Computational Topology 5.1 Homotopy

In Lecture 4, we learned about an algebraic method for describing and classifying structures. In this lecture, we look at using algebra to find combinatorial descriptions of topological spaces. We begin by looking at an equivalence relation called homotopy that gives a classification of spaces that is coarser that homeomorphism, but respects the finer classification. That is, two spaces that have the same topological type must have the same homotopy type, but the reverse does not necessarily hold. This property should remind you of our definition of a topological invariant. We then continue by looking at a powerful method for understanding topological spaces by forming algebraic images of them using functors. One functor is the fundamental group, the first group description of a space we will see. Unfortunately, this group is hard to compute and may not give us a finite description. It does, however, give us a method for proving that both the homeomorphism problem and the homotopy problem (detecting whether two spaces are homotopic) are undecidable.