2 Free actions on $(n-1)$ -connected $2n$ -manifolds

In this section, we will apply the cohomological approach outlined in [Reference Adem and Hambleton1, Section 2]. The proofs of Propositions 2.1, 2.3, and 2.5 are straightforward modifications of the results there and details are omitted. We assume that Y is a closed, orientable, $(n-1)$ -connected $2n$ -manifold with the free orientation-preserving action of a finite group G; its homology has a corresponding $\mathbb Z G$ –module structure. Both $H_{2n}(Y,\mathbb Z)$ and $H_0(Y,\mathbb Z)$ are copies of the trivial module $\mathbb Z$ , whereas $H_n(Y,\mathbb Z)$ is a free abelian group with a $\mathbb Z G$ –module structure which, by Poincaré duality, must be self–dual as a $\mathbb Z G$ –module, that is, $H_n(Y,\mathbb Z)\cong H_n(Y,\mathbb Z)^*$ . We assume, here, that Y admits a finite G–CW complex structure, with cellular chain complex denoted by $C_*(Y)$ (if the action is smooth, this is always true, and holds up to G-homotopy equivalence in the topological case).

We denote by $\Omega ^r(\mathbb Z )$ the $ \mathbb Z G$ -module uniquely defined in the stable category (where $\mathbb Z G$ -modules are identified up to stabilisation by projectives) as the r–fold dimension–shift of the trivial module $\mathbb Z$ . We refer to [Reference Adem and Milgram2] and [Reference Brown3] for background on group cohomology.

Proposition 2.1. Let Y be an $(n-1)$ -connected $2n$ -manifold with a free action of a finite group G which preserves orientation. Then there is a short exact sequence in the stable category of $\mathbb Z G$ –modules of the form

(2.1) $$ \begin{align} 0\to \Omega^{n+1}(\mathbb Z)\to H_n(Y;\mathbb Z)\to \Omega^{-n-1}(\mathbb Z)\to 0 .\end{align} $$

Corollary 2.2. The short exact sequence (2.1) yields a long exact sequence in Tate cohomology

$$\begin{align*}\dots\to\widehat H^{i+n}(G,\mathbb Z)\xrightarrow{\cup\,\sigma} \widehat H^{i-n-1}(G,\mathbb Z)\to \widehat H^i(G,H_n(Y;\mathbb Z)) \to \widehat H^{i+n+1}(G,\mathbb Z) \to\dots \end{align*}$$

determined by the class $\sigma \in \widehat H^{-2n-1}(G,\mathbb Z)$ which is the image of the generator $ 1\in \widehat H^{0}(G,\mathbb Z)\cong \mathbb Z/|G|$ .

We can analyse this sequence just as was done in [Reference Adem and Hambleton1, Section 2].

Proposition 2.3. The cohomology class $\sigma \in \widehat H^{-2n-1}(G,\mathbb Z) \cong H_{2n}(G,\mathbb Z)$ can be identified with the image of the fundamental class $c_*[Y/G]$ under the homomorphism

$$ \begin{align*}c_*\colon H_{2n}(Y/G, \mathbb Z)\to H_{2n}(BG,\mathbb Z) \end{align*} $$

induced by the classifying map $c\colon Y/G\to BG$ . Under this identification, the class $\sigma $ determines the extension (2.1).

Similarly, the map $\Omega ^{n+1}(\mathbb Z)\to H_n(Y,\mathbb Z)$ defines an extension class

$$ \begin{align*}\varepsilon_Y\in H^{n+1}(G, H_n(Y,\mathbb Z))\end{align*} $$

which appears in the long exact sequence above as the image of the generator under the map $\widehat H^0(G,\mathbb Z)\to \widehat H^{n+1}(G,\mathbb Z)$ . Algebraically, this responds to mapping the canonical defining extension for $\Omega ^{n+1}(\mathbb Z)$ (identified with the extension class for the module of cycles in $C_n(Y)$ ) to the extension obtained by reducing by the module of boundaries $B_n$ :

These two extension classes are related as follows:

Proposition 2.5. Let G denote a finite group acting freely on an $(n-1)$ -connected, orientable $2n$ -manifold Y preserving orientation, then $\varepsilon _Y\ne 0$ and $|G| = \textrm {exp} (\sigma )\cdot \textrm {exp} (\varepsilon _Y)$ . The class $\varepsilon _Y$ has exponent $|G|$ if and only if $\sigma =0$ , in which case, we have a stable equivalence

$$ \begin{align*}H_n(Y,\mathbb Z) \cong \Omega^{n+1}(\mathbb Z)\oplus \Omega^{-n-1}(\mathbb Z).\end{align*} $$

We note the standard identity $\chi (Y) = 2 + (-1)^n\dim H_n(Y,\mathbb Q)$ , and the formula $|G|\chi (Y/G) = \chi (Y)$ from the covering $Y \to Y/G$ . Since the transfer map induces an isomorphism $H_i(Y/G;\mathbb Q) \cong H_i(Y;\mathbb Q)^G$ , we have $\chi (Y/G) = 2 + (-1)^n\dim H_n(Y, \mathbb Q)^G$ . In particular

$$ \begin{align*}\dim H_n(Y,\mathbb Q)^G =(-1)^n\left ( \chi (Y/G) - 2\right).\end{align*} $$

From the stable sequence

$$ \begin{align*}0\to \Omega^{-n}(\mathbb Z)\to \Omega^{n+1}(\mathbb Z)\to H_n(Y, \mathbb Z)\to 0,\end{align*} $$

we infer the existence of projective modules $Q_r$ and $Q_s$ which fit into an exact sequence

(2.2) $$ \begin{align} 0\to \Omega^{-n}(\mathbb Z)\oplus Q_s \to \Omega^{n+1}(\mathbb Z)\oplus Q_r \to H_n(Y;\mathbb Z)\to 0 ,\end{align} $$

where $Q_i \otimes \mathbb Q \cong [\mathbb Q G]^i$ for $i=r,s$ . Here, we write $\Omega ^{j+1}(\mathbb Z)$ ( $j\geq 0$ ) for the j-th kernel in a minimal projective resolution of $\mathbb Z$ , meaning a resolution:

(2.3) $$ \begin{align} 0\to \Omega^{j+1}(\mathbb Z)\to P_j \to P_{j-1}\to\dots\to P_0\to\mathbb Z\to 0 \end{align} $$

realising $\mu ^{\prime }_j(G)$ (see [Reference Swan42, p. 193]), from which we see that

$$ \begin{align*}\operatorname{\mathrm{rank}} _{\mathbb Z} \Omega^j(\mathbb Z) + (-1)^{j-1} = |G|(\mu^{\prime}_{j-1}(G)) ~~~\textrm{and}~~~ \operatorname{\mathrm{rank}} _{\mathbb Z} \Omega^j(\mathbb Z)^G +(-1)^{j-1} = \mu^{\prime}_{j-1}(G),\end{align*} $$

where $(-1)^k|G|\mu ^{\prime }_k (G)$ is precisely the minimal value over all partial Euler characteristics of a projective resolution of $\mathbb Z$ over $\mathbb Z G$ (see [Reference Swan42, Remark, p. 195]). The corresponding invariants $\mu _k(G)$ for minimal free resolutions of $\mathbb Z$ were defined by Swan (see [Reference Swan42, p. 193]).

