Update: (9/25) I just found a nice paper by Andrew Ranicki explaining the algebraic interpretation of the finiteness obstruction.

This is the second piece of a two-part post trying to understand some of the ideas in Wall’s “Finiteness conditions for CW complexes.”

In the previous post, we considered a space ${X}$ which was a homotopy retract of an ${\leq N}$-dimensional finite CW complex (where ${N \geq 3}$), and tried to express ${X}$ itself as homotopy equivalent to one such. We built a sequence of approximations

$\displaystyle K_1 \subset K_2 \subset \dots ,$

of complexes over ${X}$, such that each ${K_i}$ was an ${i}$-dimensional finite complex and such that ${\pi_j(X, K_i) = 0}$ for ${0 \leq j \leq i}$: the maps ${K_i \rightarrow X}$ increase in connectivity at each stage. In general, we cannot make this sequence stop. However, we saw that if ${K_{N-1}}$ was chosen such that the ${\mathbb{Z}[\pi_1 X]}$-module

$\displaystyle \pi_N(X, K_{N-1}) \simeq H_N(\widetilde{X}, \widetilde{K_{N-1}})$

was free (where the tilde denotes the universal cover), then we could build ${K_N}$ from ${K_{N-1}}$ (by attaching ${N}$-cells) in such a way that ${K_N \rightarrow X}$ was a homotopy equivalence: that is, ${\pi_1 K_N \simeq \pi_1 X}$ and ${H_*( \widetilde{X}, \widetilde{K_N}) = 0}$.

The goal now is to use this requirement of freeness to build a finiteness obstruction in analogy with the algebraic situation considered in the previous post. Namely, let ${X}$ be any connected space. Then the universal cover ${\widetilde{X}}$ is a ${\pi_1 X}$-space, and the singular chain complex ${C_*(\widetilde{X})}$ is a complex of ${\mathbb{Z}[\pi_1 X]}$-modules: that is, it lives in the derived category of ${\mathbb{Z}[\pi_1 X]}$-modules. We will see below that if ${X}$ is a finite complex, then it lives in the “finitely presented” derived category introduced in the previous post—so that if ${X}$ is finitely dominated, then ${C_*(\widetilde{X})}$ is in the perfect derived category of ${\mathbb{Z}[\pi_1 X]}$.

Definition 1 The Wall finiteness obstruction of ${X}$ is the class in ${\widetilde{K}^0(\mathbb{Z}[\pi_1 X])}$ represented by the complex ${C_*(\widetilde{X})}$: that is, choose a finite complex ${P_\bullet}$ of finitely generated projective modules representing ${C_*(\widetilde{X})}$, and take ${\sum (-1)^i [P_i]}$.

It follows from the assertions made in the previous paragraph, which are proved below, that the vanishing of the finiteness obstruction is necessary for ${C_*(\widetilde{X})}$ to live in the finitely presented derived category, and for ${X}$ to be a finite cell complex. Sufficiently will take a little extra work, using the constructions of the previous post. We’ll also get to see that the above definition recovers the output of those constructions.

1. Lemmas

In order for this definition to make sense, there are a number of lemmas to check. For one thing, we need to show that if ${X}$ is a finitely dominated space, then ${C_*(\widetilde{X})}$ lives in the perfect derived category of ${\mathbb{Z}[\pi_1 X]}$. This will follow from the next lemma, which has already been alluded to a couple of times.

Lemma 1 Let ${G}$ be a discrete group. Let ${X}$ be a finite complex with a map ${X \rightarrow BG}$, classifying a principal ${G}$-bundle ${P \rightarrow X}$. Then ${C_*(P)}$ (a ${\mathbb{Z}[G]}$-module) lives in the finitely presented derived category of ${\mathbb{Z}[G]}$-modules.

Proof: In fact, the category of finite complexes over ${BG}$ is generated under finite (homotopy) colimits by the map ${\ast \rightarrow BG}$. The association ${X \mapsto P \mapsto C_*(P)}$ is the composite of two functors (pull-back ${X \mapsto X \times_{BG} EG}$ and ${P \mapsto C_*(P)}$) each of which preserves homotopy colimits. In particular, ${C_*(P)}$ is always in the smallest subcategory of ${D(\mathbb{Z}[G])}$ closed under finite colimits containing the image under the above functor of ${\ast \rightarrow BG}$. But the image of ${\ast\rightarrow BG}$ corresponds to the chain complex of the discrete ${G}$-set ${G}$, which is ${\mathbb{Z}[G]}$ concentrated in degree zero: this complex generates precisely the finitely presented derived category. $\Box$

