Related papers: The Borsuk-Ulam theorem for n-valued maps between …

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…

Borsuk-Ulam's theorem is a useful tool of algebraic topology. It states that for any continuous mapping $f$ from the $n$-sphere to the $n$-dimensional Euclidean space, there exists a pair of antipodal points such that $f(x)=f(-x)$. As for…

Let $M$ and $N$ be fiber bundles over the same base $B$, where $M$ is endowed with a free involution $\tau$ over $B$. A homotopy class $\delta \in [M,N]_{B}$ (over $B$) is said to have the Borsuk-Ulam property with respect to $\tau$ if for…

We study the Borsuk-Ulam theorem for triple (M;\tau; \R^n), where M is a compact, connected, 3-manifold equipped with a fixed-point-free involution \tau. The largest value of n for which the Borsuk-Ulam theorem holds is called the Z_2-index…

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal…

We study Borsuk-Ulam type results for the loopspace of an euclidean sphere without loops equal to their inverses.

Let $M$ and $N$ be topological spaces, let $G$ be a group, and let $\tau \colon\thinspace G \times M \to M$ 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 $N$…

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 $\tau$ with respect to the validity of the {\it Borsuk-Ulam property} that for…

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…

Suppose that $f_1,\ldots ,f_m : S(V)\to R$ are $m$ ($\geq 1$) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m+1$ satisfying $f_i(-v)=-f_i(v)$ for all $v\in S(V)$. The classical Borsuk-Ulam…

Let M be a closed, connected 3-manifold which admits Nil geometry, we determine all free involutions ${\tau}$ on M and the Borsuk-Ulam index of $(M,{\tau})$.

We observe that the classical Borsuk-Ulam theorem has an easy generalization to maps from an n-manifold M^n to R^n. We point out a geometric corollary.

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…

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…

For a finite group $H$ and connected topological spaces $X$ and $Y$ such that $X$ is endowed with a free left $H$-action $\tau$, we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the…

Let $M$ be a closed 3-manifold which admits the geometry $S^2\times \R$. In this work we determine all the free involutions $\tau$ on $M$, and the Borsuk-Ulam index of $(M,\tau)$.

Let M and N be topological spaces such that M admits a free involution $\\tau$. A homotopy class $\beta$ $\in$ [M, N ] is said to have the Borsuk-Ulam property with respect to $\\tau$ if for every representative map f : M $\rightarrow$ N of…

B\'ar\'any's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $\Delta_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $\Delta_N$ to the same point in…

We prove multiple generalizations of Fan's combinatorial labeling result for sphere triangulations. This can be seen as a comprehensive extension of the Borsuk--Ulam theorem. In typical applications, the Borsuk--Ulam theorem gives…

We consider a "twisted" noncommutative join procedure for unital $C^*$-algebras which admit actions by a compact abelian group $G$ and its discrete abelian dual $\Gamma$, so that we may investigate an analogue of Baum-Dabrowski-Hajac…