By dualising, we see that a minimal representative for $\Omega ^{-j}(\mathbb Z)$ is given by $\Omega ^j(\mathbb Z)^*$ , the dual module. Thus, for our purposes, we have

$$ \begin{align*}\operatorname{\mathrm{rank}} _{\mathbb Z} \Omega^{n+1}(\mathbb Z)^G = \mu^{\prime}_{n}(G)+ (-1)^{n+1}, ~~\operatorname{\mathrm{rank}} _{\mathbb Z} \Omega^{-n}(\mathbb Z)^G = \mu^{\prime}_{n-1}(G)+(-1)^n.\end{align*} $$

Applying invariants after tensoring over $\mathbb Q$ to the exact sequence (2.2) yields the formula

$$ \begin{align*}\mu^{\prime}_{n}(G)+ (-1)^{n+1} + r = \mu^{\prime}_{n-1}(G)+(-1)^n + s + (-1)^n[ \chi (Y/G) - 2],\end{align*} $$

whence we obtain

$$ \begin{align*}s -r = \mu^{\prime}_n(G) - \mu^{\prime}_{n-1}(G) + (-1)^{n+1} \chi (Y/G).\end{align*} $$

Theorem 2.7. If Y is a closed, $(n-1)$ -connected $2n$ -manifold with a free orientation-preserving action of G, a finite group, then for any subgroup $H\subset G$

$$ \begin{align*}\mu^{\prime}_n(H) - \mu^{\prime}_{n-1}(H) \le (-1)^n [G:H] \chi (Y/G).\end{align*} $$

Proof. We will prove this for $H=G$ by contradiction. Assume that $s-r>0$ and form the diagram

where L is the quotient of $\Omega ^{n+1}(\mathbb Z) \oplus Q_r$ in the middle vertical exact sequence. Note that this middle vertical exact sequence splits (since L is torsion-free). Hence,

$$ \begin{align*}\Omega^{n+1}(\mathbb Z) \oplus Q_r \cong L \oplus Q_s.\end{align*} $$

By Swan [Reference Swan41, Lemma 2.1], there is a projective resolution

$$ \begin{align*}0 \to L \to P_{n}\oplus Q_r \to P_{n-1} \oplus Q_s \to P_{n-2} \to \dots \to P_0\to\mathbb Z\to 0.\end{align*} $$

Since $s>r$ , this contradicts the minimality of the resolution (2.3) realising $\mu ^{\prime }_n(G)$ . Hence, we have shown that $s-r\le 0$ . The full result follows using covering spaces.

If the pair $(G,n)$ is not exceptional, the numbers $\mu _n(G)$ can be computed using group cohomology. By a result of Swan [Reference Swan42, Proposition 6.1], the invariant $\mu _n(G)$ is the least integer greater than or equal to all the numbers

$$ \begin{align*}(\dim M)^{-1}\left ( \dim H^n(G,M) - \dim H^{n-1}(G,M) + \dots + (-1)^n \dim H^0(G,M)\right )\end{align*} $$

as M ranges over all simple $\mathbb F_pG$ –modules for all primes p dividing $|G|$ . As extending the field doesn’t change dimensions, we can take $\mathbb K_p$ , an algebraically closed field of characteristic p, and restrict attention to absolutely irreducible $\mathbb K_pG$ –modules. Next we introduce

Definition 2.9. For any discrete group G of type $F_n$ , let $e_n(G)$ denote the least integer greater than or equal to all the numbers

$$ \begin{align*}\dim H^n(G,\mathbb F) - 2 \left ( \dim H^{n-1}(G,\mathbb F) - \dim H^{n-2}(G,\mathbb F) + \dots + (-1)^{n-1}\dim H^0(G,\mathbb F)\right ),\end{align*} $$

where the coefficients range over $\mathbb F = \mathbb Q$ or $\mathbb F = \mathbb F_p$ for all primes p.

We have the following elementary inequality:

Lemma 2.11. Suppose that X is a closed, orientable $2n$ -manifold with fundamental group G of type $F_n$ whose universal cover is $(n-1)$ -connected. Then, for any subgroup $H\subset G$ of finite index

$$ \begin{align*}e_n(H) \le [G:H] (-1)^n\chi (X).\end{align*} $$

Proof. Let $\mathbb F$ denote any field of coefficients. The connectivity of the universal cover implies that

$$ \begin{align*}H^i(G,\mathbb F)\cong H^i(X,\mathbb F)~~\text{for}~~ 0\le i\le n-1\end{align*} $$

and

$$ \begin{align*}\dim H^n(G,\mathbb F)\le \dim H^n(X,\mathbb F).\end{align*} $$

By Poincaré duality, we have

$$ \begin{align*}H^k(X,\mathbb F) \cong H^{2n-k}(G,\mathbb F)~~\text{for}~~ n+1\le k \le 2n.\end{align*} $$

Combining these facts and using covering space theory, we obtain the inequality.

Applying the mod p coefficient sequence yields an attractive corollary

Corollary 2.12. If X is a closed, orientable $2n$ -manifold with finite fundamental group G whose universal cover is $(n-1)$ -connected, then for all primes p dividing $|G|$ and subgroups $H\subset G$

$$ \begin{align*}\dim H^{n+1}(H,\mathbb Z)\otimes\mathbb F_p - \dim H^n(H,\mathbb Z)\otimes\mathbb F_p \le (-1)^n([G:H]\chi (X)-2).\end{align*} $$

Proof. Since $[G:H]\chi (X)$ equals the Euler characteristic of the $[G:H]$ -fold covering of X, it is enough to do the case $H=G$ . Let $h^i(G) = \dim H^i(G;\mathbb F_p)$ . From the relations noted above, and Lemma 2.11, we have the formula

$$ \begin{align*}(-1)^n(\chi(X) -2) \geq h^n(G) - 2\sum_{i =1}^{n-1} (-1)^{i+1} h^{n-i}(G).\end{align*} $$

But by the mod p coefficient sequence, we have

$$ \begin{align*}h^i(G) = \dim H^{i+1}(G,\mathbb Z)\otimes\mathbb F_p + \dim H^i(G,\mathbb Z)\otimes\mathbb F_p, \ \ \text{ for } \ 1 \leq i \leq n.\end{align*} $$

The result follows by combining these two relations.

Applying this to any subgroup $C\subset G$ of prime order, we obtain

3 Minimal $K(G,n)$ -complexes and thickenings

We now turn our attention to the existence of orientable $2n$ -manifolds having fundamental group of type $F_n$ and $(n-1)$ -connected universal cover. We recall the following well-known construction (see Kreck and Schafer [Reference Kreck and Schafer24, Section 2]):

Proof. Let K denote a finite CW complex of dimension n with $\pi _1(K) = G$ whose universal covering is $(n-1)$ -connected. For example, take a finite, cellular model for the classifying space $BG$ , and consider its n-skeleton K. Then we can construct a smooth $2n$ -manifold $Z=M(K)$ by doubling a $2n$ -dimensional handlebody thickening of K. Thus, the universal cover $\tilde {Z}$ of $M(K)$ is an $(n-1)$ -connected, closed, orientable $2n$ -manifold with a free action of G, such that

$$ \begin{align*}\pi_n(M(K)) \cong H_n(M(K); \Lambda) \cong H^n(K; \Lambda) \oplus H_n(K; \Lambda),\end{align*} $$

where $\Lambda := \mathbb Z G$ denotes the integral group ring. Moreover, the Euler characteristic $\chi (M(K)) = 2 \chi (K)$ . A variation of this construction is to let Z denote the boundary of a regular neigbourhood, for some embedding $K \subset \mathbb R^{2n+1}$ of the finite n-complex in Euclidean space.

The chain complex $C_*(\tilde {K})$ of the universal covering of a $K(G,n)$ -complex affords a free n-step resolution of the trivial $\mathbb Z G$ –module $\mathbb Z$ . Conversely, we wish to realise a given finitely generated n-step free resolution

$$ \begin{align*}\mathscr F : F_n \to F_{n-1} \to \dots \to F_1 \to F_0 \to \mathbb Z \to 0\end{align*} $$

as the equivariant chain complex of a suitable $K(G,n)$ -complex. Note that by Swan [Reference Swan42, Theorem 1.2], we have $\mu _n(G) \leq (-1)^n\chi (\mathscr F)$ and that the lower bound is attained by some resolution.

We now consider the case $n=2$ , where the argument above fails at the first step. To establish our upper bound for $q_4(G)$ , we need a more general construction and some results of C. T. C. Wall [Reference Wall45, Reference Wall46].

