On covering dimension and sections of vector bundles

TL;DR

This paper uses elementary topology and cohomology to unify and extend classical theorems in topology and combinatorics, including Lebesgue, Knaster-Kuratowski-Mazurkiewicz, and Tverberg theorems.

Contribution

It provides a new elementary approach to classical results in topology and combinatorics using mod 2 cohomology of real projective spaces.

Findings

Unified proofs of classical theorems in topology and combinatorics

Extension of the topological central point theorem

New applications to Helly-Lovász, Bárány, and Tverberg results

Abstract

An elementary result in point-set topology is used, with knowledge of the mod $2$ cohomology of real projective spaces, to establish classical results of Lebesgue and Knaster-Kuratowski-Mazurkiewicz, as well as the topological central point theorem of Karasev, which is applied to deduce results of Helly-Lov\'asz, B\'ar\'any and Tverberg

Equations

$$J(x)=\{j\in I\mid {\phi }_j(x)=\max \{{\phi }_i(x)\mid i\in I\}\}.$$

$$U_J=\{x\in X\mid \text{ for all j∈J, φj(x)>0 and φi(x)<φj(x) for all i∈I−J}\}.$$

$$s_k(x)=\sum _{\#J=k}{\psi }_J(x)s_J(x).$$

$${\pi }_V:P(\mathbb{R}[V])\to \Delta (V)$$

$${\sigma }_V:\Delta (V)\hookrightarrow S(\mathbb{R}[V]),$$

$$\Delta (V)\stackrel{{\sigma }_V}{\to }S(\mathbb{R}[V])\to P(\mathbb{R}[V])\stackrel{{\pi }_V}{\to }\Delta (V)$$

$$Y=\Delta (V_1)\times \cdots \times \Delta (V_m)$$

$${\pi }_{V_1}\times \dots \times {\pi }_{V_m}:X=P(\mathbb{R}[V_1])\times \cdots \times P(\mathbb{R}[V_m])\to Y=\Delta (V_1)\times \cdots \times \Delta (V_m)$$

$$g:Y=\Delta (V_1)\times \cdots \times \Delta (V_m)\to Z$$

$$\bigcap _{T\in {\text{SS}}_{d_l}(V_l)}g(\prod _{j<l}\Delta (V_j)\times {\Delta }_T\times \prod _{j>l}\Delta (V_j))\ne \varnothing .$$

$${\pi }_{V_1}\times \cdots \times {\pi }_{V_m}:X=P(\mathbb{R}[V_1])\times \cdots \times P(\mathbb{R}[V_m])\to Y=\Delta (V_1)\times \cdots \times \Delta (V_m)$$

$$(\sum _{v_1\in V_1}{t}_1(v_1),\dots ,\sum _{v_m\in V_m}{t}_m(v_m))\mapsto \sum _{(v_1,\dots ,v_m)\in V_1\times \cdots \times V_m}{t}_1(v_1)\cdots t_m(v_m)z_{v_1,\dots ,v_m}\in Z.$$

$$z=\sum _{(v_1,\dots ,v_m)\in V_1\times \cdots \times V_m:v_l=v}{t}_1(v_1)\cdots t_m(v_m)z_{v_1,\dots ,v_m},$$

$$(\sum _{v_1\in V_1}{t}_1(v_1),\dots ,\sum _{v_m\in V_m}{t}_m(v_m))\mapsto \sum _{(v_1,\dots ,v_m)\in V_1\times \cdots \times V_m}{t}_1(v_1)\cdots t_m(v_m)z_{v_1,\dots ,v_m}\in Z.$$

$$z=\sum _{(v_1,\dots ,v_m)\in V_1\times \cdots \times V_m:v_l=v}{t}_1(v_1)\cdots t_m(v_m)z_{v_1,\dots ,v_m},$$

$$\sum _{i\in I_s}{\lambda }_i{\phi }_{i,s}(v_i)=c\text{for s=0,…,r,}$$

Full text

1. Introduction

Throughout this note, $X$ will be a compact Hausdorff topological space and $\xi_{1}$, $\ldots$, $\xi_{n}$ ($n\geqslant 1$) will be $n$ finite-dimensional real vector bundles over $X$. We write $\xi=\xi_{1}\oplus\cdots\oplus\xi_{n}$.

