Algebraic & Geometric Topology

Splitting of Gysin extensions

A J Berrick and A A Davydov

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Abstract

Let XB 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
Received: 11 October 2000
Revised: 17 July 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
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

Subjects
Primary: 16E45: Differential graded algebras and applications 55R25: Sphere bundles and vector bundles 55S35: Obstruction theory
Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 55S20: Secondary and higher cohomology operations 55S30: Massey products

Keywords
Gysin sequence Hochschild homology differential graded algebra formal space $A_{\infty}$–structure Massey triple product

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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