String topology of Poincare duality groups

Abstract

Let G be a Poincare duality group of dimension n. For a given element g in G, let C_g denote its centralizer subgroup. Let L_G be the graded abelian group defined by (L_G)_p = oplus_{[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.