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Carmen Rovi (MPI): The signature modulo 8 of a fibration (03/11/2015)


In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature \sigma (M) in Z of an oriented 4k-dimensional geometric Poincaré complex M^4k. The precise relation between the signature modulo 8, the Arf invariant and the Brown-Kervaire invariant will be given. Furthermore we shall discuss how the relation between these invariants can be applied to the study of the signature modulo 8 of a fibration. In particular it had been proved by Meyer in 1973 that a surface bundle has signature divisible by 4. This was generalized to higher dimensions by Hambleton, Korzeniewski and Ranicki in 2007. I will explain two results from my thesis concerning the signature modulo 8 of a fibration: firstly under what conditions can we guarantee divisibility of the signature by 8 and secondly what invariant detects non-divisibility by 8 in general.

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