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 -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 be a ring, and consider the derived category of chain complexes of -modules. There are various interesting subcategories of :

- The
*finitely presented*derived category ; this is the smallest triangulated (or stable) full subcategory of containing and closed under cofiber sequences. In other words, consists of complexes which are quasi-isomorphic to finite complexes of finitely generated free modules. - The
*perfect*derived category : this is the category of objects such that 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 -category of spaces containing and closed under finite colimits. The retracts of finite cell complexes are the compact objects in this -category.) (more…)