The Borsuk-Ulam theorem for closed 3-dimensional manifolds having Nil Geometry

Borsuk-Ulam定理的覆盖与标号推广

Covering and labeling generalizations of the Borsuk-Ulam theorem

本文推广Fan的球面三角剖分标号结果，全面扩展Borsuk-Ulam定理，揭示满足复杂度界的对象结构，并应用于覆盖、着色、超图Hall定理及超平面质量划分，给出拓扑Hall定理的新证明。

Free cyclic actions on surfaces and the Borsuk-Ulam theorem

Let $M$ and $N$ be topological spaces, let $G$ be a group, and let $τ\colon\thinspace G \times M \to M$ be a proper free action of $G$. In this paper, we define a Borsuk-Ulam-type property for homotopy classes of maps from $M$ to $N$ with respect to the pair $(G,τ)$ that generalises the classical antipodal Borsuk-Ulam theorem of maps from the $n$-sphere $\mathbb{S}^n$ to $\mathbb{R}^n$. In the cases where $M$ is a finite pathwise-connected CW-complex, $G$ is a finite, non-trivial Abelian group, $τ$ is a proper free cellular action, and $N$ is either $\mathbb{R}^2$ or a compact surface without …

The Borsuk-Ulam theorem for maps into a surface

Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f:X-->S, there exists a point x belonging to X satisfying f(t(x))=f(x). In this paper, we formulate this property in terms of a relation in the 2-string braid group B_2(S) of S. If X is a compact, connected surface without boundary, we use this criterion to classify all triples (X, t, S) for which the Borsuk-Ulam property holds. We also consider various cases where…

Bourgin-Yang versions of the Borsuk-Ulam theorem for $p$-toral groups

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of $ \dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, or the $p$-torus $\mathbb Z_p^k$. This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any $p$-toral group $G$ and a $G$-map $f:S(V) \to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim Z_f= \infty$.

Metric Thickenings, Borsuk-Ulam Theorems, and Orbitopes

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds …

A Borsuk-Ulam theorem for $(\mathbb Z_p)^k$-actions on products of (mod $p$) homology spheres

It is proved that for a product action of $(\mathbb Z_p)^k$ on a product of (mod p) homology spheres $N^{n_1}\times...\times N^{n_k}$, where all $n_i$'s are assumed to be odd if $p$ is odd, and any continuous map $f\colon N^{n_1}\times...\times N^{n_k}\to \mathbb R^m$ the set $A(f)=\{x\in N^{n_1}\times...\times N^{n_k}| f(x)=f(gx) \forall g\in(\mathbb Z_p)^k\}$ has dimension at least $n_1+...+n_k-m(p^k-1)$, provided $n_i\ge mp^{i-1}(p-1)$ for all $i (1\le i\le k)$. Moreover, if $n_i\ge mp^{k-1}(p-1)$ for all $i(1\le i\le k)$ then the free action $μ$ can be assumed arbitrary.

截面范畴与参数化 Borsuk-Ulam 性质

Sectional Category and the parametrized Borsuk-Ulam property

本文通过比较二重覆盖的截面范畴，给出纤维化三元组满足参数化 Borsuk-Ulam 性质的充分条件，并应用于平凡纤维化、Hopf 纤维化及 Fadell-Neuwirth 纤维化。

Borsuk-Ulam类型定理与登山问题

Borsuk-Ulam Type Theorems and Mountain Climbing Problem

本文提出Hopf定理的新定性扩展，研究连续映射f从紧致黎曼流形M到R^n的性质，揭示了拓扑意义上的'远点'概念，并推导出Lusternik-Schnirelmann和Tucker型定理的结论，以及多维的登山引理扩展。

The F2 Cohomology of Sol^3-Manifolds

We compute the rings $H^*(N;\mathbb{F}_2)$ for $N$ a closed $\mathbb{S}ol^3$-manifold and then determine the Borsuk-Ulam indices $BU(N,ϕ)$ with $ϕ\not=0$ in $H^1(N;\mathbb{F}_2)$.

Gromov-Hausdorff 距离、Borsuk-Ulam 定理与 Vietoris-Rips 复形

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

本文利用 Vietoris-Rips 复形与等变拓扑方法，将不连续函数畸变与 Gromov-Hausdorff 距离联系起来，推广了球面间奇映射的 Borsuk-Ulam 型结果，并改进不同维数球面间 Gromov-Hausdorff 距离的上下界。

Generalizations of Tucker-Fan-Shashkin lemmas

Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.

紧度量空间上不可嵌入立方平移的等距自由Z-作用

A free Z-action by isometries on a compact metric space which is not embeddable into a cubical shift

本文构造了一个紧度量空间上的等距自由Z-作用，证明其不可嵌入立方平移，从而以存在性定理的形式强力回答了拓扑动力学中的一个著名公开问题，核心工具是Borsuk-Ulam型定理、p进完备化与等变Sullivan猜想。