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 which was a homotopy retract of an -dimensional finite CW complex (where ), and tried to express itself as homotopy equivalent to one such. We built a sequence of approximations
of complexes over , such that each was an -dimensional finite complex and such that for : the maps increase in connectivity at each stage. In general, we cannot make this sequence stop. However, we saw that if was chosen such that the -module
was free (where the tilde denotes the universal cover), then we could build from (by attaching -cells) in such a way that was a homotopy equivalence: that is, and .
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 be any connected space. Then the universal cover is a -space, and the singular chain complex is a complex of -modules: that is, it lives in the derived category of -modules. We will see below that if is a finite complex, then it lives in the “finitely presented” derived category introduced in the previous post—so that if is finitely dominated, then is in the perfect derived category of .
Definition 1 The Wall finiteness obstruction of is the class in represented by the complex : that is, choose a finite complex of finitely generated projective modules representing , and take . (more…)