An open-closed cobordism category with background space
Elizabeth Hanbury
本文引入了一个带背景空间映射的开闭配边范畴。我们确定的分类空间，这一类的某些类别的背景空间。其关键是具有扭系数的映射类群的同调稳定性。
In this paper we introduce an open-closed cobordism category with maps to a background space. We identify the classifying space of this category for certain classes of background space. The key ingredient is the homology stability of mapping class groups with twisted coefficients.
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We show an equivalence of infinite loop spaces between the classify- ing space of the cobordism category, with infinite loop space structure induced by taking disjoint union of manifolds, and the infinite loop space associated to the Madsen-Tillmann spectrum.
The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d=2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one.
The classifying space of the embedded cobordism category has been identified in by Galatius, Tillmann, Madsen, and Weiss as the infinite loop space of a certain Thom spectrum. This identifies the set of path components with the classical cobordism group. In this paper, we give a geometric interpretation of the higher homotopy groups as certain cobordism groups where all manifolds are now equipped with a set of orthonormal sections in the tangent bundle. We also give a description of the fundamental group as a free group with a set of geometrically intuitive relations.
We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.
For a finite group $G$, we define the $G$-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of $G$. Also, we find an isomorphism of its fundamental group onto the direct sum $\mathbb{Z}\oplus Ω_2^{SO}(BG)$, where $Ω_2^{SO}(BG)$ is the free oriented $G$-bordism group in dimension two, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product $G/[G,G]\times S^1\times X^G$, where $π_1(X^G)=Ω_2^{SO}(BG)$. Fin…
We prove that a pairing between the Fukaya category and the oo-category of Lagrangian cobordisms respects mapping cones. This is another step toward constructing a lift of Fukaya categories to the level of spectra (in the sense of stable homotopy theory). As corollaries, we show that the map in our previous work from cobordism groups to Floer cohomology lifts to the level of spectra, and one also recovers some results of Biran and Cornea for what we call "vertically collared" cobordisms.
Cobordism category of plumbed 3-manifolds and intersection product structures
本文引入一类具有一定代数态射的分次交换环，研究了垂直3-流形的配边范畴。特别是，我们定义了一个非结合分配代数，给出了必要条件的抽象态射之间的同调的两个铅垂3-流形几何实现的协边。本文还考虑了同调配边幺半群，利用w-不变量给出了同调3-球面属于某个同调3-球面所对应的惯性群的必要条件.
In this extended note we give a precise definition of fully extended topological field theories à la Lurie. Using complete $n$-fold Segal spaces as a model, we construct an $(\infty,n)$-category of $n$-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms $n\operatorname{Cob}$ and the cobordism bicategory $n\operatorname{Cob}^{ext}$ from it.
For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its simplicial nerve) as the fixed points of the infinite loop space of an equivariant spectrum.
We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's $S_{\bullet}$-construction and therefore it defines a model for Waldhausen $K$-theory. As an example, we discuss this model for $A$-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's $A(*)$. We also review the canonical map from the cobordism category of manifolds to $A$-theory from this viewpoint.
Universal property of graph cobordisms
本文研究了图cobordisms的普遍性质，展示了GrCob作为Gr的自由扩展，并识别了因子同调上的universal natural operations。
Fix a suitably convex, exact symplectic manifold M. We consider the stable oo-category Lag(M) of non-compact Lagrangians whose (higher) morphisms are (higher) Lagrangian cobordisms between them. We show that this oo-category pairs with the Fukaya category Fuk(M) of compact branes. In fact, we also show that there is a subcategory of Lag(M) which pairs with the wrapped Fukaya category of M. This is a first step in a project to enrich wrapped Fukaya categories over cobordism spectra. As a corollary, we show that cobordant compact branes are equivalent in the Fukaya category. We will also mention…
Cobordism categories of manifolds with corners
本文研究了带角流形的配边范畴的拓扑。具体地说，如果{Cob}_{d，<k>}是其对象是一个固定维数d的范畴，其余维数的角点小于或等于k，则我们将分类空间B{Cob}_{d，}的同伦类型确定<k>为某个Thom谱图的同伦上极限的零空间.我们还确定了同伦类型对应的配边范畴时，额外的切结构假设的流形上。这些结果推广了Galatius，Madsen，Tillmann和Weiss的结果，证明是他们方法的一种改进.作为一个应用，我们描述了同伦类型的开弦和闭弦的范畴与背景空间X，以及它的高维类似物。这概括了Baas-Cohen-Ramirez和Hanbury的工作。
For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectrum $\text{MT}(d)$ introduced in [arXiv:math/0605249]. We obtain an analogue of the Bockstein-Sullivan exact couple that arises between the classical bordism theories $MO$ and $MO_{P}$ on the level of cobordism categories.
