Simplicial Complexes

Abstract

We have been studying arbitrary spaces X using fundamental groups and homology groups, and we have been rewarded with interesting applications in the few cases in which we could compute these groups. At this point, however, we would have difficulty computing the homology groups of a space as simple as the torus T = S¹ x S¹; indeed S _n (T) is uncountable for every n ≥ 0, so it is conceivable that H _n (T) is uncountable for every n (we shall soon see that this is not so). Many interesting spaces, as the torus, can be “triangulated”, and we shall see that this (strong) condition greatly facilitates calculation of homology groups. Moreover, we shall also be able to give a presentation of the fundamental groups of such spaces.