Further Developments in Homology

Abstract

The preceding chapters have introduced homology groups for polyhedra and homotopy groups for arbitrary spaces. The homotopy groups are more general since they apply to more spaces. The process of extending homology to spaces more general than polyhedra began in the years 1921–1933 and has continued to the present day. The pioneers in this work were Oswald Veblen, Solomon Lefschetz, Leopold Vietoris, and Eduard Čech. In this chapter we shall examine some additional theory and applications of simplicial homology groups, notably the famous fixed point theorem and relative homology groups discovered by Lefschetz, and the singular homology groups, also due to Lefschetz, which extend homology theory to arbitrary spaces.