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 ${\sum (-1)^i [P_i]}$. (more…)