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