Rigorous Book on Topology of Surfaces and Simplicial Complexes

Question

In the undergraduate course I am on the topology course covered the basics of point set topogly very well and everything was done, but then moved on to simplicial complexes and surfaces which was very hand-wavy. Is there a good book on this area that does everything rigorously (i.e. doesn't define and prove things with pictures and long paragraphs)?

Answer 1

As I guess that you are rather new to topology, I highly recommend Jean Gallier’s and Dianna Xu’s A Guide to the Classification Theorem for Compact Surfaces. The book adopts a student-centered approach and offers rigorous mathematical proofs without bogging the reader down with too many details. One of the nicest things about the book is that it contains a rigorous proof of the fascinating fact that every compact surface admits a finite triangulation, i.e., is homeomorphic to a finite $ 2 $-dimensional simplicial complex (this is known in the literature as Rado’s Theorem). Most undergraduate textbooks on topology do not discuss this fact or only mention it in passing.