Equilateral $p$-gons in $\mathbb R^d$ and deformed spheres and mod $p$ Fadell-Husseini index

TL;DR

This paper establishes the existence of certain equilateral polygons in Euclidean spaces and on spheres using topological methods, extending previous work with the Fadell-Husseini index.

Contribution

It introduces the concept of r-equilateral m-gons and proves their existence in specific geometric and topological settings, advancing the application of topological tools in geometric problems.

Findings

Existence of r-equilateral p-gons in re9d if r<d

Existence of equilateral p-gons in images of injective maps from spheres

Application of Fadell-Husseini index to geometric problems

Abstract

We introduce the concept of $r$-equilateral $m$-gons. We prove the existence of $r$-equilateral $p$-gons in $\mathbb R^d$ if $r<d$ and the existence of equilateral $p$-gons in the image of continuous injective maps $f:S^d\to \mathbb R^{d+1}$. Our ideas are based mainly in the paper of Y. Soibelman \cite{soibelman}, in which the topological Borsuk number of $\mathbb{R}^2$ is calculated by means of topological methods and the paper of P. Blagojevi\'c and G. Ziegler \cite{blagojevictetrahedra} where Fadell-Husseini index is used for solving a problem related to the topological Borsuk problem for $\mathbb{R}^3$.

Equations

$${\rho }_j(f(x_0),f(x_j))={\rho }_j(f(x_j),f(x_{2j}))=\cdots ={\rho }_j(f(x_{(p-1)j}),f(x_0)),$$

$$In{d}_{G}Y\subseteq In{d}_{G}X.$$

$$In{d}_G{X}_1\cdot In{d}_G{X}_2\subseteq IndX.$$

$$In{d}_{G}A=In{d}_{G}U.$$

$$In{d}_{G}A\cdot In{d}_G(Y\setminus B)\subseteq In{d}_{G}X.$$

$$q_X^{\ast }\otimes q_Y^{\ast }:H^{\ast }(BG;\mathbb{K})\otimes H^{\ast }(BH;\mathbb{K})\to H_G^{\ast }(X;\mathbb{K})\otimes H_H^{\ast }(Y;\mathbb{K}),$$

$$\ker q_X^{\ast }\otimes q_Y^{\ast }=\ker q_X^{\ast }\otimes H^{\ast }(BH;\mathbb{K})+H^{\ast }(BG;\mathbb{K})\otimes \ker q_Y^{\ast },$$

$$In{d}_{G\times H,\mathbb{K}}X\times Y=In{d}_{G,\mathbb{K}}\otimes H^{\ast }(BH;\mathbb{K})+H^{\ast }(BG;\mathbb{K})\otimes In{d}_{H,\mathbb{K}}Y.$$

$$In{d}_{G\times H,\mathbb{K}}X=In{d}_{G,\mathbb{K}}X\otimes H^{\ast }(BH;\mathbb{K});$$

$$In{d}_{G\times H,\mathbb{K}}X\times Y=⟨f_1,\dots ,f_r,g_1,\dots ,g_s⟩\subseteq \mathbb{K}[x_1,\dots ,x_n,y_1,\dots ,y_m].$$

$$E_2^{p,q}=H^p(G;H^q(X;R)).$$

$$H^{\ast }(BG;R)\to E_2^{\ast ,0}\to E_3^{\ast ,0}\to E_4^{\ast ,0}\to \cdots \to E_{\infty }^{\ast ,0}\subseteq H^{\ast }(X_G;R).$$

$$In{d}_{\mathbb{Z}/2,{\mathbb{F}}_2}{S}^n=⟨t^{n+1}⟩\subseteq {\mathbb{F}}_2[t].$$

$$In{d}_{\mathbb{Z}/2,{\mathbb{F}}_2}X=⟨t^{n+1}⟩\subseteq H^{\ast }(B\mathbb{Z}/2;{\mathbb{F}}_2)={\mathbb{F}}_2[t].$$

$$In{d}_{\mathbb{Z}/p,{\mathbb{F}}_p}{S}^{2n-1}=⟨b^n⟩\subseteq {\mathbb{F}}_p[a,b]/⟨a^2⟩.$$

$$In{d}_{\mathbb{Z}/p,{\mathbb{F}}_p}X=⟨b^n⟩\subseteq {\mathbb{F}}_p[a,b]/⟨a^2⟩.$$

$$In{d}_{\mathbb{Z}/p,{\mathbb{F}}_p}X=⟨a{b}^n⟩\subseteq {\mathbb{F}}_p[a,b]/⟨a^2⟩.$$

$$In{d}_{G,\mathbb{K}}X\subseteq \underset{d>n}{⨁}{H}^d(BG;\mathbb{K}),$$

$$In{d}_{G,\mathbb{K}}S(U\oplus V)=⟨f\cdot g⟩\subseteq H^{\ast }(BG;\mathbb{K}).$$

$$In{d}_{(\mathbb{Z}/p{)}^n,{\mathbb{F}}_p}S(V)=⟨{\alpha }_1{b}_1+\cdots +{\alpha }_n{b}_n⟩\subseteq {\mathbb{F}}_p[b_1,\dots ,b_n]\subseteq H^{\ast }(B(\mathbb{Z}/p{)}^n;{\mathbb{F}}_p).$$

$$F(X,n)=\{(x_1,x_2,\dots ,x_n)\in X^n:x_i\ne x_j\text{ for all}i\ne j\}.$$

$$In{d}_{\mathbb{Z}/p,{\mathbb{F}}_p}F({\mathbb{R}}^d,p)=⟨a{b}^{\frac{(d-1)(p-1)}{2}},b^{\frac{(d-1)(p-1)}{2}+1}⟩\subseteq {\mathbb{F}}_p[a,b]/⟨a^2⟩.$$

$$\omega \cdot (x_0,x_1,\dots ,x_{p-1})=(x_1,x_2,\dots ,x_{p-1},x_0).$$

$$f_i(x_0,\dots ,x_{p-1})=({\rho }_i(x_0,x_{t_i}),{\rho }_i(x_{t_i},x_{2t_i}),\dots ,{\rho }_i(x_{(p-1)t_i},x_0)).$$

$$In{d}_{\mathbb{Z}/p,{\mathbb{F}}_p}S(({\Delta }^r{)}^{\perp })=⟨b^{r(p-1)/2}⟩\subseteq {\mathbb{F}}_p[a,b]/⟨a^2⟩$$

$$In{d}_{\mathbb{Z}/p,{\mathbb{F}}_p}F({\mathbb{R}}^d,p)=⟨a{b}^{(d-1)(p-1)/2},b^{(d-1)(p-1)/2+1}⟩\subseteq {\mathbb{F}}_p[a,b]/⟨a^2⟩.$$

