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…)