We show the $\TT^2$-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the $\TT^2$-cobordism classes of $\CP^2$. We construct nice oriented $\TT^2$ manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.
The homotopy type of the topological cobordism category
本文定义拓扑流形配边范畴，证明 $d\neq4$ 时其分类空间弱等价于 $Ω^{\infty-1}MTTop(d)$，并借助光滑化理论与 GMTW 的光滑情形计算推广到带切向结构及边界的版本。
This posting is INVALID and has been WITHDRAWN. Please see N. Kitchloo and JM, Spin cobordism categories in low dimensions, arXiv:0908.3114, for a replacement and revision.
The goal of this work is to describe a categorical formalism for (Extended) Topological Quantum Field Theories (TQFTs) and present them as functors from a suitable category of cobordisms with corners to a linear category, generalizing 2d open-closed TQFTs to higher dimensions. The approach is based on the notion of an n-fold category by C. Ehresmann, weakened in the spirit of monoidal categories (associators, interchangers, Mac Lane's pentagons and hexagons), in contrast with the simplicial (weak Kan and complete Segal) approach of Jacob Lurie. We show how different Topological Quantum Field T…
Bökstedt and Madsen defined an infinite loop map from the embedded $d$-dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic $K$-theory of $BO(d)$ in the sense of Waldhausen. The purpose of this paper is to establish two results in relation to this map. The first result is that it extends the universal parametrized $A$-theory Euler characteristic of smooth bundles with compact $d$-dimensional fibers, as defined by Dwyer, Weiss and Williams. The second result is that it actually factors through the canonical unit map $Q(BO(d)_+) \to A(BO(d))$.
The homotopy type of the PL cobordism category. I
本文构造分片线性配边范畴并证明其分类空间弱同伦等价于无限环路空间，为在PL拓扑中建立Madsen-Weiss型结果奠定基础。
Let $\mathcal{A}_{g,d}$ be the (topological) cobordism category of orientable surfaces whose connected components are homeomorphic to either $S^1 \times I$ with one incoming and one outgoing boundary component or the surface $Σ_{g,d}$ of genus $g$ and $d$ boundary components that are all incoming. In this paper, we study the homotopy type of the classifying space of the cobordism categories $\mathcal{A}_{g,d}$ and the associated (ordinary) cobordism categories of their connected components $\mathbb{A}_d$. $\mathcal{A}_{0,2}$ is the cobordism category of complex annuli that was considered by Co…
In this manuscript, we consider cobordism operations in the $2$-dimensional labeled open-closed topological quantum field theory for the classifying space of a connected compact Lie group in the sense of Guldberg. In particular, it is proved that the whistle cobordism operation is non-trivial in general provided the labels are in the set of maximal closed subgroups of the given Lie group. The non-triviality of cobordism operations induced by gluing the whistle with the opposite and other labeled cobordisms is also discussed.
The homotopy type of the Poincaré cobordism category for surfaces
本文定义曲面配边范畴的变体，证明其分类空间函子非1-可切除，并给出其一阶导数等价于Thom谱的结果。
TQFT string operations in open-closed string topology
开闭弦拓扑将拓扑量子场论（TQFT）操作（即弦操作）与开闭配边曲面相关联，这些操作仅依赖于曲面的同胚类型，并且满足缝合性质。我们表明，大多数TQFT字符串操作消失在开闭弦拓扑。我们描述了那些开闭协边与消失字符串操作，并给出了一个简短的列表，开闭协边与可能的非平凡字符串操作。