$${\rho }_i(x_0,x_{t_i})={\rho }_i(x_{t_i},x_{2t_i})=\dots ={\rho }_i(x_{(p-1)t_i},x_0)=c_i.$$

$$\{\begin{array}{l}{z}_0(\theta )=\cos \theta +i\sin \theta ,\\ z_1(\theta )=\cos (\theta +2\pi /3)+i\sin (\theta +2\pi /3),\\ z_2(\theta )=\cos (\theta +4\pi /3)+i\sin (\theta +4\pi /3)\end{array}$$

$$\{\begin{array}{ll}{d}_{01}(\theta )& =\Vert z_1(\theta )-z_0(\theta ){\Vert }_1=|\cos (\theta +2\pi /3)-\cos \theta |+|\sin (\theta +2\pi /3)-\sin \theta |,\\ d_{02}(\theta )& =\Vert z_2(\theta )-z_0(\theta ){\Vert }_1=|\cos (\theta +4\pi /3)-\cos \theta |+|\sin (\theta +4\pi /3)-\sin \theta |,\\ d_{12}(\theta )& =\Vert z_2(\theta )-z_1(\theta ){\Vert }_1\\ & =|\cos (\theta +4\pi /3)-\cos (\theta +2\pi /3)|+|\sin (\theta +4\pi /3)-\sin (\theta +2\pi /3)|,\end{array}$$

$$\{\begin{array}{l}{d}_{01}(\theta )=\sqrt{3}|\sin (\theta +\pi /3)|+\sqrt{3}|\cos (\theta +\pi /3)|,\\ d_{02}(\theta )=\sqrt{3}|\sin (\theta +2\pi /3)|+\sqrt{3}|\cos (\theta +2\pi /3)|,\\ d_{12}(\theta )=\sqrt{3}|\sin (\theta +\pi )|+\sqrt{3}|\cos (\theta +\pi )|.\end{array}$$

Full text

1 The first author acknowledges and thanks the finacial support of the grant * P12.160422.004/01- FAPA ANDRES ANGEL* from Vicedecanatura de Investigaciones de la Facultad de Ciencias de la Universidad de los Andes, Colombia. The second author acknowledges and thanks the financial support of the grant Proyecto Semilla 2016-2 from Vicedecanatura de Investigaciones de la Facultad de Ciencias de la Universidad de los Andes, Colombia

1. Introduction

Let $(X,\rho)$ be a metric space. For a compact subset $K$ of $X$, $b_{(X,\rho)}(K)$ denotes the Borsuk number of $K$, that is, the minimal number of parts of $K$ of smaller diameter necessary to partition $K$. The Borsuk number of $(X,\rho)$ is defined to be $B(X,\rho)=\max_{K}b_{X,\rho}(K)$. If $\Omega(\rho)$ denotes the set of metrics on $X$ equivalent to $\rho$, then the topological Borsuk number of $(X,\rho)$ is defined by $B(X)=\min_{\tau\in\Omega(\rho)}B(X,\tau)$.

The topological Borsuk problem for $\mathbb{R}^{d}$ is to estimate the topological Borsuk number of $\mathbb{R}^{d}$, $B(\mathbb{R}^{n})$ for the Euclidean metric (see [15]); one wants to know if $B(\mathbb{R}^{n})\geq n+1$. Y. Soibelman proved in [15] that $B(\mathbb{R}^{2})=3$, but it is not known if $B(\mathbb{R}^{3})\geq 4$.

A nonempty subset $S$ of a metric space $(X,\rho)$ is called equilateral (with respect to $\rho$), if there is some positive real number $c$ such that $\rho(x,y)=c$ for all $x,y\in S$, $x\neq y$. In order to prove that $B(\mathbb{R}^{d})\geq m$, it suffices to show that for any metric $\rho$ defined on $\mathbb{R}^{d}$ equivalent to the Euclidean metric, there exists some equilateral subset $S\subseteq(\mathbb{R}^{d},\rho)$ of size $m$ (see [15]). The following result is due to C. M Petty ([14]):

Theorem 1.1**.**

Every equilateral set of size 3 in a finite dimensional normed space of dimension at least 3, can be extended to an equilateral set of size 4.

Theorem 1.1 implies that for any metric $\rho$ on $\mathbb{R}^{3}$ induced by a norm we have $B(\mathbb{R}^{2},\rho)\geq 4$. In fact, restrict $\rho$ to a 2-plane $X$ in $\mathbb{R}^{3}$ and apply the result of Soibelman that the Borsuk number of $\mathbb{R}^{2}$ is 3 to obtain a $\rho$-equilateral set in $X$ of size 3. This equilateral set in $X$ is also an equilateral set of size 3 in $(\mathbb{R}^{3},\rho)$, so by theorem 1.1, it extends to a $\rho$-equilateral set of size 4. Thus, $B(\mathbb{R}^{4},\rho)\geq 4$. This gives us a partial answer to the Borsuk problem for $\mathbb{R}^{3}$.

An $m$-gon in a metric space $(X,\rho)$ is an $m$-cycle graph $G$ whose vertices are $m$ distinct elements in $X$. If $G$ has vertices $x_{0},x_{1},\ldots,x_{m-1}$ and edges $\{x_{i},x_{i+1}\}$ for $i=0,1,\ldots,m-1$ (where $x_{m}=x_{0}$), we say that $G$ is an equilateral $m$-gon in $(X,\rho)$, or a equilateral $m$-gon with respect to $\rho$, if there is some constant $c>0$ such that $\rho(x_{i},x_{i+1})=c$ for all $i=0,1,\ldots,m-1$. Given $r$ metrics on $X$, $\rho_{1},\ldots,\rho_{r}$, we say that $G$ is an $r$-equilateral $m$-gon (with respect to $\rho_{1},\ldots,\rho_{r}$) if it is equilateral for each metric $\rho_{i}$.

Let $\mathbb{Z}/m=\langle\omega|\ \omega^{m}=1\rangle$. Given an $m$-gon $G$ with vertices $x_{0},x_{1},\ldots,x_{m-1}$ and edges $\{x_{i},x_{i+1}\}$ for $i=0,1,\ldots,m-1$ (where $x_{m}=x_{0}$), $\mathbb{Z}/m$ acts naturally on the set of vertices of $G$ by $\omega\cdot x_{i}=x_{i+1}$, and this induces an action of $\mathbb{Z}/m$ on the set of edges of $G$: $\omega\cdot\{x_{i},x_{i+1}\}=\{\omega\cdot x_{i},\omega\cdot x_{i+1}\}$.

