A degree one Borsuk-Ulam theorem

$Z_2$-bordism and the Borsuk-Ulam Theorem

The purpose of this work is to classify, for given integers $m,\, n\geq 1$, the bordism class of a closed smooth $m$-manifold $X$ with a free smooth involution $τ$ with respect to the validity of the {\it Borsuk-Ulam property} that for every continuous map $ϕ: X \to R^n$ there exists a point $x\in X$ such that $ϕ(x)=ϕ(τ(x))$. We will classify a given free $Z_2$-bordism class $α$ according to the three possible cases that (a) all representatives $(X , τ)$ of $α$ satisfy the Borsuk-Ulam property; \ (b) there are representatives $(X_ 1, τ_1)$ and $(X_2, τ_2)$ of $α$ such that $(X_1, τ_1)$ satisfi…

A simple proof of the Borsuk-Ulam theorem for Z_p-actions

In this note, we give a simple proof of the Borsuk-Ulam theorem for $Z_p$-actions. We prove that, if $S^n$ and $S^m$ are equipped with free $Z_p$-actions (p prime) and $f: S^n \to S^m$ is a $Z_p$-equivariant map, then $n \leq m$.

Borsuk-Ulam property for graphs

For finite connected graphs $Γ$ and $G$, with $Γ$ admitting a free involution $τ$, we characterize the based homotopy classes $α\in[Γ,G]$ for which the Borsuk-Ulam property holds in the sense of Gonçalves, Guaschi and Casteluber-Laass, i.e., the homotopy classes $α$ so that each of its representatives $f\inα$ satisfies $f(x) = f(τ\cdot x)$ for some $x\inΓ$. This is attained through a graph-braid-group perspective aided by the use of discrete Morse theory.

The Borsuk-Ulam theorem for 3-manifolds

We study the Borsuk-Ulam theorem for triple (M;τ; \R^n), where M is a compact, connected, 3-manifold equipped with a fixed-point-free involution τ. The largest value of n for which the Borsuk-Ulam theorem holds is called the Z_2-index and in our case it takes value 1, 2 or 3. We fully discuss this index according to cohomological operations applied on the characteristic class x \in H^1(N; Z_2), where N = M/τis the orbit space. In oriented case, we obtain an expression of the index from the linking matrix of a surgery presentation of the orbit space. We illustrate our results with examples, inc…

On Borsuk-Ulam theorems and convex sets

The Intermediate Value Theorem is used to give an elementary proof of a Borsuk-Ulam theorem of Adams, Bush and Frick that, if $f: S^1\to R^{2k+1}$ is a continuous function on the unit circle $S^1$ in $C$ such that $f(-z)=-f(z)$ for all $z\in S^1$, then there is a finite subset $X$ of $S^1$ of diameter at most $π-π/(2k+1)$ (in the standard metric in which the circle has circumference of length $2π$) such the convex hull of $f(X)$ contains $0\in R^{2k+1}$.

Some remarks on the parametrized Borsuk-Ulam theorem

Given a locally trivial fibre bundle $E \to B$ (with fibres and base finite complexes), an orthogonal real line bundle $λ$ over $E$ and a real vector bundle $ξ$ over $B$, we consider a fibrewise map $f : S(λ) \to ξ$ over $B$ defined on the unit sphere bundle of $λ$. Following the fundamental work of Jaworowski and Dold on the parametrized Borsuk-Ulam theorem, we investigate lower bounds on the cohomological dimension of the set of points $v$ in $S(λ)$ such that $f(v) = f(-v)$.

The Borsuk-Ulam theorem for n-valued maps between surfaces

In this work we analysed the validity of a type of Borsuk-Ulam theorem for multimaps between surfaces. We developed an algebraic technique involving braid groups to study this problem for $n$-valued maps. As a first application we described when the Borsuk-Ulam theorem holds for splits and non-splits multimaps $ϕ\colon X \multimap Y$ in the following two cases: $(i)$ $X$ is the $2$-sphere eqquiped with the antipodal involution and $Y$ is either a closed surface or the Euclidean plane; $(ii)$ $X$ is a closed surface different of the $2$-sphere eqquiped with a free involution $τ$ and $Y$ is the …

Borsuk--Ulam theorems for elementary abelian 2-groups

Let $G$ be a compact Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere in $U$ to $V$ when $G$ is an elementary elementary abelian $p$-group for some prime $p$ or a torus. In this note, the classical Borsuk--Ulam theorem will be used to give a refinement of their result estimating the dimension of that part of the zero-set on which an elementary abelian $p$-group $G$ acts freely or a torus $G$ acts with finite isotropy groups.

