Jens Kaad
In this paper we prove that the multiplicative character of A. Connes and M. Karoubi and the determinant invariant of L. G. Brown, J. W. Helton and R. E. Howe agree up to a canonical homomorphism.
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A universal characterization of noncommutative motives and secondary algebraic K-theory
本文仿照Blumberg--Gepner--Tabuada对代数K理论的普适刻画，给出了非交换动机构造的普适刻画，并由此得到二阶K理论的可余表示定理，为构造迹映射奠定基础。
We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or alternatively, in terms of $η$-invariants of Dirac operators and local correction terms. Initially, the construction of the element in $\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the …
The first main result of this paper is to prove that the convergence of Lott's delocalized eta invariant holds for all differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle, we define a higher analogue of Lott's delocalized eta invariant and prove its convergence when the delocalized cyclic cocycle has at most exponential growth. Our second main result is to obtain an explicit formula of the delocalized Connes-Chern character of all $C^\ast$-algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an expli…
A secondary index map on relative K-homology
本文引入从相对K-同调到相对结构代数K-理论的次级指标映射，证明其与Mayer-Vietoris边界映射相容，并计算偶数维自旋流形上Dirac算子的次级指标，从而精细化非局部化APS指标定理。
Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K-theory, both in the real and complex case. We also improve several comparison results about twisted K-theory of C*-algebras to include multiplicative structures. Our results can also be interpreted in the infinity-categorical setup for parametrized spectra.
This paper is concerned with the algebraic K-theory of locally convex algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that the obstruction for the comparison map between algebraic and topological K-theory to be an isomorphism is (absolute) algebraic cyclic homology and prove the existence of an 6-term exact sequence. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Frechet algebra with bounded app. unit. This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and annou…
Relying on results of Hopkins-Morel, we show that, for $X$ a quasi-projective variety over a field of characteristic zero, the canonical map $Ω_n(X)\to MGL_{2n,n}'(X)$ is an isomorphism. Here $Ω_*(X)$ is the theory of algebraic cobordism defined by Levine-Morel, and $MGL_{*,*}'$ is the Borel-Moore homology version of the theory of algebraic cobordism defined via the algebraic Thom complex in the Morel-Voevodsky motivic stable homotopy category.
We prove that if $Γ$ is a group of polynomial growth then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocyle we thus define a higher analogue of Lott's delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if $Γ$ is of polynomial growth then there is a well defined pairing between delocalized cyclic cocyles and $K$-theory classes of $C^*$-algebraic secondary higher invariants. When this $K$-theory class is that of a…
Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of $V$, This invariant lies between $W(V)$ and the group $KO(V_{\mathbb R})$ of $\mathbb R$-linear topological vector bundles on $V_{\mathbb R}$, the set of real points of $V$. We show that the comparison maps $W(V)\to WR(V_{\mathbb C})$ and $WR(V_{\mathbb C})\to KO(V_{\mathbb R})$ that we define are isomorphisms modul…
We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (inf…
Coalgebraic $K$-theory
本文比较了代数K-理论与余代数K-理论之间的映射，揭示了在特定条件下两者等价，并重新诠释了群的特征。
We introduce a differential extension of algebraic K-theory of an algebra using Karoubi's Chern character. In doing so, we develop a necessary theory of secondary transgression forms as well as a differential refinement of the smooth Serre--Swan correspondence. Our construction subsumes the differential K-theory of a smooth manifold when the algebra is complex-valued smooth functions. Furthermore, our construction fits into a noncommutative differential cohomology hexagon diagram.
The cyclotomic trace provides a comparison of the algebraic K-theory spectrum and a pro-spectrum TR that is built from the cyclic fixed points of topological Hochschild homology. In a previous paper with Ib Madsen, we used this comparison and an approximate evaluation of the structure of the pro-spectrum TR to evaluate the p-adic K-groups of a local field K of mixed characteristic (0,p) with perfect residue field. In this paper we completly determine the structure of the pro-spectrum TR for an algebrac closure of the local field K. This leads us to formulate a conjecture for the structure of t…
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