If $K_{m}$ denotes the complete graph on vertices $x_{0},\ldots,x_{m-1}$, then the action of $\mathbb{Z}/m$ on $x_{0},x_{1},\ldots,x_{m-1}$ induces an action of $\mathbb{Z}/m$ on the edges of $K_{m}$. The set of edges of $K_{m}$ decomposes into a union of disjoint orbits; some of these orbits represent $m$-gons, some other are $n$-gons for some $n<m$ and some other orbits are not even $n$-gons for any $n$ (see figure 1 (a)). If $m=p$ is an odd prime number, then each orbit of $\mathbb{Z}/p$ acting on the set of edges of $K_{p}$ is actually a $p$-gon, and there are $(p-1)/2$ of them (see figure 1 (b)). If the subscripts of the $x_{i}$’s are thought as the elements of the finite field $\mathbb{F}_{p}$, then these $(p-2)/2$ $p$-gons, that we will denote by $C_{1},C_{2},\ldots,C_{(p-1)/2}$, can be described as follows: $C_{t}$ has edges $\{x_{0},x_{t}\},\{x_{t},x_{2t}\},\ldots,\{x_{(p-1)t},x_{0t}\}$. Our first main result is the following:

Theorem 1.2**.**

Let $p$ be an odd prime number, $\rho_{1},\rho_{2},\ldots,\rho_{r}$ be $r$ metrics on $\mathbb{R}^{d}$ that are equivalent to the Euclidean metric of $\mathbb{R}^{d}$ and fix a sequence $\{t_{1},t_{2},\ldots,t_{r}\}$, where $t_{i}\in\{1,2,\ldots,(p-1)/2\}$. If $d>r$, then there are $p$ distinct points $x_{0},x_{1}\ldots,x_{p-1}$ in $\mathbb{R}^{d}$ such each $p$-gon $C_{t_{i}}$ is $\rho_{i}$-regular, $i=1,2,\ldots,r$.

As a consequence, if we put $t_{1}=t_{2}=\cdots=t_{r}$ in theorem 1.2, then we have that there exists an $r$-equilateral $p$-gon in $\mathbb{R}^{d}$ with respect to $\rho_{1},\rho_{2},\ldots,\rho_{r}$.

To state our second main result we first introduce some terminology. Let $p$ be an odd prime and $d\geq 1$. We call the pair $(d,p)$ admissible if given any continuous injective map $f:S^{d}\to\mathbb{R}^{d+1}$ and $(p-1)/2$ metrics $\rho_{1},\ldots,\rho_{(p-1)/2}$ on $\mathbb{R}^{d+1}$, each metric equivalent to the Euclidean metric of $\mathbb{R}^{d+1}$, then there can be found $p$ different points in the image of $f$, $f(x_{0}),\ldots,f(x_{p-1})$, such that

[TABLE]

for all $j=1,2,\ldots,(p-1)/2$. In other words, the $(p-1)/2$ $p$-gons determined by $f(x_{0}),\ldots,f(x_{p-1})$, $C_{1},C_{2},\ldots,C_{(p-1)/2}$ satisfy that each $C_{i}$ is equilateral with respect to $\rho_{i}$, $i=1,2,\ldots,(p-1)/2$. Now we can state our result as follows:

Theorem 1.3**.**

If $p\leq 2d+1$ and $(p-1)^{2}/2$ is not of the form $jd+1$ for any $j=1,2,\ldots,p-1$, then the pair $(d,p)$ is admissible.

Both results, Theorem 1.2 and Theorem 1.3 are proved by means of topological methods: we use the configuration space/test map scheme and the Fadell-Husseini index with coefficients in $\mathbb{F}_{p}$ of the involved spaces to solve our problems.

2. Fadell-Husseini index

Let $G$ be a finite group and $R$ a commutative ring with unit. The Fadell-Husseini index with coefficients in $R$ assigns to each $G$-space $X$ an ideal $Ind_{G,R}X$ of the ring $H^{\ast}(BG;R)$, the cohomology ring of the classifying space $BG$ with coefficients in $R$. The important property of the Fadell-Husseini index is that it gives a necessary condition for the existence of equivariant maps between to $G$-spaces: if $X$ and $Y$ are $G$-spaces and there is some $G$-equivariant map $f:X\to Y$, then $Ind_{G,R}Y\subseteq Ind_{G,R}X$. For definitions, basic properties and some calculations see [2, 3, 4, 7, 8, 17]. Here, we want to mention the properties and calculations of the Fadell-Husseini index we will use in the proofs of our results.

For a finite group $G$ we have the associated universal $G$-bundle $EG\to BG$; if $X$ is a $G$-space $X$, then there is the associated Borel construction $X_{G}=X\times_{G}EG$ and a map $q_{X}:X_{G}\to BG$ induced by the unique map $X\to\ast$. The index $Ind_{G,R}X$ is defined as the kernel of the induced map in cohomology $q_{X}^{\ast}:H^{\ast}(BG;R)\to H^{\ast}(X_{G};R)$. The axioms of an ideal valued index theory are satisfied by Fadell-Husseini index:

Monotonicity: If there is some $G$-equivariant map $f:X\to Y$, for $G$-spaces $X$ and $Y$, then

[TABLE]

Additivity: If $\{X=X_{1}\cup X_{2},X_{1},X_{2}\}$ is excisive, where $X_{1}$ and $X_{2}$ are $G$-invariant subspaces of $X$, then

[TABLE]

Continuity: If $A$ is a closed $G$-invariant subspace of $X$, then there is a $G$-invariant neighborhood $U$ of $A$ in $X$ such that

[TABLE]

Index theorem: If $f:X\to Y$ is $G$-equivariant, $B\subseteq Y$ a closed $G$-invariant subspace and $A:=f^{-1}(B)$, then

[TABLE]

For instance, if $X$ is a $G$-space and $Y$ an $H$-space, then $X\times Y$ is a $G\times H$-space; the map $q_{X\times Y}:(X\times Y)_{G\times H}\to B(G\times H)$ can be identified with the map $q_{X}\times q_{Y}:X_{G}\times Y_{H}\to BG\times BH$ and the map $q_{X\times Y}^{\ast}$ can be identified (under Künneth isomorphisms) with

[TABLE]

where $\mathbb{K}$ is a field. The kernel of this map can be expressed as follows:

[TABLE]

that is,

[TABLE]

As particular cases, if $Y=pt$, then

[TABLE]

and if $H^{\ast}(BG;\mathbb{K})\cong\mathbb{K}[x_{1},\ldots,x_{n}],H^{\ast}(BH;\mathbb{K})\cong\mathbb{K}[y_{1},\ldots,y_{m}]$, $Ind_{G,\mathbb{K}}X=\langle f_{1},\ldots,f_{r}\rangle\text{ and }Ind_{H,\mathbb{K}}Y=\langle g_{1},\ldots,g_{s}\rangle,$ then

[TABLE]

These results on the index of products can be consulted in [7].

