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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


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


Poincaré duality group, string topology

Mathematical Subject Classification

Primary: 55P35

Secondary: 20J06


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

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