Week 02 — Simplicial Complexes and Homology

The combinatorial machinery: simplicial complexes, chain groups, boundary maps, homology as the kernel of a kernel.

Lecture

Abstract and geometric simplicial complexes · chain complexes over $\mathbb{F}_2$ and $\mathbb{Z}$ · boundary maps · cycles, boundaries, homology · Betti numbers and what they count.

Read before the lecture

• Hatcher, *Algebraic Topology*, sections 2.1–2.2 (or equivalent treatment in Edelsbrunner-Harer ch. III)

Problem set

PS1 — Computing homology by hand

1. Compute $H_0$, $H_1$, $H_2$ of the 2-sphere, 2-torus, and Klein bottle from a chosen triangulation.
2. Show that homology is independent of the chosen triangulation (sketch).
3. Implement the Smith normal form computation for homology of a small complex in Python.

Reference text for this week: chapter 02 of the bilingual notes — EN PDF · FR PDF.