Complex cobordism of involutions

Cobordism realizations of Thom polynomials of singularities

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.

Complex cobordism classes of homogeneous spaces

We consider compact homogeneous spaces G/H of positive Euler characteristic endowed with an invariant almost complex structure J and the canonical action θof the maximal torus T ^{k} on G/H. We obtain explicit formula for the cobordism class of such manifold through the weights of the action θat the identity fixed point eH by an action of the quotient group W_G/W_H of the Weyl groups for G and H. In this way we show that the cobordism class for such manifolds can be computed explicitly without information on their cohomology. We also show that formula for cobordism class provides an explicit w…

复截面的配边障碍 II：挠障碍

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。

The equivariant complex cobordism ring of a finite abelian group

We compute the equivariant (stable) complex cobordism ring $(MU_G)_*$ for finite abelian groups $G$.

Complex cobordism MU$^*[1/2]$ modulo MSU$^*[1/2]$ and related genera

This paper presents a commutative complex oriented cohomology theory with coefficients the quotient ring of complex cobordism MU$^*[1/2]$ modulo the ideal generated by any subsequence of any polynomial generators in special unitary cobordism MSU$^*[1/2]$ viewed as elements in MU$^*[1/2]$ by forgetful map.

A note on the equivariant formal group law of the equivariant complex cobordism ring

For a finite abelian group $G$, we compute the $G$-equivariant formal group law corresponding to the $G$-equivariant complex cobordism spectrum with its canonical complex orientation.

Coefficients of the $Σ_3$-equivariant complex cobordism ring

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.

Complex Cobordism of quasitoric orbifolds

We construct orbifolds with quasitoric boundary and show that they have stable almost complex structure. We show that a quasitoric orbifold is complex cobordant to finite disjoint copies of complex orbifold projective spaces. Finally some computations in the complex cobordism ring for manifolds are given.

Complex Cobordism vs. Representing Formal Group Laws

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".

Smooth Projective Toric Variety Representatives in Complex Cobordism

A general problem in complex cobordism theory is to find useful representatives for cobordism classes. One particularly convenient class of complex manifolds consists of smooth projective toric varieties. The bijective correspondence between these varieties and smooth polytopes allows us to examine which complex cobordism classes contain a smooth projective toric variety by studying the combinatorics of polytopes. These combinatorial properties determine obstructions to a complex cobordism class containing a smooth projective toric variety. However, the obstructions are only necessary conditio…

On Fredholm Index, Transversal Approximations and Quillen's Geometric Complex Cobordism of Hilbert Manifolds with some Applications to Flag Varieties of Loop Groups

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 note on the equivariant cobordism of generalized Dold manifolds

Let $(X,J) $ be an almost complex manifold with a (smooth) involution $σ:X\to X$ such that $Fix(σ)\neq \emptyset$. Assume that $σ$ is a complex conjugation, i.e, the differential of $σ$ anti-commutes with $J$. The space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,σ(x))$ is known as a generalized Dold manifold. Suppose that a group $G\cong \mathbb Z_2^s$ acts smoothly on $X$ such that $g\circ σ=σ\circ g$ for all $g\in G$. Using the action of the diagonal subgroup $D=O(1)^{m+1}\subset O(m+1)$ on the sphere $\mathbb S^{m}$ for which there are only finitely many pairs of antipodal p…

复配边模 $c_1$-球面配边及相关亏格

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。

所有紧李群的复配边理论的完备化定理与局部上同调定理

The completion and local cohomology theorems for complex cobordism for all compact Lie groups

本文利用全局函子的分裂，将等变MU-模谱的完备化定理从有限群或环面的有限扩张推广到所有紧李群，从而证明了Greenlees和May的猜想。

Infinite sums of additive unstable Adams operations and cobordism

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.

On smooth Chas-Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

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…

利用核持久性识别协边

Identifying cobordisms using kernel persistence

本文受化学应用启发，提出用核持久性定义并识别细胞复形中隧道的协边，给出出生与死亡时间的配对算法。

Darondeau-Pragacz formulas in complex cobordism

In this paper, we generalize the push-forward (Gysin) formulas for flag bundles in the ordinary cohomology theory, which are due to Darondeau-Pragacz, to the complex cobordism theory. Then we introduce the {\it universal quadratic Schur functions}, which are a generalization of the (ordinary) quadratic Schur functions introduced by Darondeau-Pragacz, and establish some Gysin formulas for the universal quadratic Schur functions as an application of our Gysin formulas.

The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces

We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for nontrivial weights) is isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant $K$-theory and complex cobordism rings of more general singular tori…

The relation between compact and non-compact equivariant cobordisms

We show that the theory of stable complex $G$-cobordisms, for a torus $G$, is embedded into the theory of stable complex $G$-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex $G$-cobordism theory does not lead to new relations.

