Regular infinite dimensional Lie groups

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply connected and the image group is regular.Comment: AmSTeX, using diag.tex with fonts lams?.ps, 38 page

Amenable groups and smooth topology of 4-manifolds

A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-dimensional topology and coarse geometry