| The construction of singular (co)homology [CSC] |
| Examples in the (co)homology of spaces [ECS] |
| Algebraic theory of abelian groups [ATAG] |
We will work through the highlights of [CSC], skipping some details. This will define the cohomology ring $H^*(X)$ of a topological space $X$, and prove some key properties (the Eilenberg-Steenrod axioms). This will rely on some material from [ATAG], which we will also review. After proving the Eilenberg-Steenrod axioms, we will be able to use them to calculate $H^*(X)$ for many spaces $X$ without further reference to the actual definition of $H^*(X)$. We will spend a large part of the course working through [ECS]. This describes the topological properties of a range of interesting spaces that occur naturally, and calculates their cohomology rings, some of which have a rich and interesting structure.
All the above files are in reasonably good shape, but nonetheless I may upload updated versions as the course progresses.
| Problems | Solutions |