A project work on algebraic topology (with categorical flavour) : suggestions for topics.

Question

As a part of my exam on Algebraic Topology, I have to prepare a brief exposition deepening a topic treated in the course.

The background is:

• basic homotopy theory (fundamental group, theory of covering spaces, Seifert Van Kampen theorem)
• basic homology theory (simplicial and singular homology, the last developed quite in detail, something about homology of spheric complexes, the axioms of Steenrod)
• very basic cohomology theory (here, just the most important definitions and the cup product cohomology ring)

Rather than a particular application or computation (we have done many in class) I would like to deepen some general result of categorical flavour, connecting different parts of the theory.

In this sense the only things that come to my mind are to read somemething about:

1. Hurewicz theorem, or other general connections between homotopy and homology.
2. Duality between homology and cohomology.

Any suggestion about these two and where to read (concisely) would be most appreciated, as any suggestion about topics I may not aware about at all.

Thanks in advance.

Answer 1

Here are some suggestions

-Spectral sequences

Serre spectral sequence in particular is a very powerful tool to compute homology, whenever you have a "fibration" $F \rightarrow E \xrightarrow \pi B$ (a fibration is a very general notion of fiber bundle) and you know $H_*(B)$ and $H_*(F)$ you can with the Serre spectral sequence compute $H_*(E)$ in favorable cases, and if you know $H_*(B)$ and $H_*(E)$ you can work backwards to compute $H_*(F)$. I would recommend this topic if your exposition has to be super brief since you don't need that much background knowledge to understand spectral sequences. Although they can be hard to understand the first time you see them. There is a section in Hatcher on spectral sequences.

-Sheaf cohomology

This is certainly of categorical nature but not obviously connected to homotopy, you define and study the sheaf cohomology of a sheaf $\mathscr F$ over a space $X$. A sheaf is a collection of groups $\mathscr FU$ for all open $U \subset X$ together with maps $\mathscr FU \rightarrow \mathscr F V$ whenever $V$ is a subset of $U$. For locally contractible spaces the singular homology of $X$ coincides with the sheaf cohomology of $X$ with respect to a specific sheaf.

-Simplicial homotopy theory

I personally really like this topic. You study "simplicial sets" which are a different form of spaces, they consist of a sequence of sets $X_n$ of $n-$simplices and face maps $X_i \rightarrow X_{i-1}$ which tell you how $i-$simplices are connected to $(i-1)-$simplices. With methods from simplicial homotopy theory you can prove that $H^i(X,G) = [X,K(G,n)]$ when $X$ is a $CW$-complex. Goerss-Jardines "simplicial homotopy theory" is a very good book for this topic.

Answer 2

You will find the proofs you are looking for in Heuts, Meier - Algebraic Topology II. Also, in the same pdf there is a proof of the representability of the cohomology functor, which is very cool.

Another topic which would be great imho is the equivalence between the standard model category of topological spaces and the one of simplicial sets, which is covered in Dwyer, Spalinski - Homotopy Theories and Model Categories. Unfortunately, I don't know if this latter topic may be covered in a brief exposition and it would require to learn a bit more than the ones you mentioned.

Let me know how things go.

EDIT: spectral sequences are a great idea, as the other user suggested. They are also covered in the first pdf I mentioned, which provides many relevant examples on how to use them to compute (co)homology groups and homotopy groups of a space making use, for example, of Postnikov towers.