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…)