Splitting of Gysin extensions

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.