Computational Topology Surface Topology Afra

Last lecture, we spent a considerable amount of effort defining manifolds. We like manifolds because they are locally Euclidean. So, even though it is hard for us to reason about them globally, we know what to do in small neighborhoods. It turns out that this ability is all we really need. This is rather fortunate, because we suddenly have spaces with more interesting structure than the Euclidean spaces to study. Recall that topology, like Euclidean geometry, is a study of the properties of spaces that remain invariant (do not change) under a fixed set of transformations. In topology, we expand the transformations that are allowed from rigid motions (Euclidean geometry) to homeomorphisms: bijective bi-continuous maps. In this lecture, we ask whether we may classify manifolds under this set of transformations, and we see that such a classification is possible for two-dimensional manifolds or surfaces.