### Cohomology and Homotopy

#### Posted by David Corfield

In posts and this $n$Lab entry, Urs has been promoting his view of cohomology as about Hom spaces between objects in certain settings, where the unknown space is on the left. Similarly homotopy is where the unknown space is on the right. This got me thinking the following thoughts during some quiet moments in a conference this morning.

- Cohomology of $X$ concerns $Hom(X, A)$, for different choices of $A$.
- Homotopy of $X$ concerns $Hom(B, X)$, for different choices of $B$.

We often choose $A$ to be homotopically simple, i.e., we want $Hom(S^m, A)$ to be simple, where $S^m$ is cohomologically simple. E.g., $A$ could be an Eilenberg-MacLane space $K(G, n)$.

We often choose $B$ to be cohomologically simple, i.e., we want $Hom(B, K(G, n))$ to be simple, where $K(G, n)$ is homotopically simple. E.g., $B$ could be a sphere $S^m$.

These choices work because $Hom(S^m, K(G, n))$ is a kind of pairing where the result is trivial for nonzero $m$ unless $m = n$.

On the one side we have cell complexes built up from attaching copies of spheres; on the other we have spaces built up as Postnikov towers out of Eilenberg-MacLane spaces.

Do we understand why Eckmann-Hilton duality breaks down?

The situation is a bit like where we have a basis for a vector space paired with the dual basis for the dual vector space. Here, however, the spheres have 1 parameter where the E-M spaces have 2 parameters. Is this why cohomology feels simpler than homotopy? Or is this also to do with the complexity of cell attachment versus complexity of building of Postnikov tower?

Just as there is arbitrariness on the choice of a basis in a vector space, could there be another pair of collections of spaces which could form a ‘duality’? In a quotation here Hatcher talks about

…what would happen if we dualized the notion of a Postnikov tower of principal fibrations, where a space is represented as an inverse limit of a sequence of fibers of maps to Eilenberg–MacLane spaces. In the dual representation, a space would be realized as a direct limit of a sequence of cofibers of maps from Moore spaces.

A Moore space is one with only nontrivial homology $G$ in dimension $n$. Was homology rather than cohomology forced here? Will that blow the chance of a pairing with homotopically simple spaces? Or would it pair with cohomotopically simple spaces?

But might there not be a very different looking pairing between sets of spaces, such that Hom spaces between them are only nontrivial if parameters match?

Better get back to my conference now.

## Re: Cohomology and Homotopy

I once asked a question like this, moore spaces are just two spheres attached by the appropriate maps, i believe. This is why cohomotopy is not the most interesting, i am told. not sure if that is helpful.