#### 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.

Notes on file formats
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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