Stable classification of four Manifolds

Riccardo Pedrotti

16 mar 2017 -- 18:30

Sala seminari, Dipartimento di Matematica, Pisa

Abstract.

The classification of smooth four manifolds up to diffeomorphism is one of nightmares of a Topologist: it’s known to be impossible (it would solve the Word Problem) in full generality, but for certain classes of fundamental groups there are important results mainly due to Freedman. If we allow more relaxed notions of classification then much more can be said and computed.

This talk is intended as an introduction to the concept of stable classification of four manifolds, which studies four manifolds up to connected sum of copies of S² x S^2. The main results are due to M. Kreck and P. Teichner. I will focus on Teichner’s approach via spectral sequences since it requires a limited background and provides quick but interesting results.

My intention is to focus on the relevant definitions and constructions necessary to understand what’s going on and what one should know in order to start computing the stable classification for a fixed fundamental group. In the end I want to discuss some interesting byproducts of such classification, which are related to the divisibility of the signature of an (almost-spin) four manifold.

Some non-trivial key-words are: almost-spin manifolds, B-bordism, James spectral sequence, Atyiah-Hirzebruch spectral sequence, \pi₁-fundamental class

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