# Homotopy Type Theory Formalized Homotopy Theory

Maybe we could figure out a way not to duplicate stuff between this page and homotopy groups of spheres?

Ali: Results about homotopy groups of spheres should go there, everything else here.

Cast of characters so far: Jeremy Avigad, Guillaume Brunerie, Favonia, Eric Finster, Chris Kapulkin, Dan Licata, Peter Lumsdaine, Mike Shulman, Evan Cavallo

## In progress

### Cohomology

cohomology

• To do cohomology with finite coefficients, all we need is the boring work of defining $\mathbb{Z}/p\mathbb{Z}$ as an explicit group.
• Calculate some more cohomology groups.
• Compute the loop space of this construction and use it to define spectra.

## At least one proof has been formalized

### K(G,n)

Eilenberg-MacLane space

cohomology

### Freudenthal Suspension Theorem

Freudenthal suspension theorem?

Implies $\pi_k(S^n) = \pi_{k+1}(S^{n+1})$ whenever $k \leq 2n - 2$

• Peter’s encode/decode-style proof, formalized by Dan, using a clever lemma about maps out of 2 n-connected types. This is the “95%” version, which shows that the relevant map is an equivalence on truncations.
• The full “100%” version, formalized by Dan, which shows that the relevant map is $2n$-connected.

### Homotopy limits

• Jeremy Avigad, Chris Kapulkin and Peter Lumsdaine arXiv Coq code

### Van Kampen

van Kampen theorem

### Covering spaces

Covering space?

• Favonia’s proofs (link in flux due to library rewrite).

### Blakers-Massey

Blakers-Massey theorem?

• Favonia/Peter/Guillaume/Dan’s formalization of Peter/Eric/Dan’s proof (link in flux due to library rewrite).
category: homotopy theory

Last revised on January 19, 2019 at 14:06:38. See the history of this page for a list of all contributions to it.