Boldizsar Kalmar
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.
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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 aim of this paper is to give an $s$-cobordism classification of topological $4$-manifolds in terms of the standard invariants using the group of homotopy self-equivalences. Hambleton and Kreck constructed a braid to study the group of homotopy self-equivalences of $4$-manifolds. Using this braid together with the modified surgery theory of Kreck, we give an $s$-cobordism classification for certain $4$-manifolds with fundamental group $π$, such that cd $π\leq 2$.
We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.
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。
We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the cobordism group of fold maps.
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallel…
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 obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by na…
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。
Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy group of the infinite dimensional real projective space onto the 2-primary part of the stable homotopy group of spheres). Analogous results - modulo small primes - are obtained for cusp maps and more complicated Morin maps as well.
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。
A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connections between virtual strings and virtual knots.
We show that any 3-dimensional homotopy lens space M^3 that is simple-homotopy equivalent to a lens space L(p,q) is topologically s-cobordant to the lens space. It follows that M has the same multi-signature as L(p,q) and the action of π_1(M) on the universal cover of M embeds in an orthogonal action on S^7.
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…
We prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory. These spaces are: - Kazarian's space (generalizing Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. - Author's space $X_τ$ giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities \cite{R--Sz}. As a consequence we obtain the ranks of cobordism groups of singular maps with a given list of allowed singularities, and also their $p$-torsion parts for big primes $p.$ Further we give complete answe…
Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the relative homotopy principle on the existence level. We also deal with the oriented version and construct a classifying space to which this oriented cobordism group is represented as the set of homotopy classes of P in the codimension n<p and n\geqq p\geqq 2.
For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fol…
We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conjecture is true in dimension 4 up to s-cobordism, assuming that the fundamental group satisfies the Farrell--Jones Conjecture.
In this paper we show that for a large class C of 4-manifolds each member of C has only finitely many simple homotopy type upto s-cobordism. This result generalizes a similar result of Hillman for certain complex surfaces. We also present a correction in the proof of Hillman's result.
A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($Σ^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($Σ^{1,0}$-singularities). Two fold maps are fold bordant if there are cobordisms between their source- and target manifolds with a fold map extending the two maps between the boundaries, if the two targets agree and the target cobordism can be taken as a product with a unit interval then the maps are fold cobordant. We compute the cobordism …
We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author's earlier works. Secondly we take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into R^n. Then we a…
We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the stable homotopy groups of spheres.
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.
In this paper we show that for m>n the set of cobordism classes of maps from m-sphere to n-sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres.
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…
Let $N$ and $P$ be smooth closed manifolds of dimensions $n$ and $p$ respectively. Given a Thom-Boardman symbol $I$, a smooth map $f:N\to P$ is called an $Ω^{I}$-regular map if and only if the Thom-Boardman symbol of each singular point of $f$ is not greater than $I$ in the lexicographic order. We will represent the group of all cobordism classes of $Ω^{I}$-regular maps of $n$-dimensional closed manifolds into $P$ in terms of certain stable homotopy groups. As an application we will study the relationship among the stable homotopy groups of spheres, the above cobordism group and higher singula…
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。
We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction, and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.
The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product structure.
Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford…
We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.
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.
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…
Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topolog…
We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two types: vanishing and non-vanishing. In our vanishing results we are able to show that the connectivity of the harmonic compactification increases with the number of incoming boundary components. Moreover, we extend the genus stabilization maps of moduli spaces to the harmonic compactification and show that the connectivity of these maps increases with the genus a…
Cobordism groups of dihedral branched covers
该研究计算了 S³ 的二歧 n 重分支覆盖的配边群，证明其为循环群，由 (2,n) 环面纽结的 n 重连通二歧覆盖生成，并用推广 Cappell-Shaneson 特征纽结的配边不变量确定其同构类型。
For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. This construction relies on another of independent interest. Take $X$ and $Y$ to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose $Y$ is simply connected and rationally homotopy equivalent to a product of Eilenberg-MacLane spaces: for exa…
