## Algebraic & Geometric Topology

### Splitting of Gysin extensions

#### 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 $A∞$–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.

#### Article information

Source
Algebr. Geom. Topol., Volume 1, Number 2 (2001), 743-762.

Dates
Revised: 17 July 2001
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882647

Digital Object Identifier
doi:10.2140/agt.2001.1.743

Mathematical Reviews number (MathSciNet)
MR1875616

Zentralblatt MATH identifier
0986.55014

#### Citation

Berrick, A J; Davydov, A A. Splitting of Gysin extensions. Algebr. Geom. Topol. 1 (2001), no. 2, 743--762. doi:10.2140/agt.2001.1.743. https://projecteuclid.org/euclid.agt/1513882647

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