There are two categorifications of the Jones polynomial: "even" discovered by M.Khovanov in 1999 and "odd" dicovered by P.Ozsvath, J.Rasmussen and Z.Szabo in 2007. The first one can be fully constructed in the category of cobordisms (strictly: in the additive closure of that category), where we can build a complex for a given tangle and show its invariance under Reidemeister moves. The even link homology is given by a monoidal functor from cobordisms into modules. However, odd link homology cannot be obtained in this way. In this paper I fill this gap. I enrich cobordisms with chronologies (pr…
This paper examines the category C^k_{d,n} whose morphisms are d-dimensional smooth manifolds that are properly embedded in the product of a k-dimensional cube with an (d+n-k)-dimensional Euclidean space. There are k directions to compose k-dimensional cubes, so C^k_{d,n} is a (strict) k-tuple category. The geometric realization of the k-dimensional multi-nerve is the classifying space BC^k_{d,n}. At the end of the paper we construct an infinite loop map to Waldhausens K-theory. ΩBC^1_{d,n}-> A(BO(d)), We believe that the map factors through Ω^\inftyΣ^\infty(BO(d)_+) and that the composite B{D…
The homotopy type of the PL cobordism category. II
本文证明了光滑配边范畴同伦型定理的 PL 类比，通过引入 PL Madsen-Tillmann 谱并建立弱同伦等价，同时讨论了拓扑版本的推广。
Open-Closed TQFT String Operations for Disc Cobordisms, Simultaneous Saddle Interactions, and Constant Homology Classes
以前，我们表明，大多数开闭拓扑量子场论（TQFT）弦操作消失，包括所有更高的亏格TQFT操作，我们描述了一个小列表亏格零开闭TQFT弦操作，可以是非平凡的。在本文中，我们考虑开闭弦运算与开闭协边同胚到磁盘。这些运算构成亏格零弦运算的主要部分，其中包括两个开弦在内点相互作用的鞍弦运算。我们不仅证明了盘弦运算与它们的半裤子分解无关，而且证明了这些盘弦运算可以通过同一点上的开弦的\emph{simultaneous}鞍相互作用来计算，并且它们在某些闭可定向子流形上的常数开弦的同调类中取值，我们将精确地确定这些值。我们还将讨论基本常数同调类在开闭弦拓扑中所起的作用。我们的主要工具是鞍形相互作用图及其变形。
We prove an h-principle with boundary condition for a certain class of topological spaces valued sheaves. The techniques used in the proof come from the study of the homotopy type of the cobordism categories, and they are of simplicial and categorical nature. Applying the main result of this paper to a certain sheaf we find another proof of the Madsen-Weiss theorem, that describes the homotopy type of the classifying space of the cobordism category as the loop space of the Thom space of the complement of the tautological bundle over the Grassmannians.
The positive scalar curvature cobordism category
本文研究正标量曲率度量空间的cobordism范畴，证明其具有无限循环空间结构，并展示微分同胚群作用的刚性定理。
A universality theorem for Voevodsky's algebraic cobordism spectrum
一个代数版本的定理由于奎伦证明。更精确地说，对于一个基域k，我们考虑具有arXiv：0709.3905v1 [math.AG]中描述的对称么半群结构的P^1-谱的运动稳定同伦范畴SH（k）。代数配边P^1-谱MGL被认为是一个具有正则定向的交换幺半群。对于SH（k）范畴中的交换幺半群E，我们用E的所有方向集来确定运动稳定同伦范畴中从MGL到E的幺半群同态集。这个结果最初由G. arXiv中的Vezzosi：math/0004050 v2 [math.AG]。
Nested cobordisms, Cyl-objects and Temperley-Lieb algebras
本文研究嵌套manifolds的离散cobordism类别，通过分层Morse理论生成生成元，并在低维例子Cyl中给出生成元关系。定义Cyl-对象及其与Temperley-Lieb代数等结构的关系，提出新的代数构造。
The topology of the category of open and closed strings
本文研究开弦和闭弦的配边范畴的拓扑。这是一个2-范畴，其中的对象是紧致的单流形，其边界分量被标记为索引集（“D-膜”的集合），1-态射是有边界流形的配边，2-态射是曲面配边的仿射。我们的方法和技术是直接推广的美国。Tillmann在她的闭弦范畴的研究中。我们输入马德森和韦斯关于稳定的映射类群的拓扑的引人注目的定理，以确定作为无限闭路空间的开弦和闭弦范畴的几何实现的同伦类型。
The homotopy category of the bordism category $hBord_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $Ω^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms wit…
Cubical cospans and higher cobordisms (Cospans in algebraic topology, III)
在分别讨论了弱立方范畴和{ \it collarable}余跨距的两篇文章之后，我们综合起来构造了拓扑空间的立方{ \it collared}余跨距的{\it weak}立方范畴.我们还建立了第二个结构，称为{ \it quasi}立方范畴，形成的任意立方cospans连接同伦推出。这种结构更简单但更弱，具有{ \it lax}恒等式。它包含了一个类似的框架，cobordisms的流形与角落，因此可能是基础，以扩大研究的TQFT的第二部分，以更高的立方度。
Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1+1
我们将同伦量子场论定义为拓扑量子场论，其流形被赋予以映入某个背景空间X的形式的额外结构。我们还建立了同伦配边范畴HCobord（n，X），使得HQFT是从该范畴到线性空间范畴的函子.然后，我们得到了HCobord}（n，X）的一些非常一般的性质，包括它只依赖于X的（n+1）-同伦类型的事实。我们还证明了一个HQFT与目标空间X和在n+1维意味着存在的几何结构在X，特别是，平面gerbes使他们的外观。本文给出了Hcobord（n，X）的一个完整刻画，其中n=1（或1+1情形），X是Eilenberg-Maclane空间K（G，2）.在最后一节中，我们推导出这些HQFT的状态和模型。
Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the classifying space of the cobordism category with objects (d-1)-dimensional manifolds embedded in R^\infty. In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius, to identify the homotopy type of the cobordism category with objects (d-1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincare duality, relating a spac…
The toric cobordisms
由环面$T^2$纤维化的光滑闭3-流形$M$由元素$ϕ\in GL(2,\mathbb{Z})$刻画.证明了$M$是曲面上环面纤维化的4-流形的边界，使得$M$上的丛结构是4-流形上的丛结构的限制当且仅当$ϕ$来自换位子群$(GL(2,\mathbb{Z}))'$ .在环面纤维化的闭三维流形类中引入了定向和无定向配边的概念。事实证明，在这种情况下，配边形成了一个群，即定向情况下的$\mathbb{Z}_{12}$和无定向情况下的$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$。当有向配边曲面是可定向曲面时，证明了它的极小亏格可以用Culler算法求出。
The circle transfer $QΣ(LX_{hS^1})_+ \to QLX_+$ has appeared in several contexts in topology. In this note we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let $C_1(X)$ denote the 1-dimensional cobordism category and let $Circ(X) \subset C_1(X)$ denote the subcategory whose objects are disjoint unions of unparametrised circles in $\mathbb{R}^\infty$. Multiplication in $S^1$ induces a functor $Circ(X) \to Circ(LX)$, and the composition of this functor with the inclusion of $Circ(LX)$ into $C_1(LX)$ is homotopic …
Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz-Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalizing Khovanov cohomology. In this paper we give mo…
We prove that Lagrangian cocores and Lagrangian linking disks of a stopped Weinstein manifold generate the Lagrangian cobordism infinity-category. As a geometric consequence, we see that any brane (after stabilization) admits a Lagrangian cobordism to a disjoint union of some standard collection of branes (cocores, linking disks, and a zero object). For example, when our stopped Weinstein manifold is a point stopped by itself, we find that any exact brane in Euclidean space admits a Lagrangian cobordism to a disjoint union of cotangent fibers and a zero object. (This is a stronger statement th…
A Heegaard-Floer TQFT for link cobordisms
该研究提出一种不依赖装饰的Heegaard-Floer同调函子，建立了从有向链环与配边范畴到二元域上多项式模范畴的对应，补充了该函子的基础性质，完善了链环配边的拓扑量子场论构造。
Symmetric monoidal extensions and graph cobordisms between finite sets
本文针对对称幺半$(\u221e,n)$-范畴，发展了一套工具以显式描述自由对称幺半扩张，并应用于构造有限集间图配边的对称幺半$(\u221e,2)$-范畴，作为有限集范畴的扩张，恢复了Barkan--Steinebrunner和Galatius的结果。
We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n \geq 4, n \neq 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.
Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
本文研究了一类特殊的二维扩展拓扑量子场论，我们称之为开-闭拓扑量子场论。这些定义在开闭协边上，我们指的是光滑紧致定向的2-流形，其角具有特定的全局结构，以模拟开弦和闭弦世界单的光滑拓扑。我们证明了开闭TQFT范畴等价于知识丰富的Frobenius代数范畴。一个有知识的Frobenius代数（A，C，i，i^*）由一个对称Frobenius代数A、一个交换Frobenius代数C和一个代数同态i：C->A与对偶i^*：A->C在一定条件下构成。这一结果是通过提供开闭配边范畴的生成元和关系描述来实现的。为了证明我们的关系的充分性，我们提供了这种配边的一个标准形式，其特征在于拓扑不变量。从一个任意的这样的配边，我们构造一个序列的移动（广义手柄滑动和手柄取消），将给定的配边转换成正常的形式。利用开闭配边范畴的生成元和关系描述，证明了它等价于由知识丰富的Frobenius代数自由生成的对称幺半范畴范畴.我们的形式然后推广到开闭协边与标记的自由边界组件的上下文中，即开闭弦世界表与D-膜标签在其自由边界。
A diagrammatic presentation of the category 3Cob
该研究用平面图表定义与三维配边范畴等价的范畴，完整描述该范畴内的合成运算，为3Cob范畴提供了直观的图表化表示方式。
(Co)bordisms of manifolds and maps are fundamental and important objects in algebraic and differential topology of manifolds and related studies were started by Thom etc.. Cobordisms of Morse functions were introduced and have been studied as a branch of the algebraic and differential topological theory of Morse functions and their higher dimensional versions, or the global singularity theory, by Kalmár, Ikegami, Sadykov, Saeki, Wrazidlo, Yamamoto etc. since 2000s. Cobordism relations are in most cases defined as the following for example; two closed manifolds of a fixed dimension or maps on t…
