Jack Morava
The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Deligne's ideas on motivic Galois groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Donaldson-Thomas invariant is expressed as the weighted Euler characteristic of the so-called Behrend (constructible) function. In \cite{Behrend} Behrend introduced a DT-type invariant for a morphism. Motivated by this invariant, we extend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues. We also discuss a categorification of the DT-type invariant for a morphism from a bivariant-theoretic viewpoint, and we finally pose some related questions for further investigations.
We show that a triangulated motivic category admits categorical Thom isomorphisms for vector bundles with an additional structure if and only if the generalized motivic cohomology theory represented by the tensor unit object admits Thom classes. We also show that the stable $\mathbb{A}^1$-derived category does not admit Thom isomorphisms for oriented vector bundles and, more generally, for symplectic bundles. In order to do so we compute the first homology sheaves of the motivic sphere spectrum and show that the class in the coefficient ring of $\mathbb{A}^1$-homology corresponding to the seco…
Motivic Steenrod algebra and Thom obstructions without desingularization
本文给出无需消解的证明，确认特征零域上双稳定模ℓ motivic上同调运算等同于motivic Steenrod代数，并应用此证明复代数环上所有Thom消解障碍消失。
Weil restriction, normal bundles and motivic Thom spaces
本文通过Weil限制研究法丛与动机Thom空间，证明Weil限制保持向量丛并扩展法丛结果，进而建立与动机Thom空间的相容性，并在有理系数动机上同调中证明Weil限制映射与范数映射一致且保持Thom类。
In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.
Let $k$ be a field, let $R$ be a commutative ring, and assume the exponential characteristic of $k$ is invertible in $R$. In this note, we prove that isomorphisms in Voevodsky's triangulated category of motives $\mathcal{DM}(k;R)$ are detected by motivic homology groups of base changes to all separable finitely generated field extensions of $k$. It then follows from previous conservativity results that these motivic homology groups detect isomorphisms between certain spaces in the pointed motivic homotopy category $\mathcal{H}(k)_*$.
Computing framed motives
本文发展计算与动机Thom谱相关的框架动机的方法，主要工具是联系框架动机与框架动机上同调的动机Atiyah-Hirzebruch谱序列，并引入扭转框架动机上同调理论，最终重构有理稳定动机同伦理论。
The real and complex Betti realizations of motivic multiplicative Thom spectra
本文研究实贝蒼生实现将motivic谱映射到拓扑谱，证明了motivic Thom谱与拓扑Thom谱在实（复）实现下的等价性，并识别了实实现的ko的E1环结构等价于实L理论谱。
Using geometric homology and cohomology we give a simple and conceptual proof of the Thom isomorphism theorem.
We establish an isomorphism between the stable homotopy groups of the 2-completed motivic sphere spectrum over the real numbers and the corresponding stable homotopy groups of the 2-completed Z/2-equivariant sphere spectrum, in a certain range of dimensions.
Fibrant resolutions for motivic Thom spectra
本研究利用Voevodsky的框架对应理论与框架动机工具，构造动机Thom谱的多种显式纤维状消解，证明相关双谱的表示性，并给出拓扑应用及代数配边谱的计算。
Let E be a circle-equivariant complex-orientable cohomology theory. We show that the fixed-point formula applied to the free loopspace of a manifold X can be understood as a Riemann-Roch formula for the quotient of the formal group of E by a free cyclic subgroup. The quotient is not representable, but (locally at p) its p-torsion subgroup is, by a p-divisible group of height one greater than the formal group of E.
Consider the Tate twist $τ\in H^{0,1}(S^{0,0})$ in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map $τ\colon S^{0,-1} \to S^{0,0}$, with cofiber $Cτ$. We show that this motivic 2-cell complex can be endowed with a unique $E_{\infty}$ ring structure. Moreover, this promotes the known isomorphism $π_{\ast,\ast} Cτ\cong \mathrm{Ext}^{\ast,\ast}_{BP_{\ast}BP}(BP_{\ast},BP_{\ast})$ to an isomorphism of rings which also preserves higher products. We then consider the closed symmetric monoidal category $({ }_{Cτ}\textbf…
A homotopy coherent Pontryagin-Thom isomorphism
构造了光滑流形上具有传递的同伦不变层的一个可表现对称幺半稳定∞-范畴，其单位恰为（结构化）几何配边，并证明单位层的自同态环可典范地等同于关联的E_∞-托姆环谱，给出了庞特里亚金-托姆同构的E_∞-提升。
We develop equivariant KK-theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K-theory.
The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample and clarify the origin of the mistake.
An equivariant Thom isomorphism theorem in operator K-theory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a non-commutative C*-algebra associated to a bundle E -> M, equipped with a compatible connection, which plays the role of the algebra of functions on the infinite dimensional total space E. If the base M is a point, we obtain the Bott periodicity isomorphism theorem of Higson-Kasparov-Trout for infinite dimensional Euclidean spaces. The construction applied to an…
Equivariant Framed 1-Manifolds and the Pontryagin-Thom Isomorphism
本文研究等变框架1-流形与等变稳定同调群之间的同构关系，明确识别了每个元素在等变同调群中的像。
Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over the sphere bundle $\mathord{\mathbb S}( \mathbb{R} \oplus V)$ in terms of vector bundles and Clifford module bundles over $…
The unit map of the algebraic special linear cobordism spectrum
本文利用框架对应技术计算动机Thom谱的无限$\mathbb{P}^1$-环空间，并证明代数特殊线性配边谱的单位映射在$\mathbb{G}_m$-同伦层上诱导同构。
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。
Motivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of Felix-Oprea-Tanré by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give a criterion of representability of classes by submanifolds, generalising results of Papadima. Along the way, we study issues of formality and give formulas for Massey products of Thom spaces. Las…
In this article we further the study of noncommutative numerical motives. By exploring the change-of-coefficients mechanism, we start by improving some of our previous main results. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP on the category of noncommutative Chow motives. This allows us to introduce the correct non…
The Pontryagin-Thom theorem for families of framed equivariant manifolds
本文证明了以紧空间为参数的带标架等变流形族的庞特里亚金-托姆定理的推广，拓展了该定理的适用范围。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。