几何 Hodge 滤过的复配边

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。

The Segal conjecture for topological Hochschild homology of complex cobordism

We study the C_p-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_+(B) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH(B) are p-adically equivalent for B = MU and BP. This generalizes the classical C_p-equivariant Segal conjecture, which corresponds to the case B = S.

Cobordism of symplectic manifolds and asymptotic expansions

The cobordism ring of symplectic manifolds defined by V.L. Ginzburg is shown to be isomorphic to the Pontrjagin ring of complex-oriented manifolds with free circle actions. This suggests an interpretation of the formal group law of complex cobordism, in terms of a composition-law on semiclassical expansions. An appendix discusses related questions about cobordism of toric varieties.

Toric Polynomial Generators of Complex Cobordism

Although it is well-known that the complex cobordism ring is a polynomial ring $Ω_{*}^{U}\cong\mathbb{Z}\left[α_{1},α_{2},\ldots\right]$, an explicit description for convenient generators $α_{1},α_{2},\ldots$ has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blo…

复配边中的 Chern-Dold 特征标与 theta 除子

Chern-Dold character in complex cobordisms and theta divisors

本文用一般主极化阿贝尔簇的光滑 theta 除子表示复配边中 Chern-Dold 特征标的系数，通过引入以对偶 Landweber-Novikov 代数为参数空间的量子化框架解释该特征标，并讨论其与 Milnor-Hirzebruch 问题的联系。

复配边拓扑周期同调的连续同调

Continuous homology of topological periodic homology of complex cobordism

本文确定了复配边谱的拓扑周期同调TP(MU)的连续模p同调，作为具有Steenrod运算的分次代数，并给出了显式代数构造C_+，其Ext-代数给出收敛到p完备化TP(MU)同伦的Adams型谱序列的E_2项。

相对分圆结构与等变复配边

Relative cyclotomic structures and equivariant complex cobordism

该研究在交换环（预）分圆谱上引入结构以构造相对$TC$，证明其与$TC$的下降关系，并探究其在$\boldsymbol{\top}$-等变复配边谱上的实例。

Hodge滤过复配边中的几何前推与次级不变量

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。

G-流形的配边

Cobordism of G-manifolds

本文综述等变配边结果，利用等变稳定同伦理论重新计算无向$C_2$-等变配边环$Ω^{C_2}_*$，该方法可推广至等变复配边并提供形式群论解释。

Equivariant cobordism of torus orbifolds

Torus orbifolds are topological generalization of symplectic toric orbifolds. We give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using toric topological method. As a result, we show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. We also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.

Persistent Homology, Matroids and Cobordisms

The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.

复配边中的$SU$-线性运算与$c_1$-球面配边理论

$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory

本文研究复配边中的$SU$-线性运算，证明其由几何运算$\partial_i$生成，并描述$c_1$-球面配边理论$W$上的$SU$-线性乘法与投影，扩展Buchstaber结果，表明$F_W$系数不生成系数环。

Multicurves and equivariant cobordism

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.

Algebraic Aspects of the Theory of Product Structures in Complex Cobordism

We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra $S$ (the Landweber-Novikov algebra) acting on its dual Hopf algebra $S^*$ with a distinguished ``topologically integral'' part $Λ$ that coincids with the coefficient ring of the complex cobordism. We describe the formal group and its logarithm in terms of representations of $S$. We introduce one-dimensional representations of a Hopf algebra. We give series of examples of such representati…

Projective toric generators in the unitary cobordism ring

By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $Ω^U_*\simeq \mathbb Z[a_1,a_2,\dots]$, ${\rm deg}(a_i)=2i$. In this paper we solve a well-known problem of constructing geometric representatives for $a_i$ among smooth projective toric varieties, $a_n=[X^{n}], \dim_\mathbb C X^{n}=n$. Our proof uses a family of equivariant modifications (birational isomorphisms) $B_k(X)\to X$ of an arbitrary smooth complex manifold $X$ of (complex) dimension $n$ ($n\geq 2$, $k=0,\dots,n-2$). The key fact is that …

The circle action on topological Hochschild homology of complex cobordism and the Brown-Peterson spectrum

We specify exterior generators for $π_* THH(MU) = π_*(MU) \otimes E(λ'_n \mid n\ge1)$ and $π_* THH(BP) = π_*(BP) \otimes E(λ_n \mid n\ge1)$, and calculate the action of the $σ$-operator on these graded rings. In particular, $σ(λ'_n) = 0$ and $σ(λ_n) = 0$, while the actions on $π_*(MU)$ and $π_*(BP)$ are expressed in terms of the right units $η_R$ in the Hopf algebroids $(π_*(MU), π_*(MU \wedge MU))$ and $(π_*(BP), π_*(BP \wedge BP))$, respectively.