In [Reference Wall45, p. 64], Wall proved that a finite complex X satisfying the D2-conditions is homotopy-equivalent to a finite $3$ -complex. We will therefore assume that all our D2-complexes have $\dim X \leq 3$ . It is not known at present whether all discrete groups have the D2-property. Note that $\mu _2(G) \leq (1- \operatorname {\mathrm {Def}}(G))$ by [Reference Swan42, Proposition 1], and equality holds if G has the D2-property.

Proposition 3.7 [Reference Hambleton12, Corollary 2.4]

Any finitely generated free resolution

$$ \begin{align*}\mathscr F : F_2 \to F_1 \to F_0 \to \mathbb Z \to 0\end{align*} $$

over $\mathbb Z G$ is chain homotopy-equivalent to $C_*(X)$ , where X is a finite $\mathrm {D2}$ -complex.

If we apply this to a minimal resolution with $\chi (\mathscr F) = \mu _2(G) = \mu ^{\prime }_2(G)$ , then if G is finite, the module $H_2(\widetilde X;\mathbb Z)$ is a minimal $\mathbb Z$ -rank representative of the stable module $\Omega ^3(\mathbb Z)$ . The following result may also be of independent interest (it applies to any finitely presented group G which is good in the sense of Freedman [Reference Freedman and Quinn9, p. 99], in particular, to poly-(finite or cyclic) groups).

For continuity, we defer the proof of this result to Appendix A.

The proof of Theorem A

Concatenating our previous results, we have obtained the estimates

$$ \begin{align*}\max \{e_n(G), \mu^{\prime}_{n}(G)-\mu^{\prime}_{n-1}(G)\}\le q_{2n}(G)\le 2 \mu_{n}(G)\end{align*} $$

for any finite group G. For the lower bound, we apply Theorem 2.7 and Lemma 2.11. For the upper bound, we apply Corollary 3.5 if $n>2$ , and Corollary 3.9 for $n=2$ .

We now prepare for the proof of Theorem B. The next result, due to Swan and Wall, shows that arbitrary periodic groups appear as fundamental groups of rational homology spheres.

Proof. We will discuss the case $k=1$ for groups of period $4$ . Swan [Reference Swan41] constructed a finitely dominated Poincaré $3$ -complex Y with $\pi _1(Y) = G$ , and Wall [Reference Wall47, Corollary 2.3.2] shows that Y is obtained from a $\mathrm {D2}$ -complex by attaching a single $3$ -cell. The chain complex $C_*(\widetilde Y)$ provides a projective resolution

$$ \begin{align*}\mathscr F' : P \to F_1 \to F_0 \to \mathbb Z \to 0\end{align*} $$

with $\chi (\mathscr F') =1$ , where P is projective, $F_1$ and $F_0$ are free and $I(G)^* =\ker d_2(\mathscr F')$ . This shows that $\mu ^{\prime }_2(G) =1$ and so $\mu _2(G) = \mu ^{\prime }_2(G) =1$ by Swan’s results.

One can give a direct argument for this last step. By adding a projective Q so that $P\oplus Q = F$ is free, we obtain a free resolution

$$ \begin{align*}\mathscr F : F \to F_1 \to F_0 \to \mathbb Z \to 0\end{align*} $$

with $I(G)^* \oplus Q = \ker d_2(\mathscr F)$ . By the ‘Roiter replacement lemma’ (see [Reference Roĭter36, Proposition 5], or [Reference Jacobinski21, Theorem 3.6]), $I(G)^* \oplus Q = J \oplus F'$ , where $F'$ is free and J is locally isomorphic to $I(G)^*$ , so $\operatorname {\mathrm {rank}}_{\mathbb Z} (J) = \operatorname {\mathrm {rank}}_{\mathbb Z}I(G)^*$ . We now divide out the image of $F'$ in F (a direct summand) to obtain a free resolution

$$ \begin{align*}\mathscr F'' : F'' \to F_1 \to F_0 \to \mathbb Z \to 0\end{align*} $$

with $J =\ker d_2(\mathscr F'')$ and $\chi (\mathscr F'') =1$ . Hence, $\mu _2(G) = 1$ .

A similar argument shows that $\mu _{2k}(G) =1$ , for all $k>1$ , if G has periodic cohomology with period dividing $2k+2$ . Details will be left to the reader.

The proof of Theorem B

By assumption, the group G is periodic of even period q. In the first case, if q divides $n+2$ , then n is even and $\mu _n(G) =1$ by Lemma 3.10. By Theorem A, we have the inequalities

$$ \begin{align*}2 \leq q_{2n}(G) \leq 2 \mu_2(G)=2,\end{align*} $$

and hence, $q_{2n}(G) =2$ .

In the second case, n is odd and the minimal Euler characteristic $q_{2n}(G)\geq 0$ by Corollary 2.13. We will show that the lower bound is realised when G is a periodic group of even period q, provided that $2q$ divides $n+1$ .

This follows from the solution of the space form problem: Madsen, Thomas and Wall [Reference Madsen, Thomas and Wall28, Theorem 1], [Reference Wall49, Corollary 12.6] proved that there exists a finite Poincaré duality complex X (called a finite Swan complex) of dimension $(2k-1)$ , with $\pi _1(X) = G$ and universal covering $\widetilde X \simeq S^{2k-1}$ , whenever $k \equiv 0 \ \pmod { e(G)}$ , where $e(G)$ is the Artin exponent of G [Reference Lam25, p. 94]. Moreover, a detailed analysis of the group cohomology of periodic groups shows that $2e(G)$ is equal to q or $2q$ , depending on the structure of its $2$ -hyperelementary subgroups (see Wall [Reference Wall49, p. 542], where the notation $2d(\pi )$ is used for the period of a periodic group $\pi $ ).

For any finite Swan complex X, there exists a degree one normal map $(f, b)\colon N \to X$ , where $N^{n}$ is a closed, topological n-manifold (see [Reference Thomas and Wall44, Corollary 3.3]). We then have a degree one normal map of pairs

$$ \begin{align*}(f\times \mathrm{id}, b\times \mathrm{id})\colon (N \times D^{n+1}, N \times S^n) \to (X \times D^{n+1}, X \times S^n).\end{align*} $$

By Wall’s ‘ $\pi $ - $\pi $ Theorem’ [Reference Wall and Ranicki50, Theorem 3.3], this normal map is normally cobordant to a homotopy equivalence of pairs. It follows that $X \times S^n$ is homotopy-equivalent to a closed, topological $2n$ -manifold. Since $X \times S^n$ has Euler characteristic zero, these examples show that $q_{2n}(G) = 0$ as required.

We conclude this section with a sample computation of the estimates for elementary abelian p-groups.

Example 3.14. If $E_k=(\mathbb Z/p\mathbb Z)^k$ then we can use the Kunneth formula to compute these invariants. The term $\mu _n(E_k)$ has a polynomial of degree n as its leading term. For $n=2, 3, 4$ , we have

$$ \begin{align*}\frac{k^2-3k+4}{2}\le q_4(E_k)\le k^2-k+2\end{align*} $$

$$ \begin{align*}\frac{k^3-3k^2+8k-12}{6}\le q_6(E_k)\le \frac{k^3+5k-6}{3}\end{align*} $$

$$ \begin{align*}\frac{k^4-2k^3+11k^2-34k+48}{24} \le q_8(E_k)\le \frac{k^4+2k^3+11k^2-14k+24}{12}\end{align*} $$

For instance, for $k=2$ , this only gives the rough estimate $1\leq q_8(E_2) \leq 6$ , but we know that $q_8(E_2) =2$ by performing surgeryFootnote ¹ on $L^7(\mathbb Z/p\mathbb Z) \times S^1$ . However, for $k=3$ , the lower bound gives $q_8(E_3) \geq 3$ , and hence, $E_3$ is not the fundamental group of a rational homology $8$ -sphere.

4 The proof of Theorem A^′

In this section, we establish a lower bound for $q_{2n}(G)$ , for G an infinite discrete group of type $F_n$ . With the results of Lemma 2.11, Corollary 3.5 and Corollary 3.9, this will complete the proof of Theorem A $'$ .

