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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Splitting of Gysin extensions

A J Berrick and A A Davydov

Algebraic & Geometric Topology 1 (2001) 743–762

arXiv: math.AT/0201135

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–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
Mathematical Subject Classification 2000
Primary: 16E45, 55R25, 55S35
Secondary: 16E40, 55R20, 55S20, 55S30
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Publication
Received: 11 October 2000
Revised: 17 July 2001
Accepted: 29 2001
Published: 4 December 2001
Authors
A J Berrick
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Singapore
A A Davydov
Department of Mathematics
Macquarie University
Sydney
NSW 2109
Australia