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Cohomology Operations Defining Cohomology Algebra of the Loop Space

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Lie Groups, Differential Equations, and Geometry

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Abstract

According to the minimality theorem (Kadeishvili, Russ Math Surv 35(3):231–238, 1980), on the cohomology H ∗(X) of a topological space X there exists a sequence of operations m i : H ∗(X)⊗i → H ∗(X), i = 2, 3, … which form a minimal A ∞ -algebra (H ∗(X), {m i }). This structure defines on the bar construction BH ∗(X) a correct differential d m so that (BH ∗(X), d m ) gives cohomology modules of the loop space H ∗(ΩX). In this paper we construct algebraic operations E p, q : H ∗(X)⊗p ⊗ H ∗(X)⊗q → H ∗(X), p, q = 0, 1, 2, 3, … which turn (H ∗(X), {m i }, {E p, q }) into a B ∞ -algebra. This structure defines on BH ∗(X) a correct multiplication, thus determines a cohomology algebra H ∗(ΩX).

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Acknowledgements

The author was supported by the European Union’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 317721.

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Authors and Affiliations

  1. A. Razmadze Mathematical Institute of Tbilisi State University, Tbilsi, Georgia

    Tornike Kadeishvili

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  1. Tornike Kadeishvili
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Correspondence to Tornike Kadeishvili .

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Editors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Palermo, Palermo, Italy

    Giovanni Falcone

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Kadeishvili, T. (2017). Cohomology Operations Defining Cohomology Algebra of the Loop Space. In: Falcone, G. (eds) Lie Groups, Differential Equations, and Geometry. UNIPA Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-319-62181-4_5

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