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String topology of Poincaré duality groups

Hossein Abbaspour, Ralph Cohen and Kate Gruher

Geometry & Topology Monographs 13 (2008) 1–10

DOI: 10.2140/gtm.2008.13.1

arXiv: math.AT/0511181

Abstract

Let G be a Poincaré duality group of dimension n. For a given element g ∈ G, let Cg denote its centralizer subgroup. Let LG be the graded abelian group defined by (LG)p = ⊕[g]Hp+n(Cg) where the sum is taken over conjugacy classes of elements in G. In this paper we construct a multiplication on LG directly in terms of intersection products on the centralizers. This multiplication makes LG a graded, associative, commutative algebra. When G is the fundamental group of an aspherical, closed oriented n–manifold M, then (LG)* = H*+n(LM), where LM is the free loop space of M. We show that the product on LG corresponds to the string topology loop product on H*(LM) defined by Chas and Sullivan.

Dedicated to Fred Cohen on the occasion of his 60th birthday

Keywords

Poincaré duality group, string topology

Mathematical Subject Classification

Primary: 55P35

Secondary: 20J06

References
Publication

Received: 7 November 2005
Revised: 19 March 2007
Accepted: 22 March 2007
Published: 22 February 2008

Authors
Hossein Abbaspour
Centre de Mathématiques
École Polytechnique
Palaiseau
France
Ralph Cohen
Department of Mathematics
Stanford University
Stanford CA 94305
USA
Kate Gruher
Department of Mathematics
Stanford University
Stanford CA 94305
USA