Geodesic
Splitting of Gysin extensions
Berrick, A J1 ; Davydov, A A2
1Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore
2Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia
Algebraic and Geometric Topology, Tome 1 (2001) no. 2, pp. 743-762
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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.

DOI : 10.2140/agt.2001.1.743
Keywords: Gysin sequence, Hochschild homology, differential graded algebra, formal space, $A_{\infty}$–structure, Massey triple product

Berrick, A J  1   ; Davydov, A A  2

1 Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore
2 Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia 
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Berrick, A J; Davydov, A A. Splitting of Gysin extensions. Algebraic and Geometric Topology, Tome 1 (2001) no. 2, pp. 743-762. doi: 10.2140/agt.2001.1.743

[1] H Cartan, S Eilenberg, Homological algebra, Princeton University Press (1956)

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