Spectral Sequence. The map $q_{X}:X_{G}\to BG$ is a fibration with fiber $X$. There is a spectral sequence $\{E_{r}^{\ast,\ast},d_{r}\}$ with $E_{2}^{p,q}=H^{p}(BG;\mathcal{H}^{q}(X;R))$ where $\mathcal{H}^{q}(X;R)$ is a system of local coefficients, this local coefficient system is determined by the action of $G=\pi_{1}(BG)$ on $H^{\ast}(X;R)$. Therefore, the $E_{2}$-page of the spectral sequence is interpreted as the cohomology of the group $G$ with coefficients in the $G$-module $H^{\ast}(X,R)$ (see for example [1, 2]),

[TABLE]

The map $q_{X}^{\ast}:H^{\ast}(BG,R)\to H^{\ast}(X_{G},R)$ can be represented as the composition (known as the edge homomorphism)

[TABLE]

Spheres, $E_{n}G$-spaces and the configuration space $F(\mathbb{R}^{d},p)$. A free $G$-space $X$ is an $E_{n}G$-space if it is an $(n-1)$-connected, $n$-dimensional CW-complex equipped with a $G$-invariant CW-structure. $E_{n}G$-spaces exist; in fact, the join $G^{\ast(n+1)}$ of $n+1$ copies of the group $G$, where $G$ is regarded as a 0-dimensional simplicial complex, is an $E_{n}G$-space.

The following properties of the Fadell-Husseini index can be found in [17].

Proposition 2.1**.**

i) For the antipodal $\mathbb{Z}/2$-action on a sphere $S^{n}$, we have that

[TABLE]

In general, if $X$ is an $E_{n}\mathbb{Z}/2$-space, then

[TABLE]

ii) For an odd prime $p$, we have $H^{\ast}(B\mathbb{Z}/p,\mathbb{F}_{p})=\mathbb{F}_{p}[a,b]/\langle a^{2}\rangle$, where $\deg(a)=1$ and $\deg(b)=2$. The unit sphere $S^{2n-1}\subseteq\mathbb{C}^{n}$ is a $\mathbb{Z}/p$-space where $\mathbb{Z}/p$ is seen as the subgroup of $S^{1}$ of $p$-th roots of 1. $S^{1}$ acts on $S^{2n-1}$ by complex multiplication and so does $\mathbb{Z}/p$. Then

[TABLE]

If $X$ is an $E_{2n-1}\mathbb{Z}/p$, then

[TABLE]

iii) For an $E_{2n}\mathbb{Z}/p$-space $X$, we have that

[TABLE]

iv) For a finite group $G$ and an $E_{n}G$-space $X$, we have

[TABLE]

*that is, for every (homogeneous) element $x\in Ind_{G}X\subseteq H^{\ast}(BG;\mathbb{K})$, $deg(x)>n$.

v) Let $U,V$ be two $G$-representations and $S(U),S(V)$ be the associated unit $G$-spheres. Assume that the vector bundles $U\to U_{G}\to BG$ and $V\to V_{G}\to BG$ are orientable over $\mathbb{K}$. If $Ind_{G,\mathbb{K}}S(U)=\langle f\rangle\subseteq H^{\ast}(BG;\mathbb{K})$ and $Ind_{G,\mathbb{K}}S(V)=\langle g\rangle\subseteq H^{\ast}(BG;\mathbb{K})$, then

[TABLE]

vi) Let $V=\mathbb{C}$ be the 1-dimensional complex $(\mathbb{Z}/p)^{k}$-representation associated with the vector $(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{F}_{p}^{n}$ (if $\omega_{i}$ is the generator of the $i^{th}$ copy of $\mathbb{Z}/p$ in $(\mathbb{Z}/p)^{n}$, then the vector $\alpha$ is characterized by the equality $\omega_{i}\cdot v=\zeta^{\alpha_{i}}v$ where $\zeta=\large{e}^{2\pi i/p}$, $v\in V$ and $i=1,\ldots,n$). Then

[TABLE]

The configuration space of $n$ distinct points in the topological space $X$ is the space

[TABLE]

The symmetric group $S_{n}$ acts on $F(X,n)$ by permuting coordinates; the cyclic group $\mathbb{Z}/n$ seen as the subgroup of $S_{n}$ generated by the permutation $(12\cdots n)$, acts on $F(X,n)$ by restricting the action of $S_{n}$. The computation of $Ind_{\mathbb{Z}/p,\mathbb{F}_{p}}F(\mathbb{R}^{d})$, where $p$ is an odd prime number and $d>1$ is made in [2] (Theorem 6.1):

[TABLE]

3. Existence of $r$-equilateral $p$-gons in $\mathbb{R}^{d}$

Assume that $p$ is an odd prime number, $r\geq 1$, $d\geq 1$, $\rho_{1},\rho_{2},\ldots,\rho_{r}$ are $r$ metrics on $\mathbb{R}^{d}$, each $\rho_{i}$ equivalent to the Euclidean metric of $\mathbb{R}^{d}$, and consider a sequence $\{t_{1},t_{2},\ldots,t_{r}\}$, where $t_{i}\in\{1,2,\ldots,(p-1)/2\}$. The problem is to show that there are $p$ distinct points $x_{0},x_{1}\ldots,x_{p-1}$ in $\mathbb{R}^{d}$ such that $C_{t_{i}}$ is $\rho_{i}$-regular for all $i=1,2,\ldots,r$.

The configuration space for our problem is the configuration space of $p$ distinct points in $\mathbb{R}^{d}$, $F(\mathbb{R}^{d},p)=\{(x_{0},x_{1},\ldots,x_{p-1})\in(\mathbb{R}^{d})^{p}:x_{i}\neq x_{j},i\neq j\}$. Consider $\mathbb{Z}/p=\langle\omega|\ \omega^{p}=1\rangle$ acting on $(\mathbb{R}^{d})^{p}$ by

[TABLE]

The group $\mathbb{Z}/p$ acts freely on $F(\mathbb{R}^{d},p)$.

