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…)

I’ve been reading Wall’s “Finiteness conditions for CW complexes.” This paper gives necessary and sufficient conditions for a space to be homotopy equivalent to a finite cell complex. Alternatively, it gives an obstruction in ${K}$-theory for when a retract (in the homotopy category) of a finite cell complex has the homotopy type of a finite cell complex. I’d like to describe this result, and try to motivate why the existence of such an obstruction is a natural thing to expect by a simpler analogy with algebra.
There is a fruitful analogy between spaces and chain complexes. Let ${R}$ be a ring, and consider the derived category ${D(R)}$ of chain complexes of ${R}$-modules. There are various interesting subcategories of ${D(R)}$:
1. The finitely presented derived category ${D_{fp}(R)}$; this is the smallest triangulated (or stable) full subcategory of ${D(R)}$ containing ${R}$ and closed under cofiber sequences. In other words, ${D_{fp}(R)}$ consists of complexes which are quasi-isomorphic to finite complexes of finitely generated free modules.
2. The perfect derived category ${D_{pf}(R)}$: this is the category of objects ${X \in D(R)}$ such that ${\hom(X, \cdot)}$ commutes with direct sums (i.e., the compact objects). It turns out that so-called perfect complexes are those that can be represented as finite complexes of finitely generated projectives.
One should think of the finitely presented objects as analogous to the finite cell complexes in topology, and the perfect objects as analogous to the retracts of finite cell complexes. (To push the analogy: the finite cell complexes are the smallest subcategory of the ${\infty}$-category of spaces containing ${\ast}$ and closed under finite colimits. The retracts of finite cell complexes are the compact objects in this ${\infty}$-category.) (more…)