Browsing by Subject "K-theory"
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Item Differential equivariant K-theory(2009-05) Ortiz, Michael Luis, 1979-; Freed, Daniel S.Following Hopkins and Singer, we give a definition for the differential equivariant K-theory of a smooth manifold acted upon by a finite group. The ring structure for differential equivariant K-theory is developed explicitly. We also construct a pushforward map which parallels the topological pushforward in equivariant K-theory. An analytic formula for the pushforward to the differential equivariant K-theory of a point is conjectured, and proved in the boundary case and for ordinary differential K-theory in general. The latter proof is due to K. Klonoff.Item Differential T-equivariant K-theory(2013-05) Alter, Mio Ilan; Freed, Daniel S.For T the circle group, we construct a differential refinement of T-equivariant K-theory. We first construct a de Rham model for delocalized equivariant cohomology and a delocalized equivariant Chern character based on [19] and [14]. We show that the delocalized equivariant Chern character induces a complex isomorphism. We then construct a geometric model for differential T-equivariant K-theory analogous to the model of differential K-theory in [27] and deduce its basic properties.Item The Goodwillie tower of free augmented algebras over connective ring spectra(2014-12) Pancia, Matthew; Blumberg, Andrew J.Let R be a connective ring spectrum and let M be an R-bimodule. In this paper we prove several results that relate the K-theory of R⋉M and T[superscript M, subscript R] to a “topological Witt vectors” construction W(R; M), where R ⋉ M is the square-zero extension of R by M and T [superscript M, subscript R] is the tensor algebra on M. Our main results include a desciption of the Taylor tower of K(R ⋉ (−)) and the derived functor of K̃(TR(−)) on the category of R-bimodules in terms of the Taylor tower of W(R;−). W(R;−) has an easily described Taylor tower, given explicitly by Lindenstrauss and McCarthy in [17]. Our main results serve as generalizations of the results for discrete rings in [17, 18] and also extend the computations by Hesselholt and Madsen [15] showing that π₀(TR(R; p)) is isomorphic to the p-typical Witt vectors over R when R a commutative ring.Item An index theorem in differential K-theory(2008-05) Klonoff, Kevin Robert, 1972-; Freed, Daniel S.We construct a geometric model for differential K-theory, and prove it is isomorphic to the model proposed in [25]. We construct differential K-orientations for families and elucidate the pushforward map given in [25] in detail. We prove a geometric index theorem for odd dimensional manifolds. Finally, using this index theorem and the holonomy theorem of Bismut and Freed from [10], we prove what may be considered a special case of a geometric refinement of the Aityah-Singer index theorem.Item K-theoretic computation of the Verlinde ring(2018-05-04) Zakharevich, Valentin; Freed, Daniel S.; Neitzke, Andrew; Ben-Zvi, David; Perutz, Timothy; Teleman, ConstantinWe compute Verlinde rings of the groups SU(3) semidirect product Z/2Z and Spin(8) semdirect product Sym(3) at level 1. We use the K-theory formulation developed by Freed, Hopkins and Teleman. More precisely, we compute the twisted equivariant K-theory of G acting on itself by conjugation. The fusion product, corresponding to the Pontryagin product on the level of K-theory, is partially computed.