Next, we define $V=\mathbb{R}^{p}\times\cdots\times\mathbb{R}^{p}$, the product of $r$ copies of $\mathbb{R}^{d}$, and define our test map $f=(f_{1},f_{2},\ldots,f_{r}):F(\mathbb{R}^{d},p)\to V$, where

[TABLE]

The group $\mathbb{Z}/p$ acts on $\mathbb{R}^{p}$ by $\omega\cdot(y_{0},y_{1},\ldots,y_{p-1})=(y_{1},y_{2},\ldots,y_{p-1},y_{0})$ and $\mathbb{Z}/p$ acts on $V$ diagonally. The map $f:F(\mathbb{R}^{d},p)\to V$ becomes $\mathbb{Z}/p$-equivariant.

The existence of $p$ distinct points $x_{0},x_{1}\ldots,x_{p-1}$ in $\mathbb{R}^{d}$ such that $C_{t_{i}}$ is $\rho_{i}$-regular for any $i$ is equivalent to the existence of an element $(x_{0},\ldots,x_{p-1})\in F(\mathbb{R}^{d},p)$ such that for each $i=1,\ldots,r$, $f_{i}(x_{0},\ldots,x_{p-1})\in\Delta$, where $\Delta=\{(t,t,\ldots,t)\in\mathbb{R}^{d}:t\in\mathbb{R}\}$; this is equivalent to say that $im\ f\cap\Delta^{r}\neq\varnothing$.

If we assume that $im\ f\cap\Delta^{r}=\varnothing$, then we get a $\mathbb{Z}/p$-equivariant map $f:F(\mathbb{R}^{d},p)\to V\setminus\Delta^{r}$. Note that the action of $\mathbb{Z}/p$ on $\mathbb{R}^{p}\setminus\Delta$ is free and so is the action of $\mathbb{Z}/p$ on $V\setminus\Delta^{r}$. Thus, we have the following:

Lemma 3.1**.**

If there is no $\mathbb{Z}/p$-equivariant map $F(\mathbb{R}^{d},p)\to V\setminus\Delta^{r}$, then there are $p$ distinct points $x_{0},x_{1}\ldots,x_{p-1}$ in $\mathbb{R}^{d}$ such that $C_{t_{i}}$ is $\rho_{i}$-regular for all $i=1,2,\ldots,r$.

Lemma 3.2**.**

Let $p$ be an odd prime number, $d\geq 1$ and $r\geq 1$. If $d>r$, then there is no $\mathbb{Z}/p$-equivariant map $F(\mathbb{R}^{d},p)\to V\setminus\Delta^{r}$.

Lemma 3.2, combined with Lemma 3.1, completes the proof of Theorem 1.2.

Proof.

Assume that there is a $\mathbb{Z}/p$-equivariant map $h:F(\mathbb{R}^{d},p)\to V\setminus\Delta^{r}$. We also have an $\mathbb{Z}/p$-equivariant map $g:V\setminus\Delta^{r}\to S((\Delta^{r})^{\perp})$ (projecting and normalizing). The composition $g\circ h$ gives a $\mathbb{Z}/p$-equivariant map $F(\mathbb{R}^{d},p)\to S((\Delta^{r})^{\perp})$. The actions of $\mathbb{Z}/p$ on $F(\mathbb{R}^{d},p)$ and $S((\Delta^{r})^{\perp})$ are free.

Since $\Delta^{r}$ is an $r$-dimensional subspace of $V=(\mathbb{R}^{p})^{r}$, we have that $(\Delta^{r})^{\perp}$ is an $r(p-1)$-dimensional subspace of $V$. Thus, $S((\Delta^{r})^{\perp})$ is a sphere of dimension $r(p-1)-1$.

We calculate Fadell-Husseini indexes. By Proposition 2.1, $ii)$,

[TABLE]

and by (2.4),

[TABLE]

The existence of a $\mathbb{Z}/p$-equivariant map $F(\mathbb{R}^{d},p)\to S((\Delta^{r})^{\perp})$ yields the inclusion $Ind_{\mathbb{Z}/p,\mathbb{F}_{p}}S((\Delta^{r})^{\perp})\subseteq Ind_{\mathbb{Z}/p,\mathbb{F}_{p}}F(\mathbb{R}^{d},p)$. From this we deduce that $r(p-1)>(d-1)(p-1)$, that is, $r>d-1$. This ends the proof. ∎

Remark 3.3**.**

The continuity of the map $f=(f_{1},f_{2},\ldots,f_{r})$ follows from the fact that a metric $\rho:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$ that induces the usual topology of $\mathbb{R}^{d}$ is a symmetric and continuous function. Nothing else about the $r$ metrics is used in the proof of theorem 1.2. So, in theorem 1.2 we can take $r$ symmetric continuous maps $\rho_{i}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$; with this generality, the conclusion of theorem 1.2 is that for each $i\in\{1,2,\ldots,r\}$ there is some $c_{i}\in\mathbb{R}$ (not necessarily positive) such that

[TABLE]

Now we list some consequences of Theorem 1.2.

Corollary 3.4**.**

Let $p$ be an odd prime number and $\rho_{1},\rho_{2},\ldots,\rho_{(p-1)/2}$ be $(p-1)/2$ metrics on $\mathbb{R}^{d}$ that are equivalent to the Euclidean metric of $\mathbb{R}^{d}$. If we have $d>(p-1)/2$, then there are $p$ distinct points $x_{0},x_{1},\ldots,x_{p-1}$ in $\mathbb{R}^{d}$ such that for each $t\in\{1,2,\ldots,(p-1)/2\}$, the $p$-gon $C_{t}$ is $\rho_{t}$-regular.

Proof.

Put $r=(p-1)/2$ and take the sequence $\{1,2,\ldots,(p-1)/2\}$ in Theorem 1.2. ∎

Corollary 3.5**.**

Let $p$ be an odd prime number and $\rho$ be a metric on $\mathbb{R}^{d}$ that is equivalent to the Euclidean metric of $\mathbb{R}^{d}$. If $d>(p-1)/2$, then there are $p$ distinct points $x_{0},x_{1},\ldots,x_{p-1}$ in $\mathbb{R}^{d}$ such that for each $t=1,2,\ldots,(p-1)/2$, the $p$-gon $C_{t}$ is regular with respect to $\rho$.

Proof.

In theorem 1.2 put $r=(p-1)/2$, $\rho_{1}=\rho_{2}=\cdots=\rho_{(p-1)/2}$ and take the sequence $\{1,2,\ldots,(p-1)/2\}$. ∎

Corollary 3.6**.**

Let $p$ be an odd prime number and $\rho_{1},\rho_{2},\ldots,\rho_{r}$ be $r$ metrics on $\mathbb{R}^{d}$ that are equivalent to the Euclidean metric of $\mathbb{R}^{d}$. If $d>r$, then there exists an $r$-equilateral $p$-gon in $\mathbb{R}^{d}$ with respect to $\rho_{1},\rho_{2},\ldots,\rho_{r}$.

