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.
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.
Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions τ on M, and the Borsuk-Ulam indice of (M,τ).
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 $\dim Z_f= \infty$.
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…
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…
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…
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 …
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 …
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…
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 $H^1(N;\mathbb{F}_2)$.
We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk--Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean sp…
We work within the framework of a program aimed at exploring various extended versions for theorems from a class containing Borsuk-Ulam type theorems, some fixed point theorems, the KKM lemma, Radon, Tverberg, and Helly theorems. In this paper we study variations of the Hopf theorem concerning continuous maps of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case of maps $f\colon M \to \mathbb{R}^m$ with $n < m$ and introduce several notions of varied types of $f$-neighbors, which is a pair of distinct points in $M$ such that $f$ takes it to a 'small' …