Motivic strict ring spectra representing semi-topological cohomology theories

We show that Shipley's "detection functor" for symmetric spectra generalizes to motivic symmetric spectra. As an application, we construct motivic strict ring spectra representing morphic cohomology, semi-topological $K$-theory, and semi-topological cobordism for complex varieties. As a further application to semi-topological cobordism, we show that it is related to semi-topological $K$-theory via a Conner-Floyd type isomorphism and that after inverting a lift of the Friedlander-Mazur $s$-element in morphic cohomology, semi-topological cobordism becomes isomorphic to periodic complex cobordism…

单纯GKM轨形复形的整等变K-理论与配边环

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。

Algebraic Cobordism of Classifying Spaces

We define algebraic cobordism of classifying spaces, Ω^*(BG) and G-equivariant algebraic cobordism Ω^*_G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted F^j(Ω^*(-)), which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We calculate Ω^*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n+1). We also calculate Ω^*(BG) when G is a finite abelian group. A finite non-abelian group for …

Equivariant complex structures on homogeneous spaces and their cobordism classes

We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the complex, almost complex and stable complex structures which are invariant under the canonical left action of the maximal torus T^k on G/H. As it is known, on G/H we may have many such structures and the computations of their toric genus in terms of fixed points for the same torus action give the constraints on possible collections of weights for the correspon…

Spaces of polytopes and cobordism of quasitoric manifolds

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By exp…

On the cobordism classification of manifolds with Z/p-action

We refer to an action of the group Z/p (p here is an odd prime) on a stably complex manifold as simple if all its fixed submanifolds have the trivial normal bundle. The important particular case of a simple action is an action with only isolated fixed points. The problem of cobordism classification of manifolds with simple action of Z/p was posed by V.M.Buchstaber and S.P.Novikov in 1971. The analogous question of cobordism classification with stricter conditions on Z/p-action was answered by Conner and Floyd. Namely, Conner and Floyd solved the problem in the case of simple actions with ident…

关于η-周期形式三元律

On $η$-periodic Formal Ternary Laws

研究η-周期动机稳定同伦范畴中Sp-定向的代数结构，通过引入带框对合等方法，公理化形式三元律构造泛沃尔特环𝒲^η，证明其与拉扎德环L在对2求逆后同构，给出分类映射，揭示整数层面恢复完整定向数据需额外二次幂级数。

Involution on the product of projective space and sphere

Let $G=\mathbb{Z}_2$ act on a finite CW-complex $X$ having mod 2 cohomology isomorphic to the product of projective space and sphere $\mathbb{F}P^n\times \mathbb{S}^m,$ where $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}.$ In this paper, we have determined the connected fixed point sets and the orbit spaces of free involutions on $X.$ As an application, we derive the Borsuk-Ulam type results.

四元数射影空间与球面乘积上的对合

Involutions on the product of Quaternionic Projective space and Sphere

本文研究模2上同调同构于HPⁿ×Sᵐ（n,m≥1）的有限CW复形上Z2自由对合的连通不动点集与轨道空间，属于代数拓扑领域的对合作用研究。

An involution on the K-theory of bimonoidal categories with anti-involution

We construct a combinatorially defined involution on the algebraic $K$-theory of the ring spectrum associated to a bimonoidal category with anti-involution. Particular examples of such are braided bimonoidal categories. We investigate examples such as algebraic K-theory of connective complex and real topological K-theory and Waldhausen's K-theory of spaces of the form BBG, for abelian groups G. We show that the involution agrees with the classical one for a bimonoidal category associated to a ring and prove that it is not trivial in the above mentioned examples.

Complex cobordism, Hamiltonian loops and global Kuranishi charts

Let $(X,ω)$ be a closed symplectic manifold. A loop $ϕ: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $π: P_ϕ \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $π$, Lalonde, McDuff and Polterovich proved that if $ϕ$ lifts to the Hamiltonian group $\mathrm{Ham}(X,ω)$, then the rational cohomology of $P_ϕ$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_ϕ$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. …

Komplexe elliptische Geschlechter und S^1-aequivariante Kobordismustheorie（复椭圆亏格与S^1-等变配边理论）

Komplexe elliptische Geschlechter und S^1-aequivariante Kobordismustheorie (Complex elliptic genera and S^1-equivariant cobordism theory)

本文引入万有复椭圆亏格并证明其S^1-作用刚性，刻画其核，构造基序列，并证明Level N亏格在特定余维数爆破下的不变性。

$KR$-theory of compact Lie groups with group anti-involutions

Let $G$ be a compact, connected, and simply-connected Lie group, equipped with an anti-involution $a_G$ which is the composition of a Lie group involutive automorphism $σ_G$ and the group inversion. We view $(G, a_G)$ as a Real $(G, σ_G)$-space via the conjugation action. In this note, we exploit the notion of Real equivariant formality discussed in \cite{Fo} to compute the ring structure of the equivariant $KR$-theory of $G$. In particular, we show that when $G$ does not have Real representations of complex type, the equivariant $KR$-theory is the ring of Grothendieck differentials of the coe…

Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes

We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably c…