Proof.

In theorem 1.2 take the constant sequence $\{1,1,\ldots,1\}$. ∎

Corollary 3.7**.**

The Borsuk number of $\mathbb{R}^{2}$ is 3.

Proof.

In theorem 1.2 take $d=2$, $r=1$ and $p=3$. Note that an equilateral $3$-gon is the same as an equilateral set of size 3. ∎

Remark 3.8**.**

Soibelman proves in [15] that the Borsuk number of $\mathbb{R}^{2}$ is 3 by the same method (the case $d=2,r=1,p=3$). He remarks that for the topological Borsuk problem in $\mathbb{R}^{n}$, $n>2$, there is an $S_{n+1}$-equivariant map $(\mathbb{R}^{n})^{n+1}\to\mathbb{R}^{n(n+1)/2}$. As in the CS/TM scheme shown above, we get an equivariant map between spheres, but the actions of various subgroups of $S_{n+1}$ are not free on the target, so Dold’s theorem cannot be applied. Our results are based on the idea of having actual free actions of some groups (i.e. $\mathbb{Z}/p$) on the unit sphere of some convenient spaces on the target. We do not get equilateral sets, but $r$-equilateral $p$-gons.

Example 3.9**.**

Bi-equilateral triangles do not always exist in $\mathbb{R}^{2}$. Theorem 1.2 or Corollary 3.6 ensures that there exist bi-equilateral triangles for any pair of metrics $\rho_{1}$ and $\rho_{2}$ in $\mathbb{R}^{d}$ whenever $d>2$. We will exhibit two metrics in $\mathbb{R}^{2}$ for which there are no bi-equilateral triangles.

Consider the Euclidean norm $\|(x,y)\|_{2}=\sqrt{x^{2}+y^{2}}$ and the norm $\|(x,y)\|_{1}=|x|+|y|$. By regarding $\mathbb{R}^{2}$ as the field of complex numbers $\mathbb{Z}$, for the Euclidean norm, we can assume, without lost of generality, that all equilateral triangles have their vertices living in the unit circle $\{z\in\mathbb{Z}:|z|=1\}$. Let $z_{0},z_{1},z_{2}$ be the cubic roots of 1: $z_{0}=1,z_{2}=\cos(2\pi/3)+i\sin(2\pi/3),z_{2}=\cos(4\pi/3)+i\sin(4\pi/3)$. Now we can describe all Euclidean equilateral triangles with their vertices in the unit circle. In fact, for $0\leq\theta\ \leq 2\pi$, the 3 complex numbers

[TABLE]

are the vertices of an equilateral triangle. It suffices to consider $0\leq\theta\leq 2\pi/3$ for obtaining all equilateral triangles with their vertices in the unit circle.

Now we calculate the length of the edges of such triangles using the taxicab norm:

[TABLE]

these expressions reduce to

[TABLE]

There is no $\theta$ between [math] and $2\pi/3$ such that $d_{01}(\theta)=d_{02}(\theta)=d_{12}(\theta)$. In fact, for $0\leq\theta\leq 2\pi/3$, the equality $d_{01}(\theta)=d_{02}(\theta)$ holds only for $\theta=0,\pi/4,\pi/2$. The common values $d_{01}(\theta)=d_{02}(\theta)$ are $\frac{\sqrt{3}}{2}(1+\sqrt{3}),3\sqrt{2}/2,\frac{\sqrt{3}}{2}(1+\sqrt{3})$ respectively; but, the respective values of $d_{12}(\theta)$ are $\sqrt{3},\sqrt{6},\sqrt{3}$. This shows that there are no equilateral triangle with respect to the Euclidean metric that is also equilateral with respect to the taxicab metric in $\mathbb{R}^{2}$. Note that the three triangles found are indeed isosceles triangles with respect to the taxicab norm. In fact, a gereral result (see [10]) shows that given a circle with two distance functions on it, $d_{1}$ and $d_{2}$, where $d_{1}$ is the restricted distance coming from a smooth embedding of $S^{1}$ into a Riemannian manifold, and $d_{2}$ is symmetric and continuous, then there are three points in $S^{1}$ forming a $d_{1}$-equilateral triangle and, at the same time, a “$d_{2}$-isosceles” triangle.

4. Existence of equilateral $p$-gons on deformed spheres

Theorem 1.2 shows the existence of equilateral $p$-gons in $(\mathbb{R}^{d},\rho)$ for $p$ prime if $p<d$; it does not apply to $m$-gons when $m$ is not prime, in particular it does not apply to quadrilaterals. For proving the existence of quadrilaterals, Fadell-Husseini index with integer coefficients is needed. In fact, we have the following result of Blagojević and Ziegler (see [3]):

Theorem 4.1**.**

For every injective continuous map $f:S^{2}\to\mathbb{R}^{3}$, there are four distinct points $y_{0},y_{1},y_{2},y_{3}$ in the image of $f$ such that

[TABLE]

and

[TABLE]

Here, $d$ is the Euclidean metric of $\mathbb{R}^{3}$. The result is still true if we change $d$ by any metric $\rho$ equivalent to $d$.

From this theorem, the existence of equilateral quadrilaterals follows for any metric $\rho$ in $\mathbb{R}^{3}$ equivalent to the Euclidean metric (take $f$ as the inclusion $S^{2}\hookrightarrow\mathbb{R}^{3}$). The proof of this theorem relies on the use of Fadell-Husseini index with integer coefficients (It is shown in [3] that the use of mod. 2 coefficients is not enough).

We generalize this result of Blagojević and Ziegler in Theorem 1.3. The problem is to determine when a pair $(d,p)$ is admissible. Consider a continuous and injective map $f:S^{d}\to\mathbb{R}^{d+1}$ and metrics $\rho_{1},\rho_{2},\ldots,\rho_{(p-1)/2}$ equivalent to the Euclidean metric on $\mathbb{R}^{d+1}$. Consider the map $F=(f_{1},f_{2},\ldots,f_{r}):(S^{d})^{p}\to(\mathbb{R}^{p})^{(p-1)/2}$ where $f_{j}(x_{0},x_{1},\ldots,x_{p-1})$ equals

[TABLE]

for $(x_{0},x_{1},\ldots,x_{p-1})\in(S^{d})^{p}$, $j=1,2,\ldots,(p-1)/2$. Let $L=\{(x,\ldots,x)\in(S^{d})^{p}:x\in S^{d}\}$ and $\Delta=\{(t,\ldots,t)\in\mathbb{R}^{p}:t\in\mathbb{R}\}$. Let us restrict $F$ to $X_{d,p}=(S^{d})^{p}\setminus L$. We assume that $im\ F\cap\Delta^{(p-1)/2}=\varnothing$ so that we can consider $F$ as a map $F:X_{d,p}\to Y_{p}:=(\mathbb{R}^{p})^{(p-1)/2}\setminus\Delta^{(p-1)/2}$. The group $\mathbb{Z}/p$ acts on these spaces by permuting coordinates and making the map $F$ equivariant.

