Computing Homology Groups

Abstract

In light of the discussion of the previous chapter, given a cubical set X we know that its homology groups H_*(X) are well defined. We have also computed H_*(X) for some simple examples and discussed the method of elementary collapse, which can be used in special cases to compute these groups. In this chapter we want to go further and argue that the homology groups of any cubical set are computable. In fact, we will derive Algorithm 3.78, which, given a cubical set X, takes as input the list of all elements of K_max(X) and as output presents the associated homology groups H_*(X). While the input is well defined, how to present the output may not be so clear at this moment. Obviously, we need to obtain a set of abelian groups. However, for this to be of use we need to know that we can present these groups in a finite and recognizable form. In particular, if X and Y are two cubical sets, it is desirable that our algorithm outputs the groups H_*(X) and H_*(Y) in such a way that it is evident whether or not they are isomorphic. With this in mind, by the end of this chapter we will prove the following result.