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Register a new userElliptic Theory on Manifolds with Corners: II. Homotopy classification and $K$-Homology
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We establish the stable homotopy classification of elliptic pseudodifferential operators on manifolds with corners and show that the set of elliptic operators modulo stable homotopy is isomorphic to the K-homology group of some stratified manifold. By way of application, generalizations of some recent results due to Monthubert and Nistor are given.
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Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the official equivariant K-homology groups) and show that these are isomorphism.
For every $infty$-category $mathscr{C}$, there is a homotopy $n$-category $mathrm{h}_n mathscr{C}$ and a canonical functor $gamma_n colon mathscr{C} to mathrm{h}_n mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Based on the idea of the homotopy $n$-category, we introduce the notion of an $n$-derivator and study the main examples arising from $infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $infty$-derivators and prove that the canonical comparison map from the Waldhausen $K$-theory of $mathscr{C}$ to the $K$-theory of the associated $n$-derivator $mathbb{D}_{mathscr{C}}^{(n)}$ is $(n+1)$-connected. We also prove that this comparison map identifies derivator $K$-theory of $infty$-derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy $n$-category, we define also a $K$-theory space $K(mathrm{h}_n mathscr{C}, mathrm{can})$ associated to $mathrm{h}_n mathscr{C}$. We prove that the canonical comparison map from the Waldhausen $K$-theory of $mathscr{C}$ to $K(mathrm{h}_n mathscr{C}, mathrm{can})$ is $n$-connected.
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.
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.
We show that Connes B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.
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