A report on recent joint work withTim Cochran.
I will report on joint work in progress with Steve Ferry which aims to characterise those open manifolds which are the compliments of finite polyhedrons K in S^n (assuming for now that K is simply connected and that 2k + 2 < n). The talk will rapidly move to measuring the twistedness of twisted doubles and the relationship of this to high codimension link theory (a la Sato and Levine).
I will characterize those maps of a closed surface to the 3-torus that are homotopic to embeddings.
The solution to the homeomorphism problem for 3-manifolds will be presented–there is now an algorithm to determine whether two given 3-manifolds are homeomorphic.
I shall present a computation of the group of isotopies of quaternionic projective space (up to a Z/2 ambiguity). Along the way, I will explain how this fits into a broader framework for manifold classification and also give an explicit representative for an exotic self-diffeomorphism of S^7.
