Isomorphism Conjecture for homotopy K-theory and groups acting on trees

代数配边与代数 K-理论的 Conner-Floyd 同构

Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory

本文研究了非A^1不变的动机谱稳定∞范畴，利用P^1和加权A^1同伦不变性定义了万有定向动机环谱MGL，并证明了任意qcqs派生概形的代数K理论可通过Conner-Floyd同构从其MGL上同调中恢复。

李、结合与交换拟同构

Lie, associative and commutative quasi-isomorphism

本文证明特征零下交换 dg 代数的交换与结合拟同构等价，并给出同伦完备 dg 李代数的 Koszul 对偶结果，从而确定有理同伦型并推出幂零李代数由泛包络代数识别。

群图的非阿贝尔同调的同伦余终结性

Homotopy cofinality for Non-Abelian homology of group diagrams

本文证明了同伦余终结函子诱导同伦余极限弱等价，据此确立群图非阿贝尔同调在同伦余终结函子原像及左 Kan 扩张下的不变性，并推广构建了单纯集与预层的非阿贝尔 Gabriel-Zisman 及 Baues-Wirsching 同调理论。

Smooth varieties up to A^1-homotopy and algebraic h-cobordisms

We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of al…

The isomorphism conjecture in L-theory: graphs of groups

We study the Fibered Isomorphism Conjecture of Farrell and Jones in L-theory for groups acting on trees. In several cases we prove the conjecture. This includes wreath products of abelian groups and free metabelian groups. We also deduce the conjecture in pseudoisotopy theory for these groups. Finally in B of Theorem 1.1 we prove the L-theory version of [[7], Theorem 1.2].

$K$-theory and homotopies of 2-cocycles on group bundles

This paper continues the author's program to investigate the question of when a homotopy of 2-cocycles $Ω= \{ω_t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, ω_0)) \cong K_*(C^*(\mathcal{G}, ω_1)).$ Building on our earlier work, we show that if $π: \mathcal{G} \to M$ is a locally trivial bundle of amenable groups over a locally compact Hausdorff space $M$, a homotopy $Ω= \{ω_t\}_{t \in [0,1]}$ of 2-cocycles on $\mathcal{G} $ gives rise to an isomorphism $K_*(C…

Tamarkin's construction is equivariant with respect to the action of the Grothendieck-Teichmueller group

Recall that Tamarkin's construction arXiv:math/9803025, arXiv:math/0003052 gives us a map from the set of Drinfeld associators to the set of homotopy classes of L-infinity quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of V. Drinfeld (Algebra i Analiz 2, (1990)) and T. Willwacher arXiv:1009.1654 both the source and the target of this map are equipped with natural actions of the Grothendieck-Teichmueller group $GRT_1$. In this paper, we use the result from arXiv:1305.4699 to prove that this map from the set of Drinfeld associators to the set of homotopy class…

Equivariant stable homotopy methods in the algebraic K-theory of infinite groups

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basic idea to the core and follow the evolution over time in order to explain the advantages of the flexible state that exists today. We end with an outline of the proof of the Borel conjecture in algebraic K-theory for groups of finite asymptotic dimension. We also discuss the relat…

Homology and homotopy in semi-abelian categories

The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups. For many years, a similar categorical framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of non-abelian groups, of rings, Lie algebras, etc. The notion of semi-abelian category due to Janelidze, Marki and Tholen allows a unified study of many important homological properties of such categories. For instance, in any semi-abelian category, the classical diagram lemmas (the…

On the Algebraic Classification of Module Spectra

Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $π_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $π_*R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of $R$-modules. T…

K-theory of toric varieties revisited

After surveying higher K-theory of toric varieties, we present Totaro's old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric schemes. In the special case of a projective simplicial toric scheme over a regular ring one obtains a rational isomorphism between the homotopy K-theory and the direct sum of m copies of the K-theory of the ground ring, m being the number of maximal cones in the underlying fan. Apart from its independent interest, in retrospect, Totaro…

Lifting graph $C^*$-algebra maps to Leavitt path algebra maps

Let $ξ:C^*(E)\to C^*(F)$ be a unital $*$-homomorphism between simple purely infinite Cuntz-Krieger algebras of finite graphs. We prove that there exists a unital $*$-homomorphism $ϕ:L(E)\to L(F)$ between the corresponding Leavitt path-algebras such that $ξ$ is homotopic to the map $\hatϕ:C^*(E)\to C^*(F)$ induced by completion. We show moreover that $\hatϕ$ is a homotopy equivalence in the $C^*$-algebraic sense if and only if $ϕ$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz-Krieger algebras is …

Motivic spectral Mackey functors

We show that if G is a finite constant group acting on a scheme X such that the order of G is invertible in the residue fields of X, then the G-equivariant motivic stable homotopy category of X is equivalent to the stabilization of the category of motivic G-spaces with finite étale transfers over X at the trivial representation sphere. Along the way we obtain several results of independent interest, among them: we construct and study norms in the motivic homotopy theory of stacks, and we extend the homotopy t-structure to DM-stacks and establish some favorable properties.

The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its étale version is …