Algebraic & Geometric Topology 1 (2001) 743–762

DOI: 10.2140/agt.2001.1.743

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_∞–structure, Massey triple product

Mathematical Subject Classification

Primary: 16E45, 55R25, 55S35

Secondary: 16E40, 55R20, 55S20, 55S30

References

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Publication

Received: 11 October 2000
Revised: 17 July 2001
Accepted: 29 2001
Published: 4 December 2001

Authors