String topology and configuration spaces of two points

Abstract

Given a closed manifold M. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincaré duality model of the manifold, and involves the configuration space of two points on M. We moreover, construct an I B L ∞ -structure on (a model of) cyclic chains on the cochain algebra of M, such that the natural comparison map to the S 1 -equivariant loop space homology intertwines the Lie bialgebra structure on homology. The construction of the coproduct/cobracket depends on the perturbative partition function of a Chern-Simons type topological field theory. Furthermore, we give a construction for these string topology operations on the absolute loop space (not relative to constant loops) in case that M carries a non-vanishing vector field and obtain a similar description. Finally, we show that the cobracket is sensitive to the manifold structure of M beyond its homotopy type. More precisely, the action of Diff(M) does not (in general) factor through aut(M). CONTENTS 1. Introduction 1.1. Statement of results 2.2. Operads, Com ∞ -and Com ∞ -algebras 2.3. Hochschild and cyclic complex 2.4. Pullback-pushout lemma 2.5. Fulton-MacPherson-Axelrod-Singer compactification of configuration spaces 2.6. The Lambrechts-Stanley model of configuration space 2.7. Graph complex models for configuration spaces 2.8. Graph complex (Lie algebra) GC H and GC M , following [9] 3. String topology operations 3.1. Preliminaries 3.2. String product (Cohomology coproduct) 3.3. String coproduct (cohomology product) 3.4. String coproduct (χ(M) = 0 case) 3.5. Definition of string bracket and cobracket 4. Cochain complex models 4.1. Iterated integrals and model for path spaces 4.2. Splitting map 4.3. Connes differential 4.4. Space of figure eights 4.5. S 1 -equivariant loops 5. Cochain zigzags for the string product and bracket 5.1. Product 5.2. Coproduct (relative to constant loops case) 5.3. Coproduct (χ(M) = 0 case) 6. Simply-connected case, and proofs of Theorems 1.2 and 1.3, and the first part of Theorem 1.5 7. Involutive homotopy Lie bialgebra structure on cyclic words 7.1. Lie bialgebra of cyclic words 7.2. Twisting by Maurer-Cartan elements 1

From gravity to string topology

“…More details on the non-triviality of this action will be given in the next section where we show that this action factors through a new dg string topology properad which has rich cohomology and which explains in a unified way many well-known universal operations in string topology introduced and studied earlier in [CS1,CS2,CEG,We,NW,Me2].…”

Invariance of the Goresky-Hingston algebra on reduced Hochschild homology

“…The reason for this name is the analogy and similarities between the properties of the algebraic product ⋆ and the geometrically defined operation of [8] of the same degree. Recently, this analogy has been made precise: it has been announced in [15] that if A is a Poincaré duality model for the rational polynomial differential forms on a simply connected oriented closed manifold M (as constructed in [14]) then the product ⋆ at the level of a relative version of Hochschild homology of A corresponds to a topologically defined version of the Goresky-Hingston product on H * (LM, M ; Q), the rational cohomology of the free loop space LM relative to constant loops.…”

“…There are different ways of making choices to construct string topology operations rigorously. Some use geometric and topological methods to describe intersections ( [8], [9], [7], [18]), others homotopy theoretic techniques [6], and others start with an algebraic chain or cochain model for a manifold, such as the commutative differential graded algebra of differential forms, and then make choices algebraically in order to construct operations using the relationship between Hochschild homology and the free loop space ( [20], [2], [19], [12], [15]). This article is concerned with an algebra structure defined through this last approach.…”

“…(1) Let M be a simply connected oriented closed manifold of dimension k and A a Poincaré duality model for the cdga of rational differential forms A(M ). The isomorphism class of the algebra structure on s The original geometric constructions for the Goresky-Hingston operation take place at the level of relative homology H * (LM, M ), as described in ( [18], [8], [9], [15]). An interesting question is to give a geometric or topological explanation of the extension of the operation on relative homology H * (LM, M ) to reduced homology H * (LM ) so that it coincides with the algebraic extension.…”

Chain-level equivariant string topology for simply connected manifolds

“…These issues have been addressed by Naef and Willwacher [33], and independently by the second author [22]. To describe the latter appproach, let us abbreviate "differential Poincaré duality algebra" by "dPD algebra".…”