Hoang Kim Nguyen
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.
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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.
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.
In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives necessary conditions for an abstract morphism between the homologies of two plumbed 3-manifolds to be realized geometrically by a cobordism. Here we also consider the homology cobordism monoid, and give a necessary condition using w-invariants for the homology 3-spheres to belong to the inertia group associated to some homology 3-spheres.
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…
In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space B{Cob}_{d,<k>} as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the pr…
The elements of the ring of bidegree (0,0) additive unstable operations in complex K-theory can be described explicitly as certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for complex cobordism MU, thus identifying the "Adams subring" of the corresponding ring of cobordism operations. We prove that the Adams subring is the centre of the ring of bidegree (0,0) additive unstable cobordism operations. For an odd prime p, the analogous result in the p-local split setting is also proved.
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.
In recent work by Clarke, Crossley and the second author, various algebras of stable degree zero operations in p-local K-theory were described explicitly. The elements are certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for p-local cobordism and for BP, thus identifying the "Adams subalgebra" of the algebras of operations. We prove that the Adams subalgebra is the centre of the ring of degree zero operations.
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…
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。
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。
We give a filtration of the unoriented cobordism ring using the infinite symplectic group, with polynomial generators given one at a time. The generating manifolds are also constructed using the cup construction.
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。
An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^1-spectra equipped with the symmetric monoidal structure described in arXiv:0709.3905v1 [math.AG]. The algebraic cobordism P^1-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation. For a commutative monoid E in the category SH(k) we identify the set of monoid homomorphisms from MGL to E in the motivic stable homotopy category with the set of all orientations of E. This result was stated originally in a s…
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。
In this paper we study the topology of the cobordism category of open and closed strings. This is a 2-category in which the objects are compact one-manifolds whose boundary components are labeled by an indexing set (the set of "D-branes"), the 1-morphisms are cobordisms of manifolds with boundary, and the 2-morphisms are diffeomorphisms of the surface cobordisms. Our methods and techniques are direct generalizations of those used by U. Tillmann in her study of the category of closed strings. We input the striking theorem of Madsen and Weiss regarding the topology of the stable mapping class gr…
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…
We attempt to answer a question of D. C. Ravenel's: "Infinite loopspace theorists, where are you?", or more prosaically: Can one start from the notion of Ring Spectrum Representing [Formal Group Laws] (the functor) and arrive at Complex Cobordism? Our answer is a tentative "sort-of".
After two papers on weak cubical categories and {\it collarable} cospans, respectively, we put things together and construct a {\it weak} cubical category of cubical {\it collared} cospans of topological spaces. We also build a second structure, called a {\it quasi} cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has {\it lax} identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT's of Part II to higher cubical degree.
We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n,X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord}(n,X), including the fact that it only depends on the (n+1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n+1 implies the existence of geometrical structures in X; in parti…
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 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 present the structure of symplectic cobordism ring $MSp_{*}$ in dimensions up to 51 and give a construction of an infinite series of elements $Γ_i$, $ i=1, 3,4, ...$, of order four in this ring, where $\operatorname{dim} \, Γ_i=8i+95$. The key element of the series is $Γ_1$ in dimension 103.
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.
We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an …
Let A be a finite abelian group. We set up an algebraic framework for studying A-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians.
In \cite{baker-ozel}, by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem \cite{Quinn}, it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied f…
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。
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。
String topology and graph cobordisms
本文通过图 cobordisms 构造了字符串操作，并利用图场论和开放-封闭场论扩展了 R-定向 M-纤维的特征类研究。
We show that the cobordism groups of negative codimensional folds maps contain direct sums of stable homotopy groups of Thom spaces of vector bundles like the circle and the infinite dimensional projective space. We give geometrical invariants which detect these direct summands.
In \cite{baker-ozel}, by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In \cite{cenap-isr}, by using Quinn's Transversality Theorem \cite{Quinn}, it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisf…
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…
We introduce the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.
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…
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