The invariants $\mu ^{\prime \prime }_k(G)$ used in the statement of Theorem A $'$ can also be defined as follows.

Definition 4.1. For $k \geq 2$ , let $\mu ^{\prime \prime }_{k}(G)=(-1)^k \cdot \min \{\chi (\mathscr F)\}$ , where $\mathscr F$ varies over all k-step resolutions

$$ \begin{align*}\mathscr F : F_k \to F_{k-1} \to \dots \to F_2 \to F_1 \to F_0 \to \mathbb Z\to 0\end{align*} $$

of $\mathbb Z$ by finitely generated free $\mathbb Z G$ -modules, which arise geometrically as the chain complex of the universal covering for a finite $CW$ -complex of dimension k with fundamental group G.

The sign $(-1)^k$ is introduced to agree with Swan’s conventions. Note the inequalities

$$ \begin{align*}\mu^{\prime}_k(G) \leq \mu_k(G) \leq \mu^{\prime\prime}_k(G)\end{align*} $$

relating these invariants to those defined by Swan. We define $\mu ^{\prime \prime }_1(G) = d(G) -1$ , where $d(G)$ denotes the minimal number of generators for G.

We now establish the lower bound for infinite groups.

Theorem 4.3. Let G be a discrete group of type $F_n$ , for $n\geq 2$ . If Y is a closed, $(n-1)$ -connected $2n$ -manifold with a free orientation-preserving action of G, a finite group, then for any subgroup $H\subset G$ of finite index

$$ \begin{align*}\mu_n(H) - \mu^{\prime\prime}_{n-1}(H) \le (-1)^n [G:H] \chi (Y/G).\end{align*} $$

Proof. It suffices to prove this inequality for $H = G$ , and then apply covering space theory. Let M denote a closed, orientable, $2n$ -dimensional manifold with $n \geq 2$ and fundamental group G of type $F_n$ , such that $\pi _i(M) = 0 $ for $1 < i <n$ . Let $K \simeq M$ be a finite $CW$ -complex homotopy-equivalent to M, and let $C:= C(K; \Lambda ) = C(\widetilde K)$ denote the chain complex of its universal covering. It is a finite chain complex, with each $C_i$ a finitely generated free $\mathbb Z G$ -module. We note that the homology of M is computed from the chain complex $C \otimes _{\mathbb Z G}\mathbb Z$ , and therefore, $\chi (M) = \sum _{i=0}^{2n} (-1)^i c_i$ , where $c_i := \operatorname {\mathrm {rank}}_{\mathbb Z G} C_i$ .

We may assume that the $(n-1)$ -skeleton $K^{(n-1)}\subset K$ has $(-1)^{n-1}\chi (\widetilde K^{(n-1)}) = \mu ^{\prime \prime }_{n-1}(G)$ , by applying Wall’s construction of a normal form to replace K by a homotopy-equivalent complex if necessary (see [Reference Wall47, p. 238]).

The long exact sequences of the triples $(K, K^{(i)}, K^{(i-1)})$ , for cohomology with $\mathbb Z G$ -coefficients gives:

$$ \begin{align*}0 \to H^{i}(K, K^{(i-1)}) \to H^{i}(K^{(i)}, K^{(i-1)}) \to H^{i +1}(K, K^{(i)}) \to H^{i +1}(K, K^{(i-1)}) \to 0.\end{align*} $$

If we let $Z^i := \ker \delta ^{i}$ and $B^i := \operatorname {\mathrm {im}} \delta ^{i-1}$ (for later use) in the cochain complex $(C^*, \delta ^*)$ , where $C^i = \operatorname {\mathrm {Hom}}_{\Lambda }(C_i, \Lambda )$ , then the sequence above becomes

$$ \begin{align*}0 \to Z^i \to C_i^* \to Z^{i+1} \to H^{i+1}(C) \to 0.\end{align*} $$

Since $H^i(C) = H_{2n-i}(C) = 0$ , for $n+1 \leq i \leq 2n-1$ , and $H^{2n}(C) = \mathbb Z$ , we can splice the short exact sequences

$$ \begin{align*}0 \to Z^i \to C_i^* \to Z^{i+1} \to 0\end{align*} $$

for $n \leq i \leq 2n-1$ , and obtain a long exact sequence

$$ \begin{align*}0 \to Z^n \to C_n^* \to C^*_{n+1} \to \dots \to C^*_{2n-1} \to C^*_{2n} \to \mathbb Z \to 0.\end{align*} $$

Since this a resolution of $\mathbb Z$ by finitely generated $\mathbb Z G$ -modules, with $\operatorname {\mathrm {rank}}_{\mathbb Z G} (C_i^*) := c_i$ , we have

$$ \begin{align*}(-1)^n\chi_{upper}(C) := (-1)^n\sum_{i=n}^{2n} (-1)^{i} c_i \geq \mu_{n}(G).\end{align*} $$

On the other hand, by the normal form construction, we have

$$ \begin{align*}(-1)^n\chi_{lower}(C) : = (-1)^n\sum_{i=0}^{n-1} (-1)^i c_i = (-1)^n(-1)^{n-1} \mu^{\prime\prime}_{n-1}(G) = -\mu^{\prime\prime}_{n-1}(G).\end{align*} $$

Therefore, $q_{2n}(G) \geq (-1) ^n\chi (M) \geq \mu _{n}(G) -\mu ^{\prime \prime }_{n-1}(G)$ , as required.

5 Rational homology 4–spheres

We now specialise our results to the case when M is a rational homology $4$ –sphere with finite fundamental group G. We would like to find restrictions on G by computing $\mu _2(G)-\mu _1(G)$ . Note that $\mu _1(G)=\mu _1'(G)$ and $\mu _2(G)=\mu _2'(G)$ , and that for any solvable finite group, $\mu _1(G)= d(G)-1$ [Reference Cossey, Gruenberg and Kovács6, Proposition 1]. Let A denote a finite abelian group minimally generated by d elements, then using Theorem 2.7, we have the estimate

$$ \begin{align*}\mu_2(A)-\mu_1(A)=\frac{d^2-3d+4}{2}\le \chi (M)=2\end{align*} $$

and so we recover the estimate proved in [Reference Teichner43, 3.4]:

Our next objective will be to consider examples where twisted coefficients can be used to establish conditions for nonabelian groups. Recall the result due to Swan [Reference Swan42, Theorem 1.2 and Proposition 6.1]: for any finite group G, $\mu _n(G)$ is the smallest integer which is an upper bound on

$$ \begin{align*}(\dim M)^{-1}\left ( h^n(G,M) - h^{n-1}(G,M) + \dots + (-1)^n h^0(G,M)\right ),\end{align*} $$

where K is a field, M is a $KG$ -module and $h^n(G;M) := \dim _KH^n(G; M)$ . Moreover, we can assume that K is algebraically closed and has characteristic p dividing $|G|$ , and it suffices to verify the upper bound on absolutely irreducible modules.

We now focus on an interesting class of nonabelian groups for which the absolutely irreducible modules are easy to determine. The following proposition follows from elementary representation theory (see [Reference Webb52, Corollary 6.2.2]).

The cyclic group C acts on the vector spaces $H^i(E_k; \mathbb K_p)$ via one-dimensional characters $\alpha \colon C \to \mathbb K_p^{\times }$ . Using the multiplicative structure in cohomology and the Bockstein, this is determined by $N_k=H^1(E_k;\mathbb K_p)\cong \operatorname {\mathrm {Hom}} (E_k, \mathbb K_p)$ as an $\mathbb K_p[C]$ –module.

