Frobenius reciprocity, we compute that the multiplicity of ρ in $\operatorname {Ind}_{H_1}^G \mathbf {1}$ is

$$\displaystyle \begin{aligned} \langle \rho, \operatorname{Ind}_{H_1}^G \mathbf{1} \rangle = \langle \operatorname{Res}_{H_1}^G \rho, \mathbf{1} \rangle = \frac{1}{|H_1|} \sum_{h_1 \in H_1} \operatorname{tr}(\rho(h_1)) \leq \frac{1}{|H_1|} |H_1| \dim \rho, \end{aligned}$$

since the trace of ρ(h ₁) is a sum of $\dim \rho$ roots of unity (as ρ(h ₁) is of finite order). □

2 The Notion of Q-Homological Wideness

We first relate Q-homological wideness (that has a transparent geometric meaning in terms of cycles) to F _ℓ-homological wideness (that is used in the proof of the main result). If the action of G on M is Q-homologically wide, there exists a non-torsion homology class $\omega \in \operatorname {H}_1(M,\mathbf {Q}) = \operatorname {H}_1(M,\mathbf {Z})\otimes \mathbf {Q}$ such that {gω} is linearly independent over Q. Fix an integer N such that $\omega ':=N\omega \in \operatorname {H}_1(M,\mathbf {Z})$; then {gω′} is a set of Z-independent non-torsion homology classes for M. These classes will remain linearly independent modulo infinitely many ℓ. In fact, we can use representation theory to say more.

Lemma 8.2.1

If the action of G on M is Q -homologically wide, then it is F _ℓ -homologically wide for all ℓ coprime to |G|.

Proof

This follows from the basic theory of modular representations in “good” characteristics. In this proof, we write “R–mod” for the category of finitely generated modules over a ring R.

If $\mathscr M$ is a Q[G]-module, then it is a Q _ℓ[G]-module (with Q _ℓ the field of ℓ-adic numbers). We let K ⊇Q _ℓ denote a splitting field for all irreducible representations of G; it suffices to assume that K contains all m-th roots of unity where m runs over all orders of elements of G. Let R ⊇Z _ℓ denote the ring of integers of K and $\mathfrak m$ its maximal ideal with residue field $k:=R/\mathfrak m$.

Fixing any R[G]-lattice $\mathscr M'$ in $\mathscr M$, we have a reduction map modulo $\mathfrak m$, producing a k[G]-module $\overline {\mathscr M'} = \mathscr M' \otimes k = \mathscr M'/\mathfrak m \mathscr M'$. The decomposition map

$$\displaystyle \begin{aligned} d \colon K_0(K[G]\mathrm{-mod}) \rightarrow K_0(k[G]\mathrm{-mod} ) \colon [\mathscr M] \rightarrow [\overline{\mathscr M'}] \end{aligned}$$

is an isomorphism for ℓ coprime to |G| and an effective map (i.e., positive integral combinations map to positive combinations) (see, e.g., [88, §15.5]; another formulation says that if ℓ is coprime to |G|, the Brauer character of the reduction of a G-representation modulo $\mathfrak m$ equals the character of the original representation, see, e.g., [54, Thm. 15.8]). By assumption, any irreducible (K-)representation of G occurs as direct summand in $\operatorname {H}_1(M,\mathbf {Z}) \otimes K$ (with multiplicity its dimension), and hence also every k-irreducible representation occurs as direct summand in $\operatorname {H}_1(M,\mathbf {Z}) \otimes k$ (with the same multiplicity). Since the regular representation Q[G] is defined over Q, we also find the regular representation F _ℓ[G] as direct summand in $\operatorname {H}_1(M,\mathbf {Z}) \otimes {\mathbf {F}}_\ell$. □

Example 8.2.2

For $G={\mathbf {Z}}/{2{\mathbf {Z}}} = \langle \left ( \begin {smallmatrix} 1 \$ 0 \\ 3 \$ -1 \end {smallmatrix} \right ) \rangle$ acting on $\mathscr M = {\mathbf {Z}}^2$, (1, 0) is a cyclic vector over Q but not over F ₃ (so ℓ = 3 is excluded by the reasoning before the lemma). The proof of the lemma does not imply that an integral cyclic vector is a cyclic vector modulo ℓ; just that if one exists, then one exists modulo ℓ, as long as ℓ is coprime to |G|. In the example, (1, 1) is a cyclic vector over both Q and F ₃. $\lozenge$