So if ${X}$ is any finite connected complex, there is a map ${X \rightarrow B \pi_1 X}$ whose (homotopy) fiber is the principal bundle ${\widetilde{X} \rightarrow X}$. The above lemma shows that ${C_*(\widetilde{X})}$ is in the finitely presented derived category ${D_{fp}( \mathbb{Z}[\pi_1 X])}$. For ${X}$retract of a finite complex, it follows that ${C_*(\widetilde{X})}$ is simply a perfect complex. In particular, we find:

Theorem 2 (Necessity) If the Wall finiteness obstruction in ${\widetilde{K}^0(\mathbb{Z}[\pi_1 X])}$ is nonzero, then ${X}$ is not of the homotopy type of a finite complex.

2. Sufficiency

Let’s now suppose that ${X}$ is finitely dominated and that its Wall finiteness obstruction vanishes. We’d like to show that it is in fact a finite complex. In order to do this, let’s combine the algebra with the homotopy-theoretic constructions of the previous post. Choose ${N \gg 0}$ such that ${H_i(\widetilde{X}; G) = 0}$ for ${i > N}$ and all groups ${G}$, and moreover such that ${H^{N+1}(X; \mathcal{F}) =0}$ for all coefficient systems ${\mathcal{F}}$ on ${X}$. We can do this either by noting the finite-dimensionality of the dominating space, or by using the fact that ${C_*(\widetilde{X})}$ is a perfect ${\mathbb{Z}[\pi_1 X]}$-complex.

As in the previous post or the beginning of this one, we construct a sequence of successive approximations

$\displaystyle K_1 \rightarrow K_2 \rightarrow \dots \rightarrow K_{N-1} \rightarrow X,$

where the one we care about is the last one. That is, ${K_{N-1}}$ is a finite ${N-1}$-dimensional complex with the property that

$\displaystyle \pi_j(X, K_{N-1}) = 0, \quad 0 \leq j \leq N-1.$

We saw in the previous post that freeness of the ${\mathbb{Z}[\pi_1 X]}$-module

$\displaystyle H_N(\widetilde{X}, \widetilde{K}_{N-1})$

would allow us to choose a ${K_N }$ (by attaching finitely many ${N}$-cells) homotopy equivalent to ${X}$.

However, let’s consider the chain complexes ${C_*(\widetilde{K}_{N-1})}$ and ${C_*(\widetilde{X})}$; these live in the perfect derived category of ${\mathbb{Z}[\pi_1 X]}$. There is a natural cofiber sequence

$\displaystyle C_*(\widetilde{K}_{N-1}) \rightarrow C_*(\widetilde{X}) \rightarrow C_*(\widetilde{X}, \widetilde{K}_{N-1}).$

We note that ${C_*(\widetilde{X}, \widetilde{K}_{N-1})}$ has no homology in indices ${\ast \leq N-1}$ as ${\pi_j(\widetilde{X}, \widetilde{K}_{N-1}) =0}$ in these dimensions, and also for indices ${\ast > N}$ because ${K_{N-1}}$ is ${N-1}$-dimensional. In particular, we have an equivalence (in the derived category)

$\displaystyle C_*(\widetilde{X}, \widetilde{K}_{N-1}) \simeq H_N(\widetilde{X}, \widetilde{K}_{N-1}) [N].$

Observe moreover that ${C_*(\widetilde{X}, \widetilde{K}_{N-1})}$ is a perfect complex, and that its ${K}$-theory invariant is the opposite to that of ${C_*(\widetilde{X})}$ (because that of ${C_*(\widetilde{K}_{N-1})}$ vanishes).

Lemma 3 The ${\mathbb{Z}[\pi_1 X]}$-module ${H_N(\widetilde{X}, \widetilde{K}_{N-1})}$ is a finitely generated projective.