Recall that by [Reference Adem and Milgram2, Corollary II.4.3, Theorem II.4.4], the mod p cohomology ring of $E_k$ is given by

$$ \begin{align*}H^*(E_k, \mathbb F_p)\cong \begin{cases} \mathbb F_2 [x_1, \dots , x_k] & \text{for } p=2\\ \Lambda (x_1, \dots, x_k)\otimes \mathbb F_p [y_1, \dots, y_k] & \text{for } p \text{ odd} \end{cases}, \end{align*} $$

where $x_1, \dots , x_k\in H^1(E_k, \mathbb F_p)$ , $y_1,\dots ,y_k\in H^2(E_k, \mathbb F_p)$ and $\Lambda (x_1, \dots , x_k)$ denotes the exterior algebra on these one-dimensional generators. Moreover, if we let $B\colon H^1(E_k, \mathbb F_p)\to H^2(E_k, \mathbb F_p)$ denote the Bockstein, then we can assume that for p odd $B(x_i)= y_i$ , whereas for $p=2$ , $B(x_i)=x_i^2$ for all $i=1, \dots , k$ . By extending coefficients, we obtain the same structure for $H^*(E_k, \mathbb K_p)$ .

The map B is compatible with respect to the C action and defines an isomorphism onto its image, thus giving rise to an exact sequence

$$ \begin{align*}0\to N_k\to H^2(E_k;\mathbb K_p)\to \Lambda^2(N_k)\to 0\end{align*} $$

as $\mathbb K_pC$ –modules. If $N_k\cong \bigoplus _{1\le i\le k} L(\alpha _i)$ then $\Lambda ^2(N_k)\cong \bigoplus _{1\le i<j\le k} L(\alpha _i\alpha _j)$ . Note that if we tensor the sequence with any other character and take C–invariants, it will still be exact, as $(|C|,p)=1$ .

Using the fact that for any $\mathbb K_pU_k$ –module M, $H^t(U_k; M) \cong H^t(E_k; M_{|_E})^C$ for every $t\ge 0$ , for any character $L(\beta )$ , we obtain the formula

$$ \begin{align*}h^2(U_k;L(\beta)) - h^1(U_k;L(\beta)) + h^0(U_k;L(\beta)) = \dim [\Lambda^2(N_k)\otimes\beta]^C + \dim L(\beta)^C.\end{align*} $$

At primes q dividing $ |C|$ , we work over the field $\mathbb K_q$ , and note that $H^i(U_k, L)= H^i(C, L^{E_k})$ . Hence, an absolutely irreducible L with some $h^i(U_k,L)\ne 0$ must also have a trivial action of $E_k$ . Arguing, as before, L is the inflation of a character $C/O_q(C)\to \mathbb K_q^{\times }$ . Thus, we have $H^i(U_k, L)= [H^i(O_q(C), \mathbb K_q)\otimes L]^{C/O_q(C)}$ . As $O_q(C)$ is cyclic, all these terms are isomorphic and of nonzero rank (equal to one) if and only if the action of $C/O_q(C)$ is trivial, and we obtain that

$$ \begin{align*}h^2(U_k;L) - h^1(U_k;L) + h^0(U_k;L)\le 1.\end{align*} $$

We apply our analysis to obtain a calculation for $\mu _2(U_k)$ and $e_2(U_k)$ :

Proposition 5.4. For $U_k = E_k\times _T C$ as above, with $N_k=H^1(E_k; \mathbb K_p)$ ,

$$ \begin{align*}\mu_2(U_k)=\max \{\dim [\Lambda^2(N_k)\otimes L(\beta)]^C + \dim L(\beta)^C\}\end{align*} $$

as $L(\beta )$ ranges over all characters $\beta \colon C \to \mathbb K_p^{\times }$ , and

$$ \begin{align*}e_2(U_k) = \dim \Lambda^2(N_k)^C - \dim N_k^C + 2.\end{align*} $$

We apply this to the special case when p is an odd prime and the action on $N_k=H^1(E_k, \mathbb K_p)$ is isotypic, that is it is the direct sum of copies of a fixed character $L(\alpha )$ .

We now apply our estimate to homology 4–spheres.

Proof. For the groups $U_k= E_k\times _TC$ , where $x^{q^2}\ne x$ for $x\in E_k$ , we consider the inequality

$$ \begin{align*}\max\{2, \frac{k(k-3)}{2}\} \le q_4(U_k) \le k(k-1).\end{align*} $$

If a rational homology 4–sphere X with fundamental group $U_k$ exists, then $q_4(U_k)=2$ and we’d have $\frac {k(k-3)}{2} \le 2$ , which implies that $k\le 4$ . Note that the upper bound implies the existence of a rational homology 4–sphere with fundamental group $U_2$ . In the case when $q=p-1$ , our estimate 5.6 shows that $2< q_4(U_k)$ for all $k>1$ .

Example 5.9. Let $\mathbb F_q$ denote a field with $q=2^k$ elements. Then the cyclic group of units $C=\mathbb Z/(q-1)\mathbb Z$ acts transitively on the nonzero elements of the underlying mod 2 vector space $E_k=(\mathbb Z/2\mathbb Z)^k$ . If we write $\mathbb F_q = \mathbb F_2[u]/(p(u))$ , where $p(u)$ is an irreducible polynomial of degree k over $\mathbb F_2$ , then the action can be described as multiplication by u. Expressing it in terms of the basis $\{1,u,\dots ,u^{k-1}\}$ , we obtain a faithful representation $C\to GL(k,\mathbb F_2)$ with characteristic polynomial $p(t)$ . This gives rise to a semidirect product $J_k= E_k\times _TC$ , where the action of C on $N_k=H^1(E_k,\mathbb K_2)$ decomposes into nontrivial, distinct characters determined by the roots of $p(t)$ . If $\alpha $ is a root of this polynomial, so are all the powers $\{\alpha ^{2^i}\}_{i=0,\dots ,k-1}$ , and these appear as a complete set of eigenvalues for the action on the k–dimensional vector space. In other words, we have $N_k\cong \bigoplus _{0\le i\le k-1} L(\alpha ^{2^i})$ . We propose to compute the invariants $\mu _2(J_k)$ and $e_2(J_k)$ .

Proposition 5.10. For the groups $J_k$ described above, we have

1. (i) $\mu _2(J_2) = 2$ , whereas $\mu _2(J_k)=1$ for all $k>2$ .

2. (ii) $e_2(J_2)=3$ , whereas $e_2(J_k)=2$ for all $k>2$ .

Proof. As $N_k\cong \bigoplus _{0\le i\le k-1} L(\alpha ^{2^i})$ , we have that $\Lambda ^2(N_k)\cong \bigoplus _{0\le i<j\le k-1} L(\alpha ^{2^i+2^j})$ . For $k>2$ , this is a sum of distinct, nontrivial characters. This follows from the fact that for $k>2$ ,

$$ \begin{align*}2^k-1 = 2^{k-1}+2^{k-2} +\dots + 2+1>2^i+2^j,\end{align*} $$

and each $\alpha ^{2^i+2^j}$ is a distinct, nontrivial $2^k-1$ root of unity. Hence, if $k>2$ , the module $\Lambda ^2(N_k)$ has no trivial summands and no repeated summands. Now, if we take any character $L(\beta )$ , we see that at most one summand in $\Lambda ^2(N_k)\otimes L(\beta )$ can be trivial. And if this occurs, then $\beta \ne 1$ . Hence, applying the formula in Proposition 5.4, we conclude that $\mu _2(J_k)=1$ . Similarly, we see that $e_2(J_k)=2$ for all $k>2$ . Also $\Lambda ^2(N_2)\cong L(1)$ , whence we see that $\mu _2(J_2)=2$ and $e_2(J_2)=3$ .

From these examples, we conclude that there exist rational homology 4–spheres with fundamental group equal to $J_k$ for $k>2$ , and so groups of arbitrarily high rank can occur as such groups, in contrast to the situation for abelian groups appearing in Corollary 5.1. For $J_2\cong A_4$ , the alternating group on four letters, we have $e_2(J_2)=3$ , $\mu _2(J_2)=2$ , and so $3\le q_4( A_4)\le 4$ . The cohomological computations also imply that $\mu _4( A_4)=1$ , whence, there does exist a rational homology $8$ –sphere with fundamental group $ A_4$ .