Algebraic cobordism via spans
本文通过跨度范畴定义了配备通用线丛数据的∞-范畴的代数cobordism，恢复了Voevodsky、Gepner和Snaith建立的Thom谱模型，并证明了在满足分裂原理性质的任何定向函子上，投影包公式和相应的Chern类和Whitney和恒等式成立，同时将跨度形式应用于完美oid几何。
Internal Reshetikhin-Turaev TQFT
该研究从带余端的ribbon范畴出发，推广Turaev的TQFT构造，通过特殊链环缠绕表示配边、结合容许着色与Kirby微积分推广，得到取值于透明对象对称幺半子范畴的内蕴TQFT，并验证其在模、预模范畴下的对应性质。
For a given list of closed manifolds $Σ_k=(P_1,...,P_k)$, we construct a cobordism category $\mathbf{Cob}_{d}^{Σ_{k}}$ of embedded manifolds with Baas-Sullivan singularities of type $Σ_k$. Our main results identify the homotopy type of the classifying spaces $B\mathbf{Cob}_{d}^{Σ_{k}}$ of these cobordism categories with that of the infinite loop-space of a certain spectrum, $\mathsf{MT}_{Σ_{k}}(d)$. Our results generalize of the work of Galatius, Madsen, Tillmann, and Weiss from arXiv:math/0605249. It also generalizes the work of Genauer on the cobordism categories of manifolds with corners fr…
Representations of the homotopy surface category of a simply connected space
我们引入空间的同伦曲面范畴，它将圆和曲面的1+1维配边范畴推广到引入背景空间的情况。我们解释了如何为一个简单的连接的背景空间，monoidal函子从这个类别的向量空间可以解释在Frobenius代数与额外的结构。
A normal form for the Thin Flat Surfaces category
该研究针对Khovanov等人引入的薄平曲面范畴TFS，证明了连通可视tf-配边的范式定理与关系充分性定理，给出了TFS的完整生成元-关系表示。
We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^\infty$-rings that is obtained by patching together homotopy zero-sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argumen…
A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov's question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds…
These lecture notes form an expanded account of a course given at the Summer School on Topology and Field Theories held at the Center for Mathematics at the University of Notre Dame, Indiana during the Summer of 2012. A similar lecture series was given in Hamburg in January 2013. The lecture notes are divided into two parts. The first part, consisting of the bulk of these notes, provides an expository account of the author's joint work with Christopher Douglas and Noah Snyder on dualizability in low-dimensional higher categories and the connection to low-dimensional topology. The cobordism hyp…
Virtual strings and their cobordisms
虚弦是曲面上闭曲线的自相交的一种形式。我们研究了弦的代数不变式以及弦集合上的两个等价关系：同伦和配边。我们证明了弦的同伦不变量构成一个无穷维李群。我们还讨论了虚弦和虚结之间的联系。
Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-closed cobordisms. Similarly, Costello introduced a chain complex of black and white graphs $BW$-Graphs, as a rational homological model of mapping class groups. We use the result on admissible fat graphs to give a new integral proof of Costellos's result that $BW$-Graphs is a homolog…
An infinity-categorical TQFT from instantons
本文将瞬子拓扑量子场论提升为从瞬子配边无穷范畴到2周期链复形无穷导出范畴的函子，简化了链复形超立方体构造，并升级了μ算子至链层次。
On the functoriality of the space of equivariant smooth $h$-cobordisms
本文构造了一个将带角光滑 $G$-流形映射到等变光滑 $h$-配边空间的 $(\infty,1)$-函子，并给出稳定类比，其构造依赖于新思想，且在非等变情形与已有构造一致。
Two-Categorical Bundles and Their Classifying Spaces
对于2-范畴2C，我们关联一个主2C-丛的概念。在M. M. Kapranov和V.A. Voevodsky给出了N.A.巴斯岛邓达斯和J.我们的主要结果是，一个好的2-范畴的几何神经是一个相关的主2-丛的分类空间。在证明这一点的过程中，我们开发了许多强大的机器，这可能是有用的，在进一步研究2-范畴拓扑。作为推论，我们得到了主丛分类的一个新的证明。基于主要定理的计算表明，在J.C. Baez和A.S. Crans分裂为普通向量丛的两个副本，直到一致。当2C是配边型2-范畴时，我们得到了配边丛的一个新概念，并证明了它是由Madsen-Weiss空间分类的。
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