Conceptos de homología simplicial

Abstract

To construct simplicial homology groups of a topological space X, it is necessary that the topological space X admits a triangulation that means that it is homeomorphic to a simplicial complex K. Once it is defined its triangulation, it is defined the p-chains C_p (K), a succession of free abelian groups that are generated by the linear combination of the p-simplejos K on the integers. Now the construction of a succession of homomorphisms border becomes (∂_p) about the p-chains K. With the property, ∂_p∘∂_(p+1) it is the trivial homomorphism. Homeomorphisms border create new groups. The group of p-cycles Z_p (K)=Ker(∂_p) of K and the group of p-border B_p (K)=∂_(p+1) (C_(p+1) (K)) of K, so that B_p (K)⊆Z_p (K)⊆C_p (K). It defines p-group Homology H_p (K) of K as the quotient group H_p (K)=(Z_p (K))/(B_p (K) ). These groups allow a triangulates classification of topological spaces and establish if two spaces are homeomorphic.