Algebraic and Geometric Topology 1 (2001), paper no. 37, pages 743-762.

Splitting of Gysin extensions

A. J. Berrick, A. A. Davydov

Abstract. 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.

Keywords. Gysin sequence, Hochschild homology, differential graded algebra, formal space, A_infty-structure, Massey triple product

AMS subject classification. Primary: 16E45, 55R25, 55S35. Secondary: 16E40, 55R20, 55S20, 55S30.

DOI: 10.2140/agt.2001.1.743

E-print: arXiv:math.AT/0201145

Submitted: 11 October 2000. (Revised: 17 July 2001.) Accepted: 29 Novemver 2001. Published: 4 December 2001.

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A. J. Berrick, A. A. Davydov
Department of Mathematics
National University of Singapore
2 Science Drive 2, Singapore 117543, SINGAPORE
and
Department of Mathematics
Macquarie University, Sydney, NSW 2109
AUSTRALIA

Email: berrick@math.nus.edu.sg, davydov@ics.mq.edu.au

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