#### Volume 1, issue 2 (2001)

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Splitting of Gysin extensions

### A J Berrick and A A Davydov

Algebraic & Geometric Topology 1 (2001) 743–762
 arXiv: math.AT/0201135
##### Abstract

Let $X\to B$ be an orientable sphere bundle. Its Gysin sequence exhibits ${H}^{\ast }\left(X\right)$ as an extension of ${H}^{\ast }\left(B\right)$–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 ${H}^{\ast }\left(B\right),$ 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