Proof: In order to see this, we need to see that maps in the derived category from ${H_N(\widetilde{X}, \widetilde{K}_{N-1})[N]}$ into ${G[N+1]}$, for any ${\mathbb{Z}[\pi_1 X]}$-module ${G}$, are zero, as these are the extension groups which we’d like to vanish. However, these are precisely maps

$\displaystyle C_*(\widetilde{X}, \widetilde{K}_{N-1}) \rightarrow G[N+1],$

or since ${\widetilde{K}_{N-1}}$ has no (possibly twisted) cohomology above level ${N-1}$, maps ${C_*(\widetilde{X}) \rightarrow G[N+1]}$. These measure the (twisted) cohomology ${H^{N+1}(X, G)}$, which we have assumed vanishes. $\Box$

In particular, we find that the Wall finiteness obstruction is, up to a sign, equal to the class of the projective module ${H_N(\widetilde{X}, \widetilde{K}_{N-1})}$ in ${K}$-theory. If the finiteness obstruction vanishes, then this ${\mathbb{Z}[\pi_1 X]}$-module is stably free. In this case, we can modify ${K_{N-1}}$ to build ${K_N}$ by first adding a bunch of ${N-1}$-spheres (which map trivially to ${X}$) so as to make a new ${N-1}$-dimensional complex ${K_{N-1}'}$. Then ${\pi_i(X, K_{N-1}') =0}$ for ${0 \leq i \leq N-1}$. Moreover, ${H_N(\widetilde{X}, \widetilde{K'}_{N-1})}$ is obtained from ${H_N(\widetilde{X}, \widetilde{K}_{N-1})}$, as a ${\mathbb{Z}[\pi_1 X]}$-module, by adding free summands, one for each ${S^{N-1}}$ added.

If we add enough of these, we can arrange it so that ${H_N(\widetilde{X},\widetilde{K_{N-1}}')}$ is actually free. Then, we are in the previous situation: we already know to add a bunch of ${N}$-cells to ${K_{N-1}'}$ to construct a ${K_N}$ mapping to ${X}$ via a homotopy equivalence.

This proves:

Theorem 4 If ${X}$ is finitely dominated and the Wall finiteness obstruction vanishes, then ${X}$ is homotopy equivalent to a finite CW complex.

3. Examples

Let’s now show that the Wall finiteness obstruction can be nonzero. More precisely, let’s show that given any finitely presented group ${G}$, and any class in ${\widetilde{K}^0(\mathbb{Z}[G])}$, we can construct a complex with ${\pi_1}$ equal to ${G}$ and with finiteness obstruction equal to this class.

Let ${P}$ be a finitely generated projective ${\mathbb{Z}[G]}$-module which is not stably free. By definition, ${P}$ is the image of an idempotent ${e: \mathbb{Z}[G]^m \rightarrow \mathbb{Z}[G]^m}$ for some ${m}$. Such an idempotent corresponds to the choice of ${a_1, \dots, a_m \in \mathbb{Z}[G]^m}$.

Start by choosing a finite complex ${X}$ with ${\pi_1 X \simeq G}$. Choose ${N \gg \dim X}$, and consider the complex ${X_1 = X \vee \bigvee_m S^N}$. We note that

$\displaystyle \pi_N X_1 \simeq \pi_N X \oplus \mathbb{Z}[G]^m, \quad \pi_N(X_1, X) \simeq \mathbb{Z}[G]^m.$

as ${\mathbb{Z}[G]}$-modules. In particular, we can choose maps ${\alpha_i : S^N \rightarrow X}$ corresponding to the generators ${a_i \in \mathbb{Z}[G]^m}$, and wedging with ${X}$, they define a map

$\displaystyle e: X_1 \rightarrow X_1$

such that ${\pi_N(e): \pi_N (X_1, X) \rightarrow \pi_N(X_1, X)}$ is given by the action of ${e}$ in ${\mathbb{Z}[G]^m}$.

Let

$\displaystyle Y = \varinjlim (X_1 \stackrel{e}{\rightarrow} X_1 \stackrel{e}{\rightarrow} \dots );$

the claim is that the finiteness obstruction of ${Y}$ is given by (up to a sign depending on the parity of ${N}$), the class of ${[P]}$. In fact, we note that ${X \rightarrow Y}$ has the property that ${C_*(X, Y)}$ is quasi-isomorphic to ${P[N]}$, by construction.

Moreover, ${Y}$ is finitely dominated (we should have checked this first): the map ${X_1 \rightarrow Y}$ admits a section up to homotopy. To see this, we can convert ${X_1 \rightarrow Y}$ into a fibration. The obstructions to finding a section are in the groups ${H^i(Y, \pi_{i}(Y, X_1))}$. For ${i > N}$, this is automatically zero since ${Y}$ has no cohomology above ${N}$. For ${i \leq N}$, the relevant homotopy groups are zero.

Note in particular that these $K$-groups can be very nonzero: if for instance $G = \mathbb{Z}/p$, the relevant $K$-groups are those of the Dedekind domain $\mathbb{Z}[\zeta_p]$ and in particular are the class groups of cyclotomic fields, which are generally nontrivial.