Homology

Abstract

The object of topology is the classification and description of the shape of a space up to topological equivalence. We have in Theorem 4.14 a technique for classifying the surfaces, but this is, as you may have noticed, rather arduous. The euler characteristic can be used to shorten the process, but for some cases a lengthy procedure is still necessary. Neither of these options provide a clear accounting for the ways in which the surfaces vary, e.g., which enclose cavities, which are non-orientable, etc., and neither can be completely generalized to higher-dimensional manifolds. Ideally, one would like some sort of algebraic invariant or computable quantity that would codify a lot of information: how many connected pieces a space has, how the gluing directions work, whether the surface is orientable or not, etc. The euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have X = 0. See Figure 6.1.

“Mathematicians are like Frenchmen; whatever you say to them they translate into their own language and forthwith it is something completely different.”

Goethe