ANRs, free isovariant ANRs and polyhedra in the uniform category

Resumen:

Since the work of Isbell in 1950s, it is commonly thought that ANRs in the uniform category are much harder to deal with than usual ANRs (in the sense of Borsuk). Indeed, not all Banach spaces are uniform ANRs. It turns out, however, that most of the technique of the topological theory of retracts does have uniform analogues, and from the geometric perspective uniform ANRs are often even more appropriate and more manageable. For instance, homotopy limits and colimits of uniform ANRs are uniform ANRs, which is not the case for usual ANRs (because e.g. the cone over the real line is not an ANR in the usual sense since it is not metrizable).

In contrast, uniform polyhedra are definitely harder to deal with that usual polyhedra or CW-complexes since they involve a lot of nontrivial combinatorics. But they are essentially used, in particular, in the proof of the following theorem ("equivariant Hilbert-Smith conjecture"): there exists no compact ANR in the category of metrizable uniform spaces with a free action of the group of p-adic integers, and uniformly continuous equivariant maps.

Most of the talk (with the exception of the free isovariant theme) is covered in the recent preprints arxiv:1106.3249 and 1109.0346.