We consider branched coverings which are simple in the sense that any point of the target has at most one singular preimage. The cobordism classes of $k$-fold simple branched coverings between $n$-manifolds form an abelian group $Cob^1(n,k)$. Moreover, $Cob^1(*,k) = \bigoplus_{n=0}^{\infty} Cob^1(n,k)$ is a module over $Ω^{SO}_*$. We construct a universal $k$-fold simple branched covering, and use it to compute this module rationally. As a corollary, we determine the rank of the groups $Cob^1(n,k)$. In the case $n = 2$ we compute the group $Cob^1(2,k)$, give a complete set of invariants and co…
In arXiv:1905.07734 we presented a construction that is an analogue of Pontryagin's for proper maps in stable dimensions. This gives a bijection between the cobordism set of framed embedded compact submanifolds in $W\times\mathbb{R}^n$ for a given manifold $W$ and a large enough number $n$, and the homotopy classes of proper maps from $W\times\mathbb{R}^n$ to $\mathbb{R}^{k+n}$. In the present paper we generalise this result in a similar way as Thom's construction generalises Pontryagin's. In other words, we present a bijection between the cobordism set of submanifolds embedded in $W\times\mat…
gl(2) foams and the Khovanov homotopy type
本文通过比较链环图的 Blanchet 和 Khovanov 链复形，利用带符号 Burnside 范畴方法，提出了 Blanchet 链环同调理论的稳定同伦细化构建方法。
Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds
本文通过 G-着色多面体和相对双曲化构造线性复杂度配边，建立 PL 流形 Wall 与 Cheeger-Gromov ρ-不变量的线性界，并应用于透镜空间 h-配边类计数、高维透镜空间复杂度及结构集非有限生成性。
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $Π_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (…
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。
Comparing combinatorial models of moduli space and their compactifications
本文比较了二维配边模空间的两种组合模型（径向狭缝配置和容许胖图），通过“临界图”映射构建了显式同伦等价，并证明该等价在它们的自然紧化（单水平调和紧化与Sullivan图）之间诱导了胞腔同胚。
We interpret heterotic M-theory in terms of h-cobordism, that is the eleven-manifold is a product of the ten-manifold times an interval is translated into a statement that the former is a cobordism of the latter which is a homtopy equivalence. In the non-simply connected case, which is important for model building, the interpretation is then in terms of s-cobordism, so that the cobordism is a simple-homotopy equivalence. This gives constraints on the possible cobordisms depending on the fundamental groups and hence provides a characterization of possible compactification manifolds using the Wh…
A 3-dimensional homotopy quantum field theory (HQFT) can be described as a TQFT for surfaces and 3-cobordisms endowed with homotopy classes of maps into a given space. For a group $π$, we introduce a notion of a modular crossed $π$-category and show that such a category gives rise to a 3-dimensional HQFT with target space $K(π,1)$. This includes numerical invariants of 3-dimensional $π$-manifolds and a 2-dimensional homotopy modular functor. We also introduce and discuss a parallel notion of a quasitriangular crossed Hopf $π$-coalgebra.
In terms of category theory, the Gromov homotopy principle for a set valued functor $F$ asserts that the functor $F$ can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor $F$ holds if the functor $F$ can be induced from a (co)homology functor. We examine the bordism principle in the case of functors given by (co)bordism groups of maps with prescribed singularities. Our main result implies that if a family $R$ of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space $B(R)$ such …
By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all non-degenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine's cusp elimination techn…
A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids
本文基于同伦理论重新审视并推广了Quinn的有限全同伦TQFT，提出了一种范畴化的一次扩展TQFT构造，适用于所有维度，并显式计算了同伦有限omega-群胚分类空间的情形，为离散高阶规范理论建模提供了工具。
We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associated with the universal complexes of singular fibers give complete invariants for all these cobordism groups. We also discuss invariants derived from hypercohomologies of the universal homology complexes of singular fibers. Finally, as an application of the theory of universal comp…
Revisiting Pontryagin's Proof of Stable Stems 1 and 2
本文借助标架配边与 Pontryagin–Thom 构造重访 Pontryagin 的证明，计算 0、1、2 维稳定茎，并指出其关于 π_2^S 的旧结论错误所在。
We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non-trivial elements in homotopy groups $π_*(B\mathrm{Diff}_{\partial}(D^d))\otimes \mathbb{Q}$ are lifted to homotopy groups of the moduli space of $h$-cobordisms $π_*(B\mathrm{Diff}_{\sqcup}(D^d\times I))\otimes \mathbb{Q}$. As a geometrical application, we show that those elements in $π_*(B\mathrm{Diff}_{\partial}(D^d))\otimes \mathbb{Q}$ for $d\geq 4$ are also lifted to the rational homotopy groups $π_*(…
We use the cobordism category constructed in arXiv:1703.01047 to the study the homotopy type of the space of positive scalar curvature metrics on a spin manifold of dimension > 4. Our methods give an alternative proof and extension of a recent theorem of Botvinnik, Ebert, and Randal-Williams from arXiv:1411.7408.
A smooth map having only fold singularities is called a fold-map. We will give effective conditions for a continuous map to be homotopic to a fold-map from the viewpoint of the homotopy principle.
Composition Maps in Heegaard Floer Homology
本文证明 Heegaard Floer 同调态射空间表述中的反向复合映射与 pair-of-pants 配边诱导映射同伦一致，并据此提出计算 hat 理论中任意配边映射的算法。
On the number of components of folds of image simple fold maps
本文研究 image simple 折叠映射奇点集分支数奇偶性的同伦不变性，证明奇数维情形不具不变性，并重证偶数维到可定向曲面情形的不变性。
We compare the invariants of flat vector bundles defined by Atiyah et al. and Jones et al. and prove that, up to weak homotopy, they induce the same map, denoted by $e$, from the $0$-connective algebraic $K$-theory space of the complex numbers to the homotopy fiber of the Chern character. We examine homotopy properties of this map and its relation with other known invariants. In addition, using the formula for $\tildeξ$-invariants of lens spaces derived from Donnelly's fixed point theorem and the $4$-dimensional cobordisms constructed via relative Kirby diagrams, we recover the formula for the…
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