Singular Homology
Satya Deo · Texts and readings in mathematics · 2003
Having defined the simplicial homology groups of a compact polyhedron (space of a finite simplicial complex) now we come to another kind of homology groups, called singular homology groups of a topological space. The first interesting feature of these homology groups is that they are defined for all topological spaces X , not only for compact polyhedra. Singular homology groups were first defined by S. Lefschetz in 1933 and were perfected in their present form by S. Eilenberg (1913–1998) in the beginning of the 1940’s. These turn out to be the most important and natural generalization of simplicial homology groups, and are most suited for the study of topological manifolds. We may recall (Theorem 4.6.3) that it took a good amount of hard work and new ideas to prove the topological invariance of simplicial homology in Chapter 4. This was first done by J.W. Alexander (1888–1971). In a contrast to this, we will see in this chapter that the topological invariance of singular homology follows almost obviously - this is another attractive feature of singular homology.