Next, I would like to describe an alternative description of *relative* K-theory which is sometimes convenient (e.g. when describing the Thom isomorphism in K-theory). Let be a compact Lie group, as always. Let be a pair of compact -spaces (with ); then we have defined the*relative* -theory via

Here is equipped with a distinguished basepoint (corresponding to ), and as a result this makes sense. As usual, we can use this definition to make into a *cohomology theory* on compact -spaces.

To describe without use of the space , we can proceed as follows.

**Definition 1** *We let be the category of **complexes* of -vector bundles

*on , which when restricted to are exact. A morphism in is a map of chain complexes.*

The idea is that we are going to assign to every element of an element of the relative K-theory , by effectively taking the alternating sum . In order to do this, we will start by modifying the complex by adding acyclic complexes. Namely, we start by adding complexes of the form

where is a -vector bundle on , to make all but the first term of trivial (i.e. coming from an -representation). With this change made, we can assume that all but the first term of is stably trivial. Then the first term of is stably trivial when restricted to by exactness of . Consequently, we can quotient all the terms by and get a complex of -vector bundles ; this is exact at the basepoint of . Now, taking the alternating sum as desired, we get a map

This map does not see *stable equivalence*; that is, if we add to a complex a complex of the form , the image in does not change. Moreover, it is homotopy invariant.

In fact, one can give a *presentation* of the group in this way. We start with the set of all such complexes in as above. We identify complexes which are *chain homotopic* to each other. Then, we mod out by the relation of (geometric) *homotopy*: if one has complexes which can be obtained by restriction to the end faces of a complex in , then they should both be identified. Given these identifications, one gets precisely the group .

I don’t really want to prove these things in detail, partially because I don’t want to get too bogged down with this project. (more…)