Suppose that $A_{1},\ldots,A_{n}$ are closed subspaces covering $X$ and that, for each $k\in\{1,\ldots,n\}$, the restriction of $\xi_{k}$ to $A_{k}$ admits a nowhere zero section, $s_{k}^{\prime}$ say. By Tietze’s theorem, $s_{k}^{\prime}$ extends to a section $s_{k}$ of $\xi_{k}$ on $X$. Then $s=(s_{1},\ldots,s_{n})$ is a nowhere zero section of $\xi$. (And conversely, if $\xi$ admits a nowhere zero section $s=(s_{k})$, we may construct such closed subspaces $A_{k}$ by choosing an inner product on each $\xi_{k}$ and setting $A_{k}=\{x\in X\mid\|s_{k}(x)\|=\max\{\|s_{j}(x)\|\mid 1\leqslant j\leqslant n\}\}$.) The main result of this paper, modelled on a cohomological lemma [7, Lemma 3.2] of Karasev and stated as Theorem 2.1, is in the same vein as this classical observation. The methods are from elementary point-set topology. As applications we derive in Sections 3 and 4, using ideas introduced by Karasev in [6, 7], a classical result of Lebesgue and Knaster-Kuratowski-Mazurkiewisz and the more recent topological central point theorem of Karasev with, as corollaries, results of Helly-Lovász and Bárány [2].

When we discuss Euler classes, we shall use representable cohomology as, for example, in [4, Section 8].

From the foregoing summary it should be clear that most of the ideas presented in this note derive from the paper [7] of Karasev. It is hoped, nevertheless, that the elementary approach taken here may have some conceptual advantages.

2. The principal result

Theorem 2.1**.**

Let $\xi_{1}$, $\ldots$, $\xi_{n}$ be $n$ finite-dimensional real vector bundles over a compact Hausdorff topological space $X$. Suppose that $(U_{i})_{i\in I}$ is a finite open cover of $X$ such that each point of $X$ lies in $U_{i}$ for at most $n$ indices $i\in I$.

Suppose that for each $i\in I$ and $k\in\{1,\ldots,n\}$ there exists a section of $\xi_{k}$ with no zeros in $U_{i}$. Then $\xi=\xi_{1}\oplus\cdots\oplus\xi_{n}$ admits a global nowhere zero section.

Proof.

We begin with an argument from [9, Lemma 2.4]. Choose a partition of unity $(\varphi_{i})_{i\in I}$ subordinate to the cover. For $x\in X$, define

[TABLE]

By assumption $\#J(x)\leqslant n$ (and $J(x)$ is non-empty). For a non-empty subset $J\subseteq I$, we now set

[TABLE]

It is clearly an open subset of $\bigcap_{j\in J}U_{j}$. Moreover, $x\in U_{J(x)}$, so that the sets $U_{J}$ cover $X$. If $J$ and $J^{\prime}$ are distinct subsets of $I$ with $\#J=\#J^{\prime}$, then $U_{J}\cap U_{J^{\prime}}=\emptyset$. (For there exist elements $j\in J$, with $j\notin J^{\prime}$, and $j^{\prime}\in J^{\prime}$, with $j^{\prime}\notin J$.)

Choose a partition of unity $(\psi_{J})$ subordinate to the open cover $(U_{J})_{J\subseteq I\,:\,1\leqslant\#J\leqslant n}$ of $X$ and, for each $J$ with $\#J=k\leqslant n$, a section $s_{J}$ of $\xi_{k}$ which is nowhere zero on $U_{J}$ (possible because $U_{J}\subseteq U_{i}$ if $i\in J$). We can then define a section $s_{k}$ of $\xi_{k}$ by

[TABLE]

Notice that, if $\psi_{J}(x)\not=0$ for some $J$ with $\#J=k$, then $s_{k}(x)\not=0$. For, if $J^{\prime}$ is a different subset with $\#J^{\prime}=k$, $\psi_{J^{\prime}}(x)=0$.

The section $s=(s_{1},\ldots,s_{n})$ of $\xi$ is nowhere zero. ∎

Corollary 2.2**.**

Suppose that $(A_{i})_{i\in I}$ is a finite closed cover of $X$ such that each point of $X$ lies in $A_{i}$ for at most $n$ indices $i\in I$.

Suppose that for each $i\in I$ and $k\in\{1,\ldots,n\}$ the restriction of $\xi_{k}$ to $A_{i}$ admits a nowhere zero section. Then $\xi$ admits a global nowhere zero section.

Proof.

It is an elementary exercise to show that there is an open cover $(U_{i})_{i\in I}$ such that (i) $A_{i}\subseteq U_{i}$, (ii) for each $(i,k)$ there exits a global section of $\xi_{k}$ with no zeros in $U_{i}$, and (iii) each point of $X$ lies in at most $n$ of the sets $U_{i}$. Then we can apply Theorem 2.1.