The orthogonal projection $Y_{p}\to\Delta^{(p-1)/2}\setminus\{0\}$ is $\mathbb{Z}/p$-equivariant, and so is the normalization map $\Delta^{(p-1)/2}\setminus\{0\}\to S(\Delta^{(p-1)/2})^{\perp}$, where $S(\Delta^{(p-1)/2})^{\perp}$ is the unit sphere of $(\Delta^{(p-1)/2})^{\perp}$. By composing, we get an equivariant map $Y_{p}\to S(\Delta^{(p-1)/2})^{\perp}\cong S^{(p-1)^{2}/1-1}$.

Proposition 4.2**.**

The pair $(d,p)$ is admissible if there is no $\mathbb{Z}/p$-equivariant map $X_{d,p}\to S(\Delta^{(p-1)/2})^{\perp}$.

Proof.

If there is no $\mathbb{Z}/p$-equivariant map $X_{d,p}\to S(\Delta^{(p-1)/2})^{\perp}$, then for the map $F:X_{d,p}=(S^{d})^{p}\setminus L\to(\mathbb{R}^{p})^{(p-1)/2}$ defined above we have $im\ F\cap\Delta^{(p-1)/2}\neq\varnothing$, that is, there exists $(x_{0},x_{1},\ldots,x_{p-1})\in X_{d,p}$ such that for every $j=1,2,\ldots,(p-1)/2$,

[TABLE]

It only remains to show that $x_{0},x_{1},\ldots,x_{p-1}$ are distinct. In fact, if $x_{r}=x_{s}$ for some $r,s$, then there are $j\in\{1,2,\ldots,(p-1)/2\}$ and $t\in\mathbb{F}_{p}$ such that $r=tj$ and $s=(t+1)j$. Then, $\rho_{j}(f(x_{tj}),f(x_{(t+1)j}))=0$ and

[TABLE]

which implies that $f(x_{0})=f(x_{1})=\cdots=f(x_{p-1})$, and since $f$ is injective, $x_{0}=x_{1}=\cdots=x_{p-1}$, which contradicts that $(x_{0},x_{1},\ldots,x_{p-1})\in X_{d,p}$. Thus, the points $x_{0},x_{1},\ldots,x_{p-1}\in S^{d}$ are distinct. ∎

We are going to consider the following problem, which is more general: to find conditions on a triplet $(d,r,p)$ such that there is no $\mathbb{Z}/p$-equivariant map $X_{d,p}\to S(\Delta^{r})^{\perp}$.

4.1. The index of the sphere $S(\Delta^{r})^{\perp}$

The unit sphere $S(\Delta^{r})^{\perp}$ is a sphere of dimension $r(p-1)-1$ with a free action of $\mathbb{Z}/p$. Then, by Proposition 2.1, $ii)$, we have

[TABLE]

We are not going to compute explicitly the index of $X_{d,p}$, but rather we are going to give conditions on $d,r,p$ such that the element $b^{r(p-1)/2}\in H^{\ast}(B\mathbb{Z}/p;\mathbb{F}_{p})$ does not belong to the index of $X_{d,p}$. We do this by using the Serre spectral sequence of the fibration $X_{d,p}\to(X_{d,p})_{\mathbb{Z}/p}\to B\mathbb{Z}/p$ to show that the element $b^{r(p-1)/2}\in H^{\ast}(B\mathbb{Z}/p;\mathbb{F}_{p})=E_{2}^{\ast,0}$ survives forever. This means that $b^{r(p-1)/2}$ is not in $Ind_{\mathbb{Z}/p,\mathbb{F}_{p}}X_{d,p}$.

The $E_{2}$-page of the Serre spectral sequence of the fibration $X_{d,p}\to(X_{d,p})_{\mathbb{Z}/p}\to B\mathbb{Z}/p$, $E_{2}^{s,t}=H^{s}(B\mathbb{Z}/p;H^{t}(X_{d,p};\mathbb{F}_{p}))$ has to be understood as the cohomology of the group $\mathbb{Z}/p$ with coefficients in the $\mathbb{Z}/p$-module $H^{\ast}(X_{d,p};\mathbb{F}_{p})$.

4.2. The cohomology $H^{\ast}(X_{d,p};\mathbb{F}_{p})$ as a $\mathbb{Z}/p$-module

We will use Lefschetz duality (see [13], theorem 70.2, page 415):

Theorem 4.3**.**

If $(X,A)$ is a compact triangulated relative homology $n$-manifold which is orientable, then there are isomorphisms

[TABLE]

for all $G$.

We apply theorem 4.3 to $((S^{d})^{p},L)$ and $G=\mathbb{Z}/p$ to have

[TABLE]

To compute the homology $H_{\ast}((S^{d})^{p},L;\mathbb{F}_{p})$ we are going to use the long exact sequence of the pair $((S^{d})^{p},L)$. Since $L\cong S^{d}$, $H_{i}(L;\mathbb{F}_{p})=\mathbb{F}_{p}$ if $i=0,d$ and $H_{i}(L;\mathbb{F}_{p})=0$ otherwise.

An induction argument using the the Künneth isomorphism yields that the only nonzero homology groups $H_{i}((S^{d})^{p};\mathbb{F}_{p})$ occurs when $i=0,d,2d,\ldots,pd$; besides, $H_{jd}((S^{d})^{p};\mathbb{F}_{p})$ is the direct sum of all terms of the form $A_{1}\otimes A_{2}\otimes\cdots\otimes A_{p}$, where $j$ of the $A_{l}$’s are equal to $H_{d}(S^{d};\mathbb{Z}/p)$ and the remaining $A_{l}$’s are equal to $H_{0}(S^{d};\mathbb{Z}/p)$. If $N:=(\mathbb{F}_{p})^{\oplus(p)}$ with the action of $\mathbb{Z}/p$ given by permuting the $p$ copies of $\mathbb{F}_{p}$ in the obvious way, then we have

[TABLE]

The long exact sequence in homology of the pair $((S^{d})^{p},L)$ gives isomorphisms