Proof. By the estimates above, for $G= A_4$ , we only need to rule out $q_4(G) = 3$ , so suppose that there exists $M^4$ with $\pi _1(M) = A_4$ and $\chi (M) = 3$ . Applying the universal coefficient theorem, we can use the computation at $p=2$ (see [Reference Adem and Milgram2, Theorem 1.3, Chapter III]) to show that $H_4(A_4;\mathbb Z) = 0$ . Hence, $\widehat H^{-5}(A_4,\mathbb Z)=0$ and applying Proposition 2.5, we infer that $ \pi _2(M)$ is stably isomorphic to $J \oplus J^*$ , where J denotes a minimal representative of $\Omega ^3(\mathbb Z)$ . Since $\chi (M) = 3$ , we have $H^0(G; \pi _2(M)) = \mathbb Z$ . From the exact sequence for Tate cohomology [Reference Brown3, Chapter IV.4], we have a surjection $\mathbb Z = H^0(G;\pi _2(M)) \twoheadrightarrow \widehat H^0(G;\pi _2(M))$ . However,

$$ \begin{align*}\widehat H^0(G;\pi_2(M)) = \widehat H^0(G;J \oplus J^*) = \widehat H^{-3}(G;\mathbb Z)\oplus \widehat H^{3}(G;\mathbb Z) \cong H_2(G;\mathbb Z) \oplus H_2(G;\mathbb Z)\end{align*} $$

and since $H_2(G;\mathbb Z) = \mathbb Z/{2}\mathbb Z$ , this is impossible. For $G= A_5$ , we apply the fact that for every nonperiodic finite subgroup G of $SO(3)$ , $\mu _2(G)=2$ (see Remark 6.3). The rest of the argument is analogous to that for $A_4$ , since the restriction map $H^*(A_5;\mathbb Z/2\mathbb Z) \to H^*(A_4; \mathbb Z/2\mathbb Z)$ is an isomorphism. This is true because both groups share the same $2$ –Sylow subgroup $(\mathbb Z/2\mathbb Z)^2$ , with normaliser $A_4$ (see [Reference Adem and Milgram2, Theorem 6.8, Chapter II]). This implies that $e_2(A_5)=3$ , $H_4(A_5;\mathbb Z) = 0$ and $H_2(A_5;\mathbb Z) = \mathbb Z/{2}\mathbb Z$ (note that the other two p–Sylow subgroups are cyclic, so don’t contribute to even degree homology). Therefore, we can rule out $q_4(A_5)=3$ whence $q_4(A_5)=4$ .

6 Some further remarks and questions in dimension four

In this section, we will briefly discuss some questions about rational homology $4$ -spheres whose fundamental groups are finite.

Section 6A. Existence via surgery. The main open problem is to characterise the finite groups G for which $q_4(G)=2$ . To make progress, we need more constructions of rational homology $4$ -spheres.

Examples of $\mathbb Q S^4$ -manifolds can be constructed by starting with a rational homology $3$ -sphere X, forming the product $X \times S^1$ , and then doing surgery on an embedded $S^1 \times D^3 \subset X \times S^1$ representing a generator of $\pi _1(S^1) = \mathbb Z$ . This construction is equivalent to the ‘thickened double’ construction $Z = M(K)$ for a finite $2$ -complex of Proposition 3.1 (compare [Reference Hambleton and Kreck13, Section 4]).

Since the quotient of a free finite group action on a rational homology $3$ -sphere is again a rational homology $3$ -sphere, one could use the examples $X=Y/G$ studied by [Reference Adem and Hambleton1], where Y is a $\mathbb Q S^3$ and G is a finite group acting freely on Y. However, to obtain a $\mathbb Q S^4$ with finite fundamental group by this construction, Y must, itself, have finite fundamental group.

Section 6B. Groups of deficiency zero. There are many finite groups with deficiency zero: for example, Wamsley [Reference Wamsley51] showed that a metacyclic group G with $H_2(G;\mathbb Z) =0$ has $\operatorname {\mathrm {Def}}(G) = 0$ . In particular, the class of finite groups arising as fundamental groups of rational homology 4–spheres includes groups with periodic cohomology of arbitrarily high period. There is an extensive literature on this problem: for example, see [Reference Campbell and Robertson4, Reference Campbell and Robertson5, Reference Cossey, Gruenberg and Kovács6, Reference Epstein8, Reference Johnson and Wamsley22, Reference Mennicke and Neumann30, Reference Neumann33, Reference Sag and Wamsley37, Reference Sag and Wamsley38, Reference Searby and Wamsley39].

According to Swan, $1\leq \mu _2(G) \leq 1 - \operatorname {\mathrm {Def}}(G)$ (see [Reference Swan42, Proposition 1, Corollary 1.3]), hence, if G is a finite group of deficiency zero, we have $\mu _2(G)=1$ . Thus, for such groups by Proposition 3.1, we can construct an orientable 4–manifold M with $\pi _1(M)=G$ and $\chi (M)=2$ . More generally, this can be done whenever $\mu _2(G) =1$ by Theorem B (see Corollary 3.9 and the series of groups $J_k$ considered in Example 5.9). Then M is a rational homology 4–sphere, and in these cases, there is a minimal representative J for the stable module $\Omega ^3(\mathbb Z)$ with $\operatorname {\mathrm {rank}}_{\mathbb Z}(J) = |G|-1$ (compare [Reference Hambleton and Kreck13, Corollary 4.4]). For example, if G is the fundamental group of a closed, oriented $3$ -manifold, then $J \cong I(G)^*$ .

Section 6C. Algebraic questions. For the rational homology $4$ -spheres M with $\pi _1(M)=G$ constructed in Theorem B, we have $\pi _2(M) = H_2(\widetilde M;\mathbb Z) = J \oplus J^*$ , where J is a minimal representative for $\Omega ^3(\mathbb Z)$ over $\mathbb Z G$ , with $\operatorname {\mathrm {rank}}_{\mathbb Z}(J) = |G|-1$ .

Moreover, J is locally, and hence, rationally isomorphic to the augmentation ideal $I(G)$ , and the equivariant intersection form $s_M$ on $\pi _2(M) = J \oplus J^*$ is metabolic, with totally isotropic submodule $0 \oplus J^*$ . Similar results hold for the higher-dimensional examples constructed in Proposition 3.1.

More generally, for any finite group G, the existence of a representative J for the stable module $\Omega ^3(\mathbb Z)$ with $\operatorname {\mathrm {rank}}_{\mathbb Z}(J) = |G|-1$ is equivalent to the condition $\mu _2(G) =1$ (see Proposition 3.7).

For any closed, oriented $4$ -manifold M with finite fundamental group G, we have seen in 2.3 that $\pi _2(M)$ is stably given by an extension of $\Omega ^{-3}(\mathbb Z)$ by $\Omega ^3(\mathbb Z)$ (see also [Reference Hambleton and Kreck13, Proposition 2.4]) and that the extension class in $\operatorname {\mathrm {Ext}}^1_{\mathbb Z G}(\Omega ^{-3}(\mathbb Z), \Omega ^{3}(\mathbb Z)) \cong H_4(G; \mathbb Z)$ is given by the image of the fundamental class of M. For any rational homology $4$ -sphere M with finite fundamental group G, the condition $\chi (M) =2$ implies that $\operatorname {\mathrm {rank}} _{\mathbb Z} (\pi _2(M)) = 2(|G| -1)$ and $H^0(G; \pi _2(M)) = 0$ .

Finally, we point out that many questions in the representation theory of finite groups can be investigated by induction and restriction to proper subgroups. At present, we do not see how to apply this technique in our setting.

7 Appendix A: The Proof of Theorem 3.8

In this section, we give a direct construction of the minimal $4$ -manifold needed for Theorem 3.8. The idea is to use a handlebody thickening (see Definition 7.4) of a finite $2$ -complex K instead of starting with an embedding of K in $\mathbb R^5$ . The advantage of this thickening is that we can identify the intersection form of its $4$ -manifold boundary, and then apply a recent refinement of Freedman’s work due to Teichner, Powell and Ray (see [Reference Powell, Ray and Teichner35, Corollary 1.4]).

