2.1. Simplicial and Singular Homology

The most important homology theory in algebraic topology, and the one we shall be studying almost exclusively, is called singular homology. Since the technical apparatus of singular homology is somewhat complicated, we will first introduce a more primitive version called simplicial homology in order to see how some of the apparatus works in a simpler setting before beginning the general theory.

The natural domain of definition for simplicial homology is a class of spaces we call \(\Delta\)-complexes, which are a milde generalization of the more classical notion of a simplicial complex. Historically, the modern definition of singular homology was first given in [Eilenberg 1944], and \(\Delta\)-complexes were introduced soon thereafter in [Eilenberg-Zilber 1950] where they were called semisimplicial complexes. Within a few years this term came to be applied to what Eilenberg and Zilber called complete semisimplicial complexes, and later there was yet another shift in terminology as the latter objects came to be called simplicial sets. In theory this frees up the term semisimplicial complex to have its original meaning, but to avoid potential confusion it seems best to introduce a new name, and the term \(\Delta\)-complex has at least the virtue of brevity.

• \(\Delta\)-Complexes
• Simplicial Homology
• Singular Homology
• Homotopy Invariance
• Exact Sequences and Excision
• The Equivalence of Simplicial and Singular Homology