String topology of Poincaré duality groups

Abstract

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/ D HC 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. 55P35; 20J06 Dedicated to Fred Cohen on the occasion of his 60th birthday