String topology of Poincare duality groups

Details String topology of Poincare duality groups String topology of Poincare duality groups

2005-11-07

arxiv.org/abs/math/0511181v2

Geom. Topol. Monogr. 13 (2008) 1-10

Mathematics

Algebraic Topology

This is the version published by Geometry & Topology Monographs on 22 February 2008

Scientific paper

10.2140/gtm.2008.13.1

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.

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