The following document aims to give an introduction to simplicial homology. I discuss the basics of some of the most important techniques for computing homology, including collapses, discrete Morse theory, and long exact sequences. The various techniques are illustrated with worked-out examples. The text does not cover homotopy theory or cohomology theory.
Chapter 0: Algebraic preliminaries
Chapter 1: Simplicial homology
Chapter 2: Combinatorial techniques
Chapter 3: Algebraic techniques