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Register a new userA decomposition of equivariant K-theory in twisted equivariant K-theories
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For G a finite group and X a G-space on which a normal subgroup A acts trivially, we show that the G-equivariant K-theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized by the orbits of the conjugation action of G on the irreducible representations of A. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of A to the subgroup of G which fixes the isomorphism class of the irreducible representation.
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We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy K-theory of G-schemes (which we construct as an E-infinity-ring) is stable under arbitrary base change, and we deduce that homotopy K-theory of G-schemes satisfies cdh descent.
Let G=SU(2) and let Omega G denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H^*(Omega G) is a divided polynomial algebra Gamma[x]. The algebra Gamma[x] can be described as an inverse limit as k goes to infinity of the symmetric subalgebra in the exterior algebra Lambda(x_1, ...,x_k) in the variables x_1, ..., x_k. We compute the R(G)-algebra structure of the G-equivariant K-theory of Omega G in a way which naturally generalizes the classical computation of the ordinary cohomology ring of Omega G as a divided polynomial algebra Gamma[x]. Specifically, we prove that K^*_G(Omega G) is an inverse limit of the symmetric (S_{2r}-invariant) subalgebra of K^*_G((P^1)^{2r}), where the symmetric group S_{2r} acts in the natural way on the factors of the 2r-fold product (P^1)^{2r} and G acts diagonally via the standard action on each complex projective line P^1.
We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $alpha: X to K(ZZ, 3)$. We introduce the topological index and the analytical index on the bordism group of $alpha$-twisted $Spin^c$-manifolds over $(X, alpha)$, taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topological index and the analytical index. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of $(X, a)$ analogous to Baum-Douglass geometric cycles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold $(X, F)$ with a twisting $alpha: X to K(ZZ, 3)$, which generalizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer families index theorem to twisted cases.
Let $X$ be a compact Hausdorff space, let $Gamma$ be a discrete group that acts continuously on $X$ from the right, define $widetilde{X} = {(x,gamma) in X times Gamma : xcdotgamma= x}$, and let $Gamma$ act on $widetilde{X}$ via the formula $(x,gamma)cdotalpha = (xcdotalpha, alpha^{-1}gammaalpha)$. Results of P. Baum and A. Connes, along with facts about the Chern character, imply that $K^i_Gamma(X) otimes mathbb{C} cong K^i(widetilde{X}slashGamma) otimes mathbb{C}$ for $i = 0, -1$. In this note, we present an example where the groups $K^i_Gamma(X)$ and $K^i(widetilde{X}slashGamma)$ are not isomorphic.
We use the geometry of the space of fields for gauged supersymmetric mechanics to construct the twisted differential equivariant K-theory of a manifold with an action by a finite group.
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