Loop homology algebra of a closed manifold

2002/03/14 by Yves Félix, Jean‐Claude Thomas, Jean-Claude Thomas +4 · 1 citation
Mathematics · Physics and Astronomy · #17A65 #17B55 #54N45 #55N33 #55P35 #81T30 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories #math.AT #msc:17A65 #msc:17B55 #msc:54N45 #msc:55N33 #msc:55P35 #msc:81T30

paper · pdf · doi:10.48550/arxiv.math/0203137

New version 19 pages

openalex publication_date 2002/03/14 · arxiv created 2003/06/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The loop homology of a closed orientable manifold M of dimension d is the ordinary homology of the free loop space M^S1 with degrees shifted by d, i.e. \mathbb H_*(M^S1) = H_*+d(M^S1). Chas and Sullivan have defined a loop product on \mathbb H_*(M^S1) and an intersection morphism I : \mathbb H_*(M^S1) → H_*(ΩM). The algebra \mathbb H_*(M^S1) is commutative and I is a morphism of algebras. In this paper we produce a model that computes the algebra \mathbb H_*(M^S1) and the morphism I. We show that the kernel of I is nilpotent and that the image is contained in the center of H_*(ΩM), which is in general quite small.

Cited by

• A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology of Truncated Polynomials

Related

• The loop homology algebra of spheres and projective spaces
• Homology transfer products on free loop spaces: orientation reversal on spheres
• Free Loop Space Homology of Highly Connected Manifolds
• Spaces of self-equivalences and free loops spaces
• Free loop space and homology