Section 7A. Metabolic forms. To analyse the intersection form of the handlebody thickening, we will need some algebraic preparations.

Definition 7.1. Let $(E, [q])$ denote a quadratic metabolic form on a $\Lambda $ -module $E = N \oplus N^*$ , where N is a left $\Lambda $ -module and $N^*$ inherits a left $\Lambda $ -module structure via the standard anti-involution $ a \mapsto \bar a$ on $\Lambda = \mathbb Z G$ . Then

$$ \begin{align*}q((x, \phi), (x', \phi')) = \phi(x') + g(\phi, \phi'),\end{align*} $$

where $x, x' \in N$ , $\phi , \phi ' \in N^*$ and $g \in \operatorname {\mathrm {Hom}}(N^* \otimes N^*, \Lambda )$ is a sesquilinear form. We use the notation $(E, [q]) = \operatorname {\mathrm {Met}}(N, g)$ for this metabolic form (see [Reference Hambleton and Kreck14, Section 2] for metabolic forms defined on a nonsplit extension of N and $N^*$ ).

The associated Hermitian form $h = q + q^*$ is nonsingular, and $N \oplus 0 \subset E$ is a totally isotropic direct summand. More explicitly,

$$ \begin{align*}h((x, \phi), (x', \phi')) = \phi(x') + \overline{\phi'(x)} + g(\phi, \phi') + \overline{g(\phi', \phi)}.\end{align*} $$

In our geometric setting, the metabolic forms arise on modules $E = H^2(K) \oplus H_2(K)$ , where K is a finite $2$ -complex with fundamental group G (take coefficients in $\Lambda = \mathbb Z G$ ). If G is finite, then $H^2(K; \Lambda ) \cong \operatorname {\mathrm {Hom}}_{\Lambda }(H_2(K), \Lambda )$ , and the definition above applies. If G is infinite, then we slightly generalise our notion of metabolic form.

Definition 7.2. Let $E = N \oplus \widehat N$ , and let $\alpha \colon \widehat N\to N^*$ be a $\Lambda $ -module homomorphism. Define a generalised metabolic form $(E, [q]):= \operatorname {\mathrm {Met}}(N, \widehat N, \alpha , g)$ by the formula

$$ \begin{align*}q((x, \phi), (x', \phi')) = \alpha(\phi)(x') + g( \phi, \phi'),\end{align*} $$

where $x, x' \in N$ , $\phi , \phi ' \in \widehat N$ , and $g \in \operatorname {\mathrm {Hom}}(\widehat N \otimes \widehat N, \Lambda )$ is a given sesquilinear form.

Here are some preliminary remarks.

• • Let $(E, [q])$ be any quadratic form, and suppose that U is a finitely generated submodule on which the restriction $\lambda _0$ of $\lambda = q + q*$ to U is nonsingular. Then there is is orthogonal splitting $(E, [q]) \cong U \perp L$ .

  Proof. Consider the following sequence

  $$ \begin{align*}0 \to U \to E \xrightarrow{\operatorname{\mathrm{ad}} \lambda} E^* \to U^* \to 0,\end{align*} $$

  where the composition $\operatorname {\mathrm {ad}} \lambda _0\colon U \to U^*$ is an isomorphism by assumption. Therefore, the inclusion $U \subset E$ is a split injection, and $E = U \perp L$ , where $L := U^{\perp }$ . To check this last point, note that a splitting map for the inclusion $i \colon U \to E$ is given by

  $$ \begin{align*}r := (\operatorname{\mathrm{ad}} \lambda_0)^{-1} \circ i^* \circ \operatorname{\mathrm{ad}} \lambda \circ i.\end{align*} $$

  For $e \in E$ , we compute

  $$ \begin{align*}\hphantom{xxxxx} \lambda(e - i(r(e)), i(h)) = \lambda (e, i(h)) - \lambda (i(r(e)), i(h)) = \lambda (e, i(h)) - \lambda_0(r(e), h) = 0\end{align*} $$

  after substituting the formula for r. Therefore, $E = U + U^{\perp }$ , and $U \cap U^{\perp } = 0$ since $\lambda _0$ is nonsingular.

• • Let $(E, [q]) = \operatorname {\mathrm {Met}}(H,g)$ be a metabolic quadratic form on $E = H\oplus H^*$ , where $H = \Lambda ^r$ is a finitely generated free $\Lambda $ -module. Then $(E, [q]) \cong H(\Lambda ^r)$ is a hyperbolic form.

  Proof. This is a standard fact (see [Reference Wall and Ranicki50, Lemma 5.3]).

Section 7B. A handlebody thickening. Let K be a finite $2$ -complex with $\pi _1(K) = G$ . We construct a suitable thickening of K to be used in the proof of Theorem 3.8.

Then $ M:= \partial N(K)$ has the intersection form $\lambda _M =\operatorname {\mathrm {Met}}(H_2(K), g)$ , since $H_2(\partial N(K)) = H^2(K) \oplus H_2(K)$ and the direct summand $H^2(K)$ is totally isotropic (compare [Reference Kreck and Schafer24, Section 2]). All the homology groups have coefficients in $\Lambda := \mathbb Z G$ .

Remark 7.5. Note that the quadratic intersection form $\operatorname {\mathrm {Met}}(H_2(K), g)$ is a generalised metabolic form (see Example 7.3). It is nonsingular if $\pi _1(K) = G$ is a finite group. If G is infinite, this form has radical $H^2(G; \Lambda )$ , and the cokernel of its adjoint is $H^3(G; \Lambda )$ by the exact sequence

$$ \begin{align*}0 \to H^2(G;\Lambda) \to H^2(M;\Lambda) \xrightarrow{\operatorname{\mathrm{ad}} \lambda_M} \operatorname{\mathrm{Hom}}_{\Lambda}(H_2(M), \Lambda) \to H^3(G;\Lambda) \to 0\end{align*} $$

arising from the universal coefficient theorem.

Section 7C. A self-homotopy equivalence. Let $N(K)_r := N(K) \natural \, r(S^2 \times D^3)$ denote this new thickening of $K \vee r(S^2)$ . We recall the construction of a useful homotopy self-equivalence of $K \vee r(S^2)$ .

Proof. We recall some of the notation from [Reference Hambleton12, Section 2]. There is an identification

(7.1) $$ \begin{align} \pi_2(K \vee r(S^2)) \cong \pi_2(K) \oplus \Lambda^r\cong \pi_2(X) \oplus C_3(X) \oplus F, \end{align} $$

and we fix free $\Lambda $ -bases $\{e_1, \dots , e_r\}$ for $C_3(X) \cong \Lambda ^r$ and $\{f_1, \dots , f_r\}$ for $F \cong \Lambda ^r$ . The same notation $\{e_i\}$ and $\{f_j\}$ will also be used for continuous maps $S^2 \to K \vee r(S^2)$ in the homotopy classes of $ \pi _2(K \vee r(S^2))$ defined by these basis elements. Notice that the maps $f_j\colon S^2 \to K \vee r(S^2)$ may be chosen to represent the inclusions of the $S^2$ wedge factors.

An examination of the proof of [Reference Hambleton12, Lemma 2.1] shows that the simple homotopy equivalence $f\colon X \vee r(S^2) \simeq K$ is obtained by extending a certain simple homotopy equivalence $h \colon K \vee r(S^2) \to K \vee r(S^2)$ over the (stabilised) inclusion

by attaching the $3$ -cells of X in domain by the maps $e_i =[\partial D^3_i]$ , $1\leq i \leq r$ , and $3$ -cells in the range via the maps $f_i =[\partial D^3_i]$ , $1\leq i \leq r$ , which homotopically cancel the $S^2$ wedge factors, to obtain a complex $K' \simeq K$ .