The Borsuk-Ulam theorem for planar polygon spaces

The moduli space of planar polygons with generic side lengths is a closed, smooth manifold. Mapping a polygon to its reflected image across the $X$-axis defines a fixed-point-free involution on these moduli spaces, making them into free $\mathbb{Z}_2$-spaces. There are some important numerical parameters associated with free $\mathbb{Z}_2$-spaces, like index and coindex. In this paper, we compute these parameters for some moduli spaces of polygons. We also determine for which of these spaces a generalized version of the Borsuk-Ulam theorem hold. Moreover, we obtain a formula for the Stiefel-Wh…

Borsuk-Ulam type theorems for manifolds

This paper establishes a Borsuk-Ulam type theorem for PL-manifolds with a finite group action, depending on the free equivariant cobordism class of a manifold. In particular, necessary and sufficient conditions are considered for a manifold with a free involution to be a Borsuk-Ulam type.

Borsuk-Ulam type theorems for G-spaces with applications to Tucker type lemmas

In this paper we consider several generalizations of the Borsuk-Ulam theorem for G-spaces and apply these results to Tucker type lemmas for G-simplicial complexes and PL-manifolds.

A Borsuk--Ulam theorem for cyclic $p$-groups

We describe a connective $K$-theory Borsuk--Ulam/Bourgin--Yang theorem for cyclic groups of order a power of a prime $p$. Consider two finite dimensional complex representations $U$ and $V$ of the cyclic group $Z /p^{k+1}$ of order $p^{k+1}$, where $k\geq 0$. For $0\leq l\leq k$, we write $V_l$ for the subspace of $V$ fixed by the cyclic subgroup of order $p^l$, and require that the fixed subspace, $V_{k+1}$, be zero and that $V_k$ be non-zero. Put $δ(V)=\sum_{l=0}^k p^l dim_C (V_l/V_{l+1})-(p^k-1)$. Then the zero-set of any $Z /p^{k+1}$-map $S(U) \to V$ from the unit sphere in $U$ (for some i…

Theorems of Borsuk-Ulam type for flats and common transversals

In this paper some results on the topology of the space of $k$-flats in $\mathbb R^n$ are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in $\mathbb R^n$, and on measure partitions by hyperplanes, are deduced.

Borsuk - Ulam Type spaces

We consider spaces with free involutions that satisfy the Borsuk - Ulam theorems (BUT-spaces). There are several equivalent definitions for BUT-spaces that can be considered as their properties. Our main technical tool is Yang's cohomological index.

关于良分离映射的 Borsuk--Ulam 定理

A Borsuk--Ulam theorem for well separated maps

本文基于经典 Borsuk-Ulam 定理，通过引入良分离性质，证明满足特定奇性条件的连续映射的像必然包含一个 $m$-立方体，推广了原有定理的结论。

Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited

We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb{Z}/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in a real $(\mathbb{Z}/2)^k$-representation of the same dimension. Our proof method allows us to derive Borsuk--Ulam theorems for certain equivariant maps from Stiefel manifolds, from the corresponding results about products of spheres, leading to alternative proofs and extensions of some results of Fadell and Husseini.

Parametrized Borsuk-Ulam problem for projective space bundles

Let $π: E \to B$ be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let $π^{'}: E^{'} \to B$ be vector bundle such that $\mathbb{Z}_2$ acts fiber preserving and freely on $E$ and $E^{'}-0$, where 0 stands for the zero section of the bundle $π^{'}:E^{'} \to B$. For a fiber preserving $\mathbb{Z}_2$-equivariant map $f:E \to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \{x \in E | f(x)= 0\}.$ As an application, we also estimate the cohomological dimension of the $\mathbb{Z}_2$-coincidence set $A_f=\{x \in E | f(x)…

Borsuk-Ulam性质与截面范畴

Borsuk-Ulam property and Sectional Category

本文发现了双重覆盖的截面范畴与三元组Borsuk-Ulam性质之间的联系，证明了当$q$的截面范畴大于$q^Y$时该三元组满足BUP，并证明了仿紧空间中$(X,τ)$的指数与商映射截面范畴减1相一致。

度量空间的Borsuk-Ulam型定理

Borsuk-Ulam type theorems for metric spaces

该文研究拓扑空间中的Borsuk-Ulam型问题，探寻连续映射下距离足够远的点被映射到同一点的充分条件，拓展了相关定理的适用场景。

Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem

We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete, we show that the exact version is much harder. Specifically, finding a solution with $n$ cuts is FIXP-hard, and deciding whether there exists a solution with fewer than $n$ cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem e…

丛上同伦类的Borsuk-Ulam性质、参数化辫子群及其在曲面丛中的应用

The Borsuk-Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles

该研究在纤维丛框架下推广Borsuk-Ulam性质，将其判定问题转化为涉及基本群、轨道空间与参数化辫子群的代数问题，并确定了一类S¹上2-环面丛满足该性质的自映射同伦类。

