Splitting of Gysin extensions

Let $X \to B$ 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

Mathematical Subject Classification 2000

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

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

Volume 1, issue 2 (2001)

A J Berrick and A A Davydov