Then we have $h \circ [\partial D^3_i ]= f_i$ by the construction of h (see [Reference Hambleton12, p. 364]). Hence, we can extend h over X by the identity on the $3$ -cells attached in domain and range along the maps $\{f_i\colon S^2 \to K \vee r(S^2)\}$ . We obtain a map

$$ \begin{align*}h'\colon X\vee r(S^2) \to K' := K \vee r(S^2)\cup \bigcup\{ D^3_i : [\partial D^3_i] = f_i, 1\leq i \leq r\}\end{align*} $$

extending h. From the construction of the map h (see [Reference Hambleton12, p. 364]), it follows that $h'$ is a (simple) homotopy equivalence, which induces a simple homotopy equivalence $f\colon X \vee r(S^2) \simeq K$ , after composition with the obvious projection $K' \to K$ .

Section 7D. Topological surgery. We will now apply some results of topological surgery due to Freedman. Recall that $N(K)$ is the $5$ -dimensional thickening of K constructed above and $N(K)_r = N(K) \natural \, r(S^2 \times D^3)$ is its stabilisation. We have introduced the notation $M = \partial N(K)$ , and let $M_r = \partial N(K)_r = \partial N(K) \# r(S^2 \times S^2)$ .

We now combine these ingredients. Recall that $X \vee r(S^2) \simeq K$ , so that $H_2(K) \cong H_2(X)\oplus H$ , where $H \cong \Lambda ^r$ . We have the isomorphism

(7.2) $$ \begin{align} H_2(N(K)_r) = H_2(K \vee r(S^2)) \cong H_2(K) \oplus F\cong H_2(X) \oplus H \oplus F, \end{align} $$

where $F\cong \Lambda ^r$ . We fix free $\Lambda $ -bases $\{e_1, \dots , e_r\}$ for $H \cong \Lambda ^r$ , and $\{f_1, \dots , f_r\}$ for $F \cong \Lambda ^r$ . It follows that $M_r := \partial N(K)_r$ has intersection form

$$ \begin{align*}\lambda_{M_r} = \lambda_{M} \oplus H(F) = \operatorname{\mathrm{Met}}(H_2(K), g) \oplus H(\Lambda^r),\end{align*} $$

where the classes $\{f_1, f_2, \dots , f_r\}$ and their duals provide a standard hyperbolic base for the second summand $H(F)$ . By construction, $h_*(e_i) = f_i$ , $h_*(f_i) = e_i$ for $1 \leq i \leq r$ , and $h_*(x) = x$ for all $x \in H_2(X)$ . Note that $H^2(K) = H^2(X)\oplus H^*$ is totally isotropic under $\lambda _M$ , and orthogonal to, the summand $H(F)$ .

Proof. We have the decomposition:

$$ \begin{align*}H_2(M_r) = H^2(X) \oplus H_2(X) \oplus H \oplus H^*\oplus F\oplus F^*\end{align*} $$

in the notation introduced in (7.2).

The metabolic intersection form $\lambda _{M_r} = \lambda _{M} \oplus H(F)$ admits a self-isometry $\beta _*$ (induced from the map $\beta $ constructed in Lemma 7.7) extending the map $h_*\colon H_2(K) \oplus \Lambda ^r \to H_2(K) \oplus \Lambda ^r$ constructed above. Since the images of the basis elements $h_*(e_i) = f_i \in F$ have dual classes $f^*_i \in F^*$ , it follows that

(7.3) $$ \begin{align} \lambda_{M_r}(\beta_*(f^*_i), e_j) = \lambda_{M_r}(\beta_*(f^*_i), \beta_*(f_j)) = \lambda_{M_r}(f^*_i, f_j) = \delta_{ij}. \end{align} $$

Similarly, we have the formulas

(7.4) $$ \begin{align} \lambda_{M_r}(\beta_*(f^*_i), \beta_*(f^*_j)) = \lambda_{M_r}(f^*_i, f^*_j) = 0, \ \text{ for all } \ 1\leq i, j \leq r, \end{align} $$

and

(7.5) $$ \begin{align} \lambda_{M_r}(e_i, e_j) = \lambda_{M_r}(\beta_*(f_i),\beta_*( f_j)) = \lambda_{M_r}(f_i,f_j)= 0, \ \text{ for all } \ 1\leq i, j \leq r. \end{align} $$

Let $U = \langle \beta _*(F^*); H; F\oplus F^*\rangle $ denote the submodule of $H_2(M_r)$ generated by $H(F) = F \oplus F^*$ , together with the classes $\{\beta _*(f^*_i)\}$ , and the classes $\{e_i\}$ , for $1\leq i \leq r$ . Then we claim that $U \cong \beta _*(F^*) \oplus H \oplus H(F)$ is a free direct summand of $H_2(M_r)$ , with indicated basis elements, on which the restriction of $\lambda _{M_r}$ is a nonsingular form.

We check that $U \cong \beta _*(F^*) \oplus H \oplus F\oplus F^*$ is a free submodule (of rank $4r$ ) in $H_2(M_r)$ by first showing that

$$ \begin{align*}\beta_*(F^*) \cap (H \oplus F \oplus F^*) = 0.\end{align*} $$

We then observe that the restriction $\lambda _U$ of the intersection form to $U \subset H_2(M_r)$ is nonsingular. It follows that $(U, \lambda _U)$ is an orthogonal direct summand of $(H_2(M_r), \lambda _{M_r})$ .

Here are the details: suppose that $u \in \beta _*(F^*) \cap (H \oplus F \oplus F^*)$ . We can express

$$ \begin{align*}u = \sum a_i \beta_*(f_i^*) = \sum b_i e_i +\sum c_i f_i + \sum d_i f_i^*\end{align*} $$

as $\Lambda $ -linear combinations of the basis elements. Then by the formula (7.3) above, we have $ \lambda _{M_r}(\beta _*(f^*_i), e_j) = \delta _{ij}$ , and hence, $\lambda _{M_r}(u, e_i) = a_i$ . Since H is totally isotropic by (7.5), the summand $H \oplus F \oplus F^*$ is orthogonal to H, and it follows that $\lambda _{M_r}(u, e_i) = 0$ . Hence, all the $a_i$ are zero and $u=0$ .

Now let $\lambda _U$ denote the restriction of $\lambda _{M_r}$ to U. The submodule $H \oplus F$ is a totally isotropic-based free direct summand of rank $2r$ in U, and the dual basis elements under $\lambda _U$ form the basis of the complementary direct summand $\beta _*(F^*) \oplus F^*$ . Hence, $\lambda _U$ is nonsingular, and in fact, $\lambda _U \cong \operatorname {\mathrm {Met}}(H \oplus F, g)$ , where g encodes the intersections of $\beta _*(F^*)$ with $F^*$ (which may be nonzero). In this situation, it follows that $\lambda _U \cong H(\Lambda ^{2r})$ is isomorphic to a nonsingular hyperbolic form (see [Reference Wall and Ranicki50, Lemma 5.3]).

Hence, there is a splitting for the intersection form

$$ \begin{align*}\lambda_{M_r} = \operatorname{\mathrm{Met}}(H_2(K), g) \perp H(F) \cong (E, \lambda_0)\perp \lambda_U\end{align*} $$

with respect to the orthogonal complement $(E, \lambda _0)=(\lambda _U)^{\perp }$ . Since M has good fundamental group and $\lambda _{M_r}$ contains the hyperbolic subform

$$ \begin{align*}\lambda_U \cong H(\Lambda^{2r}) \cong H(\Lambda^r) \perp H(F),\end{align*} $$

topological surgery [Reference Powell, Ray and Teichner35, Corollary 1.4] shows that $M \approx M_0 \operatorname {\mathrm {\sharp }} 2r(S^2 \times S^2)$ . The resulting closed, topological $4$ -manifold $M_0$ has $\chi (M_0) = 2 \chi (X)$ .

The construction of the manifold $M(X):= M_0$ completes the proof of Theorem 3.8.