In the present paper we study bundles equipped with extra homotopy conditions, in particular so-called simplicial $n$-bundles. It is shown that (under some condition) the classifying space of 1-bundles is the double coset space of some finite dimensional Lie group. We also establish some relation between our bundles and C*-algebras.
We establish a general "affine representability" result in ${\mathbb A}^1$-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in ${\mathbb A}^1$-homotopy theory. Our results simplify and significantly generalize F. Morel's ${\mathbb A}^1$-representability theorem for vector bundles.
In their construction of the topological index for flat vector bundles, Atiyah, Patodi and Singer associate to each flat vector bundle a particular $\mathbb{C/Z}$-$K$-theory class. This assignment determines a map, up to weak homotopy, from $K_{a}\mathbb{C}$, the algebraic $K$-theory space of the complex numbers, to $F_{t,\mathbb{C/Z}}$, the homotopy fiber of the Chern character. In this paper, we give evidence for the conjecture that this map can be represented by an infinite loop map. The result of the paper implies a refined Bismut-Lott index theorem for a compact smooth bundle $E\rightarro…
We establish a relative version of the abstract "affine representability" theorem in ${\mathbb A}^1$--homotopy theory from Part I of this paper. We then prove some ${\mathbb A}^1$--invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass--Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in ${\mathbb A}^1$--homotopy theory.
We show that each of the three $K$-theory multifunctors from small permutative categories to $\mathcal{G}_*$-categories, $\mathcal{G}_*$-simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these $K$-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe that the right-induced homotopy theory of Bohmann-Osorno $\mathcal{E}_*$-categories is equivalent to the homotopy theory of pointed simplicial categories.
Birational Motivic Homotopy Theories and the Slice Filtration
本文是努力定义一个类似的切片过滤不稳定motivic同伦类别的一部分。我们的方法的灵感来自于这样一个事实，即三角结构不发挥相关作用的建设双有理同伦范畴，以及工作的卡恩-Sujatha \cite{K-theory/0596}双有理动机，其中存在的连接层之间的切片过滤和双有理不变量明确建议。我们的主要结果表明，本文所构造的双有理motivic稳定同伦范畴与切片滤子的正交分量范畴是等价的。依赖于这个等价性，我们能够描述射影空间（包括$\mathbb P ^{\infty}$），Thom空间和爆破的切片。
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…
The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism. We introduce several kinds of bivariant theory associated with a suitable ring spectrum and we construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes. This fundamental classes satisfies all the expected properties from classical intersection theory and lead to Gysin morphisms, Rieman…
