Computational Topology 6.1 Chains and Cycles

In Lecture 3, we learned about a combinatorial method for representing spaces. In Lecture 4, we studied groups and equivalence relations implied by their normal subgroups. In this lecture, we look at a combinatorial and computable functor called homology that gives us a finite description of the topology of a space. Homology groups may be regarded as an algebraization of the first layer of geometry in cell structures: how cells of dimension n attach to cells of dimension n− 1 [1]. Mathematically, the homology groups have a less transparent definition than the fundamental group, and require a lot of machinery to be set up before any calculations. We focus on a weaker form of homology, simplicial homology, that both satisfies our need for a combinatorial functor, and obviates the need for this machinery. Simplicial homology is defined only for simplicial complexes, the spaces we are interested in. Like the Euler characteristic, however, homology is an invariant of the underlying space of the complex. Indeed, the invariance of the Euler characteristic is often derived from the invariance of homology. Homology groups, unlike the fundamental group, are abelian. In fact, the first homology group is precisely the abelianization of the fundamental group. We pay a price for the generality and computability of homology groups: homology has less differentiating power than homotopy. Once again, however, homology respects homotopy classes, and therefore, classes of homeomorphic spaces.