Andrei Kustarev
We define complex cobordism realizations of cohomological Thom polynomials and study their existence, uniqueness and other features. We show that problem is non-trivial on the example of $Σ^1$ singularity.
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We correct the proof in the unoriented case of Theorem 1.2 in the paper "Cobordisms of maps with singularities of given class"
We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stacks. We demonstrate that the resulting theory generalizes the parametrized cobordism of Galatius-Madsen-Tillman-Weiss. The theory has the feature of being both versatile and general, allowing for the inclusion of families of various geometric data, such as metrics on manifolds and…
Cobordism of nested manifolds
研究嵌套流形的配边，通过推广Pontryagin-Thom构造，建立嵌套配边与链环配边之间的联系，并给出Wall定理的替代证明。
We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that our construction recovers the construction of Wasserman, providing a new perspective on equivariant bordism. We connect the results with bordisms of global quotient orbifolds, utilizing the machinery of Gepner-Henriques to describe bordisms of framed orbifolds in terms of equivaria…
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…
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…
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.
A refinement of the Pontryagin-Thom theorem for unstable Thom spectra and its applications
本文改进了Pontryagin-Thom定理，揭示了不稳定Thom谱的同调群结构，并应用于计算新的cobordism群及确定特定Spin流形的存在条件。
On the cokernel of the Thom morphism for compact Lie groups
通过研究 Atiyah-Hirzebruch 谱序列与 Steenrod 代数作用，完整描述了紧致李群上 Thom 态射满射性失效的情况，并利用特殊正交群的几何构造了微分 Thom 态射核中的非平凡配边类。
René Thom's remarkable and far-reaching concept of transversality has found numerous powerful applications. Most importantly, it allowed Thom to develop cobordism theory, which led to a piercing insight into the topology of smooth manifolds, when he generalized Lev Pontryagin's earlier concept of framed cobordism. For his profound findings, in 1958 he was awarded the Fields Medal. The present paper provides a down-to-earth approach to transversality theory which only assumes basic knowledge about smooth manifolds. Following an idea of Andrew Putman, we then use transversality to give a novel a…
Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.
In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. We also make some remarks on general methods of calculating certain pullbacks of rings which typically occur in calculations of equivariant cobordism.
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 define an invariant for the existence of r pointwise linearly independent sections in the tangent bundle of a closed manifold. For low values of r, explicit computations of the homotopy groups of certain Thom spectra combined with classical obstruction theory identifies this invariant as the top obstruction to the existence of the desired sections. In particular, this shows that the top obstruction is an invariant of the underlying manifold in these cases, which is not true in general. The invariant is related to cobordism theory and this gives rise to an identification of the invariant in …
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…
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 Obstructions to Complex Sections II: Torsion Obstructions
本文研究几乎复流形上复截面存在性的挠障碍，利用 Adams-Novikov 谱序列计算 Thom 谱，证明对素数 p≥3，当 r<p²−p 时 p 阶挠障碍消失。
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。
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。
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…
In this paper we derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularities Σ^{i,j}. The formulas are given as linear combinations of Schur polynomials, and all coefficients are nonnegative.
Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterate…
Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algeb…
Thom polynomials relative to prescribed maps around the boundary
本文建立了一种相对版本的Thom多项式理论，用于研究可微映射中奇点出现的上同调障碍，并将其应用于浸入的不变量。
We prove a formula for Thom polynomials of Morin (or A_d) singularities in any codimension. We use a combination of the test-curve method of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and they are computationally efficient for d less than 7.
Geometric Hodge filtered complex cobordism
本文为具有复系数 de Rham 复滤过的光滑流形构造了复配边 Hodge 滤过扩展的几何圈模型，并利用细化的 Pontryagin-Thom 构造，证明了该模型与 Hopkins-Quick 抽象模型间的显式同构。
Geometric pushforward in Hodge filtered complex cobordism and secondary invariants
本研究构建了任意复流形间真全纯映射下几何Hodge滤过复配边的函子性前推同态，改进了已有结论，定义了相关基本类并给出几何描述，还刻画了零配边映射Abel-Jacobi不变量的配边类似物并给出新构造。
Combining the Kazarian approach to Thom polynomials via classifying spaces of singularities with the Fulton-Lazarsfeld theory of numerical positivity for ample vector bundles, we show that the coefficients of various Schur function expansions of the Thom polynomials of stable and unstable singularities are nonnegative.
