Homology manifolds and Euclidean bundles
Fabian Hebestreit (University of Aberdeen)
Monday 9th October 16:00-17:00 Maths 311B
It is a curious fact of life in geometric topology, that the classification of closed manifolds by surgery theory becomes easier as one passes from smooth to piecewise linear and finally to topological manifolds. It was long conjectured that an even cleaner statement should be expected in the somewhat arcane world of homology manifolds of the title, which ought to fill the role of some "missing manifolds". This was finally proven by Bryant–Ferry–Mio–Weinberger in the 90's in the form a surgery sequence for homology manifolds, building on an earlier theorem of Ferry and Pedersen that any homology admits admits a degree one (normal) map from a topological manifold.
In the talk I will try to explain this surgery sequence, and further that its existence is incompatible with the theorem of Ferry–Pedersen. The latter is therefore incorrect and/or the proof the former incomplete. This is joint work with M. Land, M. Weiss, and C. Winges.