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 bordism and cobordism. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of $({\Bbb Z}_2)^k$-equivariant unoriented bordism and can still derive the classical result of tom Dieck.
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}…
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…
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 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 $MU_{\mathbb{R}}$.
Homological Description of Equivariant Geometric Bordism
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；College of Mathematics and Sciences, Shanghai Normal University（上海师范大学数理学院）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
We propose a dimensional reduction procedure in the Stolz--Teichner framework of supersymmetric Euclidean field theories (EFTs) that is well-suited in the presence of a finite gauge group or, more generally, for field theories over an orbifold. As an illustration, we give a geometric interpretation of the Chern character for manifolds with an action by a finite group.
The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operatio…
We calculate the equivariant motivic Chern class for configuration space of a quasiprojective (maybe singular) variety and the space of vectors with different directions. We prove the formulas for generating series of these classes. We generalize the localization theorems results about Bialynicki-Birula decomposition to acquire some stability for the motivic Chern classes of configuration spaces.
We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging $G$-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group $G$ symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space $BG$. We also consider unoriented analogues, that is $G$-equivariant invertible pin$^\pm$-TQFTs (fermionic time-reversal-SPTs) and their g…
S^1-Equivariant simplicial algebras and de Rham theory
这是我们以前提交的“\infty -categories monoidales rigides et caracteres de Chern”的姊妹篇，其中我们给出了特征零点的光滑概形的导出闭路空间上的函数与其代数de Rham上同调之间的比较。因此，我们得到函和乘法版本的HKR分解定理有关霍赫希尔德同调和霍奇上同调。
