Heat kernels and the range of the trace on completions of twisted group algebras

The Completion Theorem in twisted equivariant $K$-Theory for proper and discrete actions

We compare different algebraic structures in twisted equivariant K-Theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-Theory, we prove a completion Theorem of Atiyah-Segal type for twisted equivariant K-Theory. Using a Universal coefficient Theorem, we prove a cocompletion Theorem for Twisted Borel K-Homology for discrete Groups.

The Atiyah--Segal completion theorem in twisted K-theory

In this note we prove the analogue of the Atiyah-Segal completion theorem for equivariant twisted K-theory in the setting of an arbitrary compact Lie group G and an arbitrary twisting of the usually considered type. The theorem generalizes a result by C. Dwyer, who has proven the theorem for finite G and twistings of a more restricted type. While versions of the general result have been known to experts, to our knowledge no proof appears in the current literature. Our goal is to fill in this gap. The proof we give proceeds in two stages. We first prove the theorem in the case of a twisting ari…

Operator-valued pseudo-differential operators and the twisted index pairing

The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincaré dual to the continuous trace algebra. Index formulas for pseudo-differential operators twisted by a bundle on the opposite continuous trace algebra are obtained in terms of an Atiyah-Singer type index formula describing the twisted index pairing as index theory for elliptic operators.

The symplectic Verlinde algebras and string K-theory

We construct string topology operations in twisted K-theory. We study the examples given by symplectic Grassmannians, computing the twisted K-theory of the loop spaces of quaternionic projective spaces in detail. Via the work of Freed-Hopkins-Teleman, these computations are related to completions of the Verlinde algebras of Sp(n). We compute these completions, and other relevant information about the Verlinde algebras. We also identify the completions with the twisted K-theory of the Gruher-Salvatore pro-spectra. Further comments on the field theoretic nature of these constructions are made.

A completion theorem for fusion systems

We show that the twisted K-theory of the classifying space of a p-local finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.

扭曲双范畴影子与迹

Twisted Bicategorical Shadows and Traces

本文引入带$G$-扭曲数据的双范畴，构造$g$-扭曲双范畴影子与迹，证明其$G$-Morita不变性，可恢复$g$-扭曲Hattori–Stallings迹与零次扭曲Dennis迹，适用于$C_n$-Green函子等对象。