Bernardo Uribe
The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring on even-dimensional generators. In this paper we review the cases in which the conjecture is known to hold and we highlight the properties that permit one to prove the conjecture in these cases.
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We give a description of real equivariant bordism for elementary abelian 2-groups, which is similar to the description of complex equivariant bordism for the group S^1 x ... x S^1 given by Hanke in 2005.
We restrict geometric tangential equivariant complex $T^n$-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant $K$-theory characteristic numbers, that the information encoded in the oriented torus graph associated to a stably complex torus manifold completely describes its equivariant bordism class. We also consider the role of omnioriented quasitoric manifolds in this description.
In this paper we compute homotopical equivariant bordism for the group ${\bf Z/2}$, namely $MO^{\bf Z/2}$, geometric equivariant bordism $Ω^{\bf Z/2}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. In doing these computations, we use the techniques the author developed in the complex setting. Because we are working in the real setting only with Z/2, these techniques simplify greatly.
We study the fixed points of the universal G-equivariant n-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe the fixed points of the complex equivariant K-theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.
In this paper we survey results and recent progresses on the equivariant bordism classification of 2-torus manifolds and unitary toric manifolds.
In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of S^1 actions on stably complex four-manifolds with precisely three…
Chern classes in equivariant bordism
本文在$U(m)$-等变同伦配边中引入细化Conner-Floyd-Chern类的Chern类，证明其构成正则序列生成增广理想，导致Greenlees-May局部同调谱序列坍塌，并据此重新证明了$MU$-完备化定理。
The Homology of Complex Equivariant Bordism
研究阿贝尔紧致李群\(A\)及定向\(A -\)谱\(E\)，通过两种方式计算tom Dieck的同伦\(A -\)等变复配边谱\(MU_A\)的\(E -\)同调并修正错误，还计算了几何\(A -\)等变复配边谱\(mU_A\)的\(E -\)同调。
The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group $\mathbb{Z}_2^n$ and torus group $T^n$, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group $\mathcal{Z}_n(\mathbb{Z}_2^n)$ of 2-torus manifolds, and determine the dimention of $\mathcal{Z}_n(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups $\mathcal{Z}…
Equivariant Milnor map
将Milnor映射推广到等变情形，证明圆群和二阶循环群的等变酉群上存在该映射，并关联其核与自由共轭的磁等变酉配边群。
Reduced characteristic number criteria for $T^k$- and $(\mathbb Z_2)^k$-equivariant bordisms
本文为具有孤立不动点的酉T^k-流形和闭光滑(Z_2)^k-流形建立了简化的等变配边判据，分别仅依赖于单个等变陈类多项式和最高等变Stiefel-Whitney类的幂，并引入最小区分度解决了Kosniowski猜想中欧拉示性数的线性下界问题。
We define equivariant homology theories using bordism of stratifolds with a G-action, where G is a discrete group. Stratifolds are a generalization of smooth manifolds which were introduced by Kreck. He defines homology theories using bordism of suitable stratifolds. We develop the equivariant generalization of these ideas.
Oriented and unitary equivariant bordism of surfaces
本研究显式计算了有限群$G$作用下光滑紧曲面的定向与酉等变配边群，刻画了其秩与挠子群的结构，并给出相关配边性质的新证明，建立了二维SK群与等变配边群的联系。
By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. This is used in order to determine the image of the Pontrjagin-Thom map for toral G.
Equivariant bordism rigidity for toric manifolds
本文提出等变配边方法，证明环面流形同构等价于弱等变酉配边，并给出全定向拟环面流形的刚性结果，以拓扑框架检测几何与组合刚性。
Equivariant bordism classification of five-dimensional $(\mathbb{Z}_2)^3$-manifolds with isolated fixed points
本文针对具有有效光滑$(\mathbb{Z}_2)^3$-作用且不动点孤立的五维闭光滑流形，确定其等变无定向配边群$\mathcal{Z}_5((\mathbb{Z}_2)^3)$的$\mathbb{Z}_2$维数为77并给出基，完成该类流形的等变无定向配边分类。
Non-finitely generated $(\mathbb{Z}_2)^k$-equivariant bordism ring
本文研究了$(\mathbb{Z}_2)^k$-equivariant bordism环的有限生成性和不可分解元素的存在性，证明对于$k\geqslant 3$，该环不是有限生成的，并构造了无限族不可分解元素。
Local cohomology techniques in equivariant homotopy theory, introduced by John Greenlees, may be applied to understand homology of classifying spaces through other equivariant data. In this paper we relate the local cohomology filtration to the families filtration. By doing so, we may identify geometry codified by the local cohomology filtration in the setting of equivariant bordism. The constructions which arise are naturally analyzed by localized K-theory machinery due to Atiyah and Segal, which we review.
The universal property of bordism of commuting involutions
本文提出oriented el$_2^{RO}$-代数这一形式体系以刻画交换对合流形的等变配边环结构，并证明初等阿贝尔2-群的等变配边是该代数的初始对象，同时推广至其他等变同调理论的泛性质刻画。
This paper shows that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of …
We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$.
For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian 2-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A = C_2$ due to Hanke-Wiemeler (2018). We work in the framework of global homotop…
Homological Description of Equivariant Geometric Bordism
本文利用偶性准则与对偶原理构造有限链复形$\frak{B}_\bullet((\bZ_2)^k)$，证明特定等变无向配边群与该复形同调群同构，并通过谱序列坍缩得到配边群维数的显式公式。
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…
Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented …
In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed points sets equipped with their tubular neighborhoods. It is given a description of the equivariant vector bundles with action of a discrete group $G$ for the case when the action over the base is proper quasi-free, i.e. the stationary subgroup of any point is finite. The description is given in terms of some classifying space.
A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, introduced by tom Dieck (1970), with the $\mathbb{Z}/2$-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). Our proof combines a computation of the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring due to Strickland (2001) with a detailed investigati…
Representation characterization of equivariant geometric bordism
该研究针对$G_k=(\mathbb Z_2)^k$，建立了忠实$G_k$表示作为光滑闭$G_k$流形不动点数据的刻画方法，证明其与偶性条件等价，还应用该结果得到$\dim_{\mathbb Z_2}\mathcal Z_4(G_3)=32$及对应的基元。
In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the $C_2$-Segal conjecture. We compute the localized slice spectral sequence of the $C_4$-norm of $BP_\mathbb{R}$ in a range and show that the Hill--Hopkins--Ravenel slice differ…
On the slice spectral sequence for quotients of norms of Real bordism
本文研究了 Real 边界同调谱乘法范数的等变商，引入了计算其等变同伦群的新技术，并完整计算了 $BP^{((C_{2^n}))}\langle m,m\rangle$ 等理论的局部化切片谱序列。
This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples are equivariant stable homotopy, equivariant $K$-theory or equivariant bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., the adjective `global' refers to simultaneous and compatible actions of all compact Lie groups. We give a self-contained treatment of unstable and stable global h…
We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{…
Invariant prime ideals in equivariant Lazard rings
本文计算了阿贝尔紧致李群$A$的等变Lazard环的不变素理想谱，并证明其与紧$A$-谱的Balmer谱同胚，比较映射由等变复配边同调诱导。
Universal Embedding spaces for $G$-manifolds
本文为任意紧李群 G 构造了万有 G-流形 U_n(G)，使所有 n 维光滑 G-流形都能以平凡法丛嵌入其中且嵌入在等变合痕下唯一，并证明其上同调逆极限给出 G-流形的示性类类比，上同伦群等于等变配边群。
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