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

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