$S^2 \times S^3$上的自由对合

S^6上具有三维不动点集的对合

Involutions on S^6 with 3-dimensional fixed point set

本文对S^6上所有具有三维不动点集的对合进行了分类。特别地，我们讨论了纽结3-球面上具有不动点集的对合的分类与同伦CP ^3上自由对合的分类之间的关系。

关于$\mathbb Z_2$ -等变Montgomery-Yang对应的注记

A note on the $\mathbb Z_2$-equivariant Montgomery-Yang correspondence

本文给出了三维同伦复射影空间上自由对合的一种分类。利用$\mathbb Z_2$ -等变Montgomery-Yang对应，得到了不动点集嵌入$S^3$的$S^6$上的所有光滑对合.

The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero

Let M and N be topological spaces such that M admits a free involution $\τ$. A homotopy class $β$ $\in$ [M, N ] is said to have the Borsuk-Ulam property with respect to $\τ$ if for every representative map f : M $\rightarrow$ N of $β$, there exists a point x $\in$ M such that f ($\τ$ (x)) = f (x). In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N , and of the fundamental groups o…

The Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle -- part 2

Let $M$ be a topological space that admits a free involution $τ$, and let $N$ be a topological space. A homotopy class $β\in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $τ$ if for every representative map $f: M \to N$ of $β$, there exists a point $x \in M$ such that $f(τ(x))= f(x)$. In this paper, we determine the homotopy class of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to any free involution of $T^2$ for which the orbit space is $K^2$. Our results are given in terms of a certain family of homomorphisms i…

Kervaire流形上的有限群作用

Finite group actions on Kervaire manifolds

（4k+2）维Kervaire流形是一个闭的、分段线性（PL）流形，其Kervaire不变量为1，并且与两个（2k+1）维球面的乘积具有相同的同调。证明了奇阶有限群自由作用在Kervaire流形上当且仅当它自由作用在相应的球面积上。如果Kervaire流形M是光滑的，则M上的每个光滑结构都允许一个自由光滑对合。如果k + 1不是2次幂，则4k+2维的Kervaire流形不允许任何自由TOP对合。自由“奇异”（PL）对合是在30、62和126维的Kervaire流形上构造的。30维Kervaire流形上的每个光滑结构都允许一个自由的Z/2 x Z/2作用。

扭共轭子拟群与 Alexander 拟群的优良对合

Good involutions of twisted conjugation subquandles and Alexander quandles

本文完整刻画了自由拟群、扭共轭子拟群及所有 Alexander 拟群的优良对合，并枚举线性拟群的对合，给出阶数不超过 23 的显式映射，同时刻画了连通对合型 Alexander 拟群。

Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$-equivariant homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivia…

Anti-symplectic involutions for Lagrangian spheres in a symplectic quadric surface

We show that the space of anti-symplectic involutions of a monotone $S^2\times S^2$ whose fixed points set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic.

4-流形的 $\mathbb{Z}_2$-指数的阻碍理论

Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds

本文为具有自由对合的紧连通4维流形的 $\mathbb{Z}_2$-指数发展了完整的阻碍理论，利用带扭系数的上同调及计算轨道空间扭上同调的谱序列，实现了该指数的两步计算，并通过实例进行了验证。

The Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle

Let $M$ be a topological space that admits a free involution $τ$, and let $N$ be a topological space. A homotopy class $β\in [ M,N ]$ is said to have {\it the Borsuk-Ulam property with respect to $τ$} if for every representative map $f: M \to N$ of $β$, there exists a point $x \in M$ such that $f(τ(x))= f(x)$. In this paper, we determine the homotopy classes of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to a free involution $τ_1$ of $T^2$ for which the orbit space is $T^2$. Our results are given in terms of a certain family of hom…

绝对伽罗瓦群中对合的Artin-Schreier广群

Artin-Schreier quandles of involutions in absolute Galois groups

本研究提出与绝对伽罗瓦群相关的新域不变量Artin-Schreier quandle，明确其在形式实数域中的自由生成结构，给出洛朗级数域下的计算方式，证实一般情形下该结构存在关系，细化了域的实谱研究。

关于二维环面相空间上的时间约束等变Hermite算子值映射

On Time-Reversal Equivariant Hermitian Operator Valued Maps from a 2D-Torus Phase Space

本文对环面相空间$X$到$\mathcal{H}_n^*$的映射、$n \times n$的空间、非奇异厄米特算子到等变同伦的映射进行了分类。该等方差是关于定义某个对称类的$X$上的时间反演对合和$\mathcal{H}_n^*$上的对合。此外，我们研究了某些控制退化等地图，并描述了如何可以实现跳跃从一个同伦类到另一个，而只允许“小”退化。

The Chekanov torus in $S^2\times S^2$ is not real

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $\mathbb{T}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.

可缩流形上伪自由群作用的刚性

Rigidity of pseudo-free group actions on contractible manifolds

本文宣布与弗兰克康诺利和吉姆戴维斯的联合工作。通过研究伴随有挠的无限群在泛覆盖中的作用，推广了n-环面上伪自由对合的分类。包括一个非黎曼的例子，从限制的作用Coxeter群的CAT（0）流形。

四维流形上的非动力学同伦相干作用

Non-kinetic homotopy coherent actions on four-manifolds

该研究给出闭单连通光滑四维流形上二阶非动力学光滑同伦相干作用的首个例子，通过限制离散圆周群在稳定化\(K3\)曲面上的作用获得，还构造相关扩张并展示特殊边界对合，证明了关于整同调球上自由对合光滑扩张的定理。

实现为实矩角流形上非自由 $\boldsymbol{\rm \bb Z_2^k}$ 作用轨道空间的流形

Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds

该研究针对实矩角流形上的 $\mathbb Z_2^m$ 子群作用，给出轨道空间为拓扑流形的准则，构造了多类球面与超椭圆流形，并完成3维情形下相关子群与超椭圆对合的分类及等价刻画。

Borsuk-Ulam型空间

Borsuk - Ulam Type spaces

我们考虑具有满足Borsuk-Ulam定理的自由对合的空间（BUT-空间）。BUT-空间有几个等价的定义可以被认为是它们的性质。我们的主要技术工具是Yang的上同调指数。