Goodwillie关于长纽结空间的余单纯模型及其应用

The topology of spaces of knots

We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial. These models build on the calculus of isotopy functors and are weakly homotopy equivalent to knot spaces when the ambient dimension is greater than three. The mapping space model, and the evaluation map on which it builds, is suitable for analysis through differential topology. The cosimplicial model gives rise to spectral sequences which converge to cohomol…

纽结空间的伽罗瓦对称性

Galois symmetries of knot spaces

该研究利用小圆盘operad的伽罗瓦对称性，证明Goodwillie-Weiss谱序列的众多微分在素数p处消失，结合嵌入演算与有限型理论的关联结果，推导出特定阶数的Goodwillie-Weiss逼近是p-局部泛Vassiliev不变量。

Coformality and the rational homotopy groups of spaces of long knots

We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in ${\mathbb R}^d$, where $d\ge 4$, collapses at the $E^2$ page. The main ingredients in the proof are Sinha's cosimplicial model for the space of long knots and a coformality result for the little balls operad.

Operads and chain rules for the calculus of functors

We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct…

Embedding calculus knot invariants are of finite type

We show that the map on components from the space of classical long knots to the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n-1) knot invariant. We also compute the second page in total degree zero for the spectral sequence converging to the components of this tower as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to the…

纽结空间的模型与 Atiyah 对偶

Models for knot spaces and Atiyah duality

本文利用 Atiyah 对偶研究圆周到高维闭流形光滑嵌入空间的余模模型，引入新谱序列计算其同调，并证明相关同伦群同构与上链复形等价。

Homotopy Bott-Taubes integrals and the Taylor tower for spaces of knots and links

This work continues the study of a homotopy-theoretic construction of the author inspired by the Bott-Taubes integrals. Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots. Their techniques were later used by Cattaneo et al. to construct real "Vassiliev-type" cohomology classes in spaces of knots in higher-dimensional Euclidean space. By doing this integration via a Pontrjagin-Thom construction, we constructed cohomology classes in the knot space with arbitrary coefficients. We later showed that a refinement of this …

Sinha谱序列的Fox-Neuwirth基

A Fox-Neuwirth Basis for the Sinha Spectral Sequence

本文通过引入基于Fox-Neuwirth树索引的正则CW复形上的等价余单纯结构，为Sinha谱序列提供了一种适用于所有维度m>=2及所有系数的组合表示。

Knots, operads and double loop spaces

We show that the space of long knots in an euclidean space of dimension larger than three is a double loop space, proving a conjecture by Sinha. We construct also a double loop space structure on framed long knots, and show that the map forgetting the framing is not a double loop map in odd dimension. However there is always such a map in the reverse direction expressing the double loop space of framed long knots as a semidirect product. A similar compatible decomposition holds for the homotopy fiber of the inclusion of long knots into immersions. We show also via string topology that the spac…

嵌入学与结的grope同调

Embedding calculus and grope cobordism of knots

本文证明了嵌入学不变量在长结中的满射性，并揭示了其在计算和理论上的更强能力。

纽结圆周的空间与奇异光滑结构

Spaces of knotted circles and exotic smooth structures

本文证明同胚光滑流形上的纽结空间具有相同的同伦 $(2n-7)$ 型，给出四维情形下基本群的新结果，并利用 Goodwillie-Weiss 塔的新模型得到 $\pi_2$ 的下界。

Poisson structures on the homology of the space of knots

In this article we study the Poisson algebra structure on the homology of the totalization of a fibrant cosimplicial space associated with an operad with multiplication. This structure is given as the Browder operation induced by the action of little disks operad, which was found by McClure and Smith. We show that the Browder operation coincides with the Gerstenhaber bracket on the Hochschild homology, which appears as the E2-term of the homology spectral sequence constructed by Bousfield. In particular we consider a variant of the space of long knots in higher dimensional Euclidean space, and…

A classification of Taylor towers of functors of spaces and spectra

We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this…

关于嵌入空间的Weiss-Williams定理和长结空间的同伦类型

A Weiss-Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots

本文建立了嵌入空间的伪同调结果，描述了块嵌入与普通嵌入在同伦度范围内的差异，并通过分析得出长结空间的同伦类型，包括同调群的计算。

Spaces of smooth embeddings and configuration categories

In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots.