A duality between string topology and the fusion product in equivariant K-theory

Details A duality between string topology and the fusion product in equivariant K-theory A duality between string topology and the fusion product in equivariant K-theory

2006-08-14

arxiv.org/abs/math/0608366v1

Mathematics

Algebraic Topology

10 pages

Scientific paper

Let G be a compact Lie group. Let E be a principal G-bundle over a closed manifold M, and Ad(E) its adjoint bundle. In this paper we describe a new Frobenius algebra structure on h_*(Ad(E)), where h_* is an appropriate generalized homology theory. Recall that a Frobenius algebra has both a product and a coproduct. The product in this new Frobenius algebra is induced by the string topology product. In particular, the product can be defined when G is any topological group and in the case that E is contractible it is precisely the Chas-Sullivan string product on H_*(LM). We will show that the coproduct is induced by the Freed-Hopkins-Teleman fusion product. Indeed, when M is replaced by BG and h_* is K-theory the coproduct is the completion of the Freed-Hopkins-Teleman fusion structure. We will then show that this duality between the string and fusion products is realized by a Spanier-Whitehead duality between certain Thom spectra of virtual bundles over Ad(E).

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