Let be an orientable sphere bundle. Its Gysin sequence exhibits as an extension of –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 corresponding to a component of its –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 is a formal space; the second, with integer coefficients, is where is a torus.
"Splitting of Gysin extensions." Algebr. Geom. Topol. 1 (2) 743 - 762, 2001. https://doi.org/10.2140/agt.2001.1.743