Simplicially controlled Algebra

Spiros Adams-Florou (University of Glasgow)

Wednesday 26th November, 2014 16:00-17:00 Maths 516


A classic question in Topology is: "When is a map homotopic to a homeomorphism?" One approach to studying this problem is to look at the point preimages - a not unreasonable hope is that if the preimages are 'close' to being points, in some suitable sense, then the map might be 'close' to a homeomorphism. The approach in controlled topology is to say that the point preimages are 'close' to points if they are small with respect to some metric. More generally, in controlled topology one aims to associate a 'size' to various topological invariants and ideally prove that if an invariant has sufficiently small size then it must vanish. Therefore, whether an invariant vanishes or not can be a delicate issue requiring a careful mixture of geometry and algebra. In this talk I will discuss controlled topology and in particular the use of controlled algebra in determining when a simplicial complex is a (homology) manifold. If time permits I may also mention how this fits into the surgery programme.

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