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。
Complex cobordism modulo $c_1$-spherical cobordism and related genera
本文证明复配边环中由 c1-球面配边生成元生成的理想为素理想，并利用 Baas-Sullivan 理论构造模 c1-球面配边的复定向上同调理论，进而联系 Abel 上同调与 Krichever-Hoehn 复椭圆亏格。
For any (n+1)-dimensional weight vector χ of positive integers, the weighted projective space P(χ) is a projective toric variety, and has orbifold singularities in every case other than CP^n. We study the algebraic topology of P(χ), paying particular attention to its localisation at individual primes p. We identify certain p-primary weight vectors π for which P(π) is homeomorphic to an iterated Thom space over S^2, and discuss how any P(χ) may be reconstructed from its p-primary factors. We express Kawasaki's computations of the integral cohomology ring H^*(P(χ);Z) in terms of iterated Thom is…
We describe the positivity of Thom polynomials of singularities of maps, Lagrangian Thom polynomials and Legendrian Thom polynomials. We show that these positivities come from Schubert calculus.
The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra in various ways. In this work we first establish, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let $R$ be an $E_\infty$-ring spectrum and $\mathrm{Pic}(R)$ denote its Picard $E_\infty$-group.…
We combine recently developed intersection theory for non-reductive geometric invariant theoretic quotients with equivariant localisation to prove a formula for Thom polynomials of Morin singularities. These formulas use only toric combinatorics of certain partition polyhedra, and our new approach circumvents the poorly understood Borel geometry of existing models.
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…
Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes
本文定义了单纯GKM轨形复形，引入了可构建与可分性质，并计算了可分单纯GKM轨形复形的整等变上同调环、等变K-理论环及等变配边环。
Grothendieck polynomials were introduced by Lascoux and Schützenberger, and they play an important role in K-theoretic Schubert calculus. In this paper, we give a new definition of double stable Grothendieck polynomials based on an iterated residue operation. We illustrate the power of our definition by calculating the Grothendieck expansion of K-theoretic Thom polynomials of $A_2$ singularities. We present the expansion in two versions: one displays its expected stabilization, while the other displays its expected finiteness property.
We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur function expansions. We describe several recursions for the coefficients. For some singularities, we give old and new computations of their Thom polynomials.
The rationality of the singularities of the $A_n$-loci is the natural question that arises in the papers devoted to the study of the Thom polynomials and $K$-theoretic invariants of the said loci. In this paper we prove that, in general, the $A_2$-loci have singularities worse than rational, and therefore they can not be presented as a GIT quotient of a smooth variety with respect to a reductive group.
Geometric representation of cohomology classes for the Lie groups Spin(7) and Spin(8)
本文利用八元数对称性构造复定向映射，刻画 Spin(7) 与 Spin(8) 第三整系数上同调生成元及其八倍在 Thom 态射下的像，并几何实现 Hopkins–Singer 微分 Thom 态射核中的非平凡类。
In this paper we propose a systematic study of Thom polynomials for group actions defined by M. Kazarian. On one hand we show that Thom polynomials are first obstructions for the existence of a section and are connected to several problems of topology, global geometry and enumerative algebraic geometry. On the other hand we describe a way to calculate Thom polynomials: the method of restriction equations. It turned out that though the idea is quite simple the method is very powerful. We reproduced and improved earlier result in several directions (singularities, Schubert calculus, quivers). Ho…
Bordism and resolution of singularities
本文通过改编奇点消解算法，证明复bordism到复导出orbifold及等变bordism的同调理论映射可分裂，进而为任意闭辛流形定义复cobordism取值的Gromov-Witten不变量，并约束Hamilton纤维丛空间拓扑。
Thom condition and Monodromy
本文定义了Thom条件，证明在特定条件下复解析函数芽存在无不动点的几何局部单调性，推广了已有定理并应用于奇点不合并的研究。
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