Splitting of Gysin extensions

Details Splitting of Gysin extensions Splitting of Gysin extensions

2002-01-16

arxiv.org/abs/math/0201145v1

Algebr. Geom. Topol. 1 (2001) 743-762

Mathematics

Algebraic Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-37.abs.html

Scientific paper

Let X --> B be an orientable sphere bundle. Its Gysin sequence exhibits H^*(X) as an extension of H^*(B)-modules. We prove that the class of this extension is the image of a canonical class that we define in the Hochschild 3-cohomology of H^*(B), corresponding to a component of its A_infty-structure, and generalizing the Massey triple product. We identify two cases where this class vanishes, so that the Gysin extension is split. The first, with rational coefficients, is that where B is a formal space; the second, with integer coefficients, is where B is a torus.

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