In this lecture we define topological manifolds and then study their "top"-dimensional homology groups (i.e. homology in the dimension of the manifold).

We construct a covering space $\bigsqcup_{x \in M} H_n(M , M \setminus x) \to M$ (for $M$ $n$-dimensional), and show that if $K \subset M$ a closed subset, then there is a natural isomorphism between the relative homology group $H_n(M, M \setminus K)$ and the space of compactly supported sections of this covering space over $K$.

Along the way we introduce presheaves and pre-cosheaves, which are further fundamental objects of study in algebraic topology, category theory, and beyond.

Next lecture we will define what it means for a manifold to be orientable, and show how the main result from today allows one to compute the top-dimensional homology groups of a manifold.

Remark

The new forum site is accessible here (there is also a direct link below to this lecture's typo-reporting-thread), and this has now replaced the old commenting system. The forum is open to anyone with an ethz.ch or a uzh.ch email address (you do not have to be registered for the course).