2006/08/14 by Kate Gruher, Gruher, Kate
Mathematics · Physics and Astronomy · #55P25 #55P35 #55R10 #Algebra over a field #Algebra representation #Algebraic Topology (math.AT) #Black Holes and Theoretical Physics #Combinatorics #Contractible space #Coproduct #Duality (order theory) #Equivariant map #FOS: Mathematics #Frobenius algebra #Geometry #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Nonlinear Waves and Solitons #Product (mathematics) #Pure mathematics #Topology (electrical circuits) #math.AT #msc:55P25 #msc:55P35 #msc:55R10
paper · pdf · doi:10.48550/arxiv.math/0608366
10 pages
arxiv created 2006/08/14 · openalex publication_date 2006/08/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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).