[TABLE]

for $j=2,3,\ldots,p$, and an exact sequence

[TABLE]

where $\iota_{\ast}$ is the map induced by the inclusion $L\hookrightarrow(S^{d})^{p}$. We have that $H_{i}((S^{d})^{p},L;\mathbb{F}_{p})=0$ in any other case. If $x$ is the generator of $H_{d}(L;\mathbb{F}_{p})$ and $x_{i}$ the generator of $H_{d}((S^{d})^{p};\mathbb{F}_{p})$ carried by the $i$-th copy of $S^{d}$ in $(S^{d})^{p}$, then we have that $\iota_{\ast}(x)=x_{1}+\cdots+x_{p}$. Thus, $\iota_{\ast}$ is injective, $H_{d+1}((S^{d})^{p},L;\mathbb{F}_{p})=0$ and

[TABLE]

Thus, we have that

[TABLE]

and by theorem 4.3 and using that ${p\choose i/d}={p\choose p-i/d}$):

[TABLE]

4.3. The Serre spectral sequence of the fibration $X_{d,p}\to(X_{d,p})_{\mathbb{Z}/p}\to B\mathbb{Z}/p$

The $E_{2}$-page of the Serre spectral sequence of the fibration $X_{d,p}\to(X_{d,p})_{\mathbb{Z}/p}\to B\mathbb{Z}/p$ is given by:

[TABLE]

We have that $H^{t}(X_{d,p};\mathbb{F}_{p})\neq 0$ only for $t=0,d,2d,\ldots,(p-1)d$. For $t=0$, $H^{0}(X_{d,p};\mathbb{F}_{p})=\mathbb{F}_{p}$ so that $E_{2}^{\ast,0}=H^{\ast}(\mathbb{Z}/p;\mathbb{F}_{p});\mathbb{F}_{p}[a,b]/\langle a^{2}\rangle$. To compute the cohomology groups $H^{s}(\mathbb{Z}/p;N)$ we consider the norm map $\nu:N\to N$ given by multiplication by $1+\omega+\omega^{2}+\cdots+\omega^{p-1}$ (where $\mathbb{Z}/p=\langle\omega|\omega^{p}=1\rangle$) and have that for $s\geq 2$ even $H^{s}(\mathbb{Z}/p;N)=coker\ \overline{\nu}$ and for $s$ odd $H^{s}(\mathbb{Z}/p;N)=ker\ \overline{\nu}$, where $\overline{\nu}:N_{\mathbb{Z}/p}\to N^{\mathbb{Z}/P}$ is induced by $\nu$ (see [5]).This map $\overline{\nu}$ is an isomorphism, so we get the following:

[TABLE]

For $t=d,2d,\ldots,(p-2)d$ we have

[TABLE]

We also have $E_{2}^{s,(p-1)d}=H^{s}(\mathbb{Z}/p;M)$. Otherwise, $E_{2}^{s,t}=0$. The $E_{2}$-term of the spectral sequence is shown in figure 2.

The element $b^{r(p-1)/2}$ lives in $E_{2}^{r(p-1),0}=H^{r(p-1)}(\mathbb{Z}/p;\mathbb{F}_{p})$. With our description of $E_{2}^{\ast,\ast}$ we see that there are only two differentials that might hit elements in $E_{2}^{r(p-1),0}$, that are $d_{r(p-1)}:E_{2}^{0,r(p-1)-1}\to E_{2}^{r(p-1),0}$ and $d_{(p-1)d+1}:E_{2}^{(r-d)(p-1)-1,d(p-1)}\to E_{2}^{r(p-1),0}$.

Proposition 4.4**.**

If the two differentials $d_{r(p-1)}:E_{2}^{0,r(p-1)-1}\to E_{2}^{r(p-1),0}$ and $d_{(p-1)d+1}:E_{2}^{(r-d)(p-1)-1,d(p-1)}\to E_{2}^{r(p-1),0}$ are zero, then $b^{r(p-1)/2}$ does not belong to $Ind_{\mathbb{Z}/p,\mathbb{F}_{p}}X_{d,p}$, that is, $Ind_{\mathbb{Z}/p,\mathbb{F}_{p}}S(\Delta^{r})^{\perp}\nsubseteq Ind_{\mathbb{Z}/p,\mathbb{F_{p}}}X_{d,p}.$ In particular, there is no $\mathbb{Z}/p$-equivariant map $X_{d,p}\to S(\Delta^{r})^{\perp}$.

If $r\leq d$, then $E_{2}^{(2r-d)(p-1)-1,d(p-1)}=0$. On the other hand, $E_{2}^{0,r(p-1)-1}\neq 0$ implies that $r(p-1)-1=jd$ for some $j=1,2,\ldots,p-1$. The result is that if $r\leq d$ and $r(p-1)$ is not of the form $jd+1$ for any $j=1,2,\ldots,p-1$, then the two differentials above are zero. This implies that $b^{r(p-1)/2}$ is not in the index $Ind_{\mathbb{Z}/p,\mathbb{Z}/p}X_{d,p}$. Thus we have:

Proposition 4.5**.**

If $r\leq d$ and $r(p-1)$ is not of the form $jd+1$ for any $j=1,2,\ldots,p-1$, then there is no $\mathbb{Z}/p$-equivariant map $X_{d,p}\to S(\Delta^{r})^{\perp}$.

Theorem 1.3 follows from proposition 4.2 and proposition 4.5 by taking $r=(p-1)/2$. A direct consequence of Theorem 1.3 is:

Corollary 4.6**.**

If $p\leq 2d+1$ and $d$ is even, then $(d,p)$ is admissible.

Example 4.7**.**

Corollary 4.6 implies that $(2,5)$ is an admissible pair. In particular, taking the Euclidean metric $d$ of $\mathbb{R}^{3}$ and a continuous injective map $f:S^{2}\to\mathbb{R}^{3}$, there are 5 distinct points $y_{0},y_{1},y_{2},y_{3},y_{4}$ in the image of $f$ such that

[TABLE]

and

[TABLE]

Example 4.8**.**

The triplet $(3,5)$ is admissible. In fact we have that $(p-1)^{2}/2=8$ and for $j=1,2,3,4$; $jd+1$ gives the values $4,7,10,13$.

Thus, for any continuous and injective map $f:S^{3}\to\mathbb{R}^{4}$, two metrics $\rho_{1}$ and $\rho_{2}$ that induce the usual topology of $\mathbb{R}^{4}$, there are 5 distinct points in the image of $f$, $y_{0}=f(x_{0}),y_{1}=f(x_{1}),y_{2}=f(x_{2}),y_{3}=f(x_{3}),y_{4}=f(x_{4})$, such that

[TABLE]

[TABLE]