博苏克-乌拉姆定理的选择结构推广

Selection-structure generalizations of the Borsuk-Ulam theorem

研究博苏克-乌拉姆定理的选择结构推广，受弗里克和韦尔纳工作启发，证明相关类似物及沃洛维科夫定理版本，还给出公平划分应用，如火腿三明治和项链分割定理的选择结构版本。

零伦等变移位与Borsuk-Ulam相容紧致阿贝尔群

Nullhomotopic equivariant shifts and Borsuk-Ulam-compatible compact abelian groups

本文否证了Borsuk-Ulam风格猜想，指出其仅在无挠profinite紧致阿贝尔群情形下失效，并给出精确刻画。

诺林反极着色猜想的链级博苏克 - 乌拉姆障碍证明

A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture

研究\(n\geq2\)时\(n\)维超立方体红 - 蓝边着色中连接顶点与其对映点的单色路径问题，通过构造特定链映射并利用链级博苏克 - 乌拉姆障碍予以证明。

图的Borsuk--Ulam性质 II：$\boldsymbol{\rm \bb{Z}}_n$-作用

Borsuk--Ulam property for graphs II: The $\mathbb{Z}_n$-action

本研究针对带有限群自由作用的拓扑空间，给出Borsuk--Ulam性质的几何与代数等价刻画，并将其应用于一维图的循环群作用情形，借助离散Morse理论有效控制图辫群要素。

Borsuk-Ulam Theorems for Complements of Arrangements

In combinatorial problems it is sometimes possible to define a $G$-equivariant mapping from a space $X$ of configurations of a system to a Euclidean space $\mathbb{R}^m$ for which a coincidence of the image of this mapping with an arrangement $\mathcal{A}$ of linear subspaces insures a desired set of linear conditions on a configuration. Borsuk-Ulam type theorems give conditions under which no $G$-equivariant mapping of $X$ to the complement of the arrangement exist. In this paper, precise conditions are presented which lead to such theorems through a spectral sequence argument. We introduce a…

A parametrized version of the Borsuk Ulam theorem

The main result of this note is a parametrized version of the Borsuk-Ulam theorem. We show that for a continuous family of Borsuk-Ulam situations, parameterized by points of a compact manifold W, its solution set also depends continuously on the parameter space W. Continuity here means that the solution set supports a homology class which maps onto the fundamental class of W. When W is a subset of Euclidean space, we also show how to construct such a continuous family starting from a family depending in the same way continuously on the points of the boundary of W. This solves a problem related…

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。

正交群的Borsuk-Ulam型定理与正交四分割

Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions

本文给出Makeev提出的正交四分割猜想的首个完整证明，核心是证明正交群上的Borsuk-Ulam型定理，通过构造显式模型映射并利用等变度原理完成证明。

Beyond the Borsuk-Ulam theorem: The topological Tverberg story

Bárány's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $Δ_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $Δ_N$ to the same point in $\mathbb{R}^d$. The proof of this result for the case when $r$ is a prime, as well as some colored version of the same result, using the results of Borsuk-Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem." In this paper we show how advanced equivariant topology meth…

Index与截面范畴

Index and Sectional Category

该研究针对自由有限群作用空间引入第q个index概念，结合商映射的截面范畴给出其通用估计，建立了Borsuk-Ulam理论经典问题与截面范畴研究的关联，还得到特定假设下的等式关系并产出两领域新结果。

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。

Stiefel流形的Borsuk-Ulam型定理与正交质量分割

Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

本文推广Borsuk-Ulam定理到Stiefel流形，并应用于证明在给定m个测度时存在k个相互正交的超平面，使得任意n个超平面将每个测度分成2^n等份，并给出d的下界。

Generalized Kneser coloring theorems with combinatorial proofs

The Kneser conjecture (1955) was proved by Lovász (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matoušek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lovász, Sarkaria, and Kriz. We also give a comb…

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。

Waist of the sphere for maps to manifolds

We generalize the sphere waist theorem of Gromov and the Borsuk--Ulam type measure partition lemma of Gromov--Memarian for maps to manifolds.

Colorful subgraphs in Kneser-like graphs

Combining Ky Fan's theorem with ideas of Greene and Matousek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.

球丛的 Ky Fan 定理

Ky Fan theorem for sphere bundles

本文将经典 Ky Fan 定理从三角剖分球面推广到三角剖分球丛的全空间，扩展了关于对跖着色的组合拓扑结论。

广义Z_p-图克勒引理的组合度版本及其组合证明

Combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma with a combinatorial proof

本文提出了广义Z_p-图克勒引理的组合度版本，并通过纯粹组合的方法证明了该定理，同时引入了一个独立的组合霍普夫追踪公式。