(Here are the details when $\#I>n$. Consider a subset $J\subseteq I$ with $\#J>n$. The open sets $X-A_{j}$, $j\in J$, cover $X$. Choose a partition of unity $(\chi_{j})_{j\in J}$ subordinate to this cover. Then we can define $U_{j}^{J}=\{x\in X\mid\#J\cdot\chi_{j}(x)<1\}$. By construction $\bigcap_{j\in J}U_{j}^{J}=\emptyset$ (because $\sum_{j\in J}\chi_{j}(x)=1$ for any $x$) and $A_{j}\subseteq U_{j}^{J}$ for $j\in J$.

Every nowhere zero section of $\xi_{k}$ over the closed set $A_{i}$ extends to a global section of $\xi_{k}$ and such a section will be nowhere zero on an open neighbourhood of $A_{i}$. So it is easy to choose $U_{i}$ in the intersection of the sets $U_{i}^{J}$ with $\#J>n$ and $i\in J$ to satisfy (i) and (ii). And then, for any set $J$ with $\#J>n$, we have $\bigcap_{i\in J}U_{i}=\emptyset$.) ∎

3. A theorem of Lebesgue

If the mod $2$ cohomology Euler class $e(\xi)$ of $\xi$ is non-zero, the zero-set ${\rm Zero}(s)=\{x\in X\mid s(x)=0\}$ of any section $s$ of $\xi$ is non-empty.

Proposition 3.1**.**

Let $p:X\to Y$ be a continuous map from $X$ to a compact Hausdorff space $Y$. Suppose that $(B_{i})_{i\in I}$ is a finite cover of $Y$ by closed sets such that any point of $Y$ lies in at most $n$ of the sets $B_{i}$.

If the mod $2$ cohomology Euler class $e(\xi)$ of $\xi$ is non-zero, then there exist $i\in I$ and $k\in\{1,\ldots,n\}$ such that $p({\rm Zero}(s))\cap B_{i}$ is non-empty for each section $s$ of $\xi_{k}$.

Proof.

We take $A_{i}=p^{-1}(B_{i})$ in Corollary 2.2. Since $e(\xi)\not=0$, $\xi$ does not admit a nowhere zero section. Hence there is a pair $(i,k)$ such that ${\rm Zero}(s)\cap B_{i}\not=\emptyset$ for every section $s$ of $\xi_{k}$. ∎

For a finite set $V$ of cardinality $n+1$, we write ${\mathbb{R}}[V]$ for the $(n+1)$-dimensional real vector space of maps $t:V\to{\mathbb{R}}$, $\Delta(V)$ for the $n$-simplex of maps $t$ such that $t(v)\geqslant 0$ for all $v\in V$ and $\sum_{v}t(v)=1$, $S({\mathbb{R}}[V])$ for the unit $n$-sphere in ${\mathbb{R}}[V]$ of vectors $t$ with $\sum_{v}t(v)^{2}=1$ and $P({\mathbb{R}}[V])$ for the $n$-dimensional real projective space of lines $[t]$ in ${\mathbb{R}}[V]$ (where $t\in{\mathbb{R}}[V]$ is non-zero). There is a surjective map

[TABLE]

defined by $(\pi_{V}[t])(v)=t(v)^{2}$ for $t\in S({\mathbb{R}}[V])$. There is also an embedding

[TABLE]

defined by $(\sigma_{V}(t)(v)=\sqrt{t(v)}$, such that the composition

[TABLE]

is the identity. The Hopf line bundle over $P({\mathbb{R}}[V])$, with fibre ${\mathbb{R}}t\subseteq{\mathbb{R}}[V]$ at $[t]$, is denoted by $H$.

The support, ${\rm supp}(t)$, of $t\in{\mathbb{R}}[V]$ is the set of points $v\in V$ such that $t(v)\not=0$. For an integer $d$, $0\leqslant d\leqslant\#V$, we write $\SS_{d}(V)$ for the set of finite subsets $T$ of $\Delta(V)$ such that any two elements of $T$ have disjoint supports and $\#V-\#T=d$. For $T\in\SS_{d}(V)$, we write $\Delta_{T}$ for the convex hull of $T$; it is a simplex of codimension $d$ in $\Delta(V)$.

Lemma 3.2**.**

For $T\in\SS_{d}(V)$, the simplex $\Delta_{T}$ can be expressed as $\pi_{V}({\rm Zero}(s))$ for some section $s$ of the vector bundle $dH={\mathbb{R}}^{d}\otimes H$.

Proof.

Let $E_{T}$ be the codimension $d$ vector subspace of ${\mathbb{R}}[V]$ spanned by $\sigma_{V}(T)$. The section $s_{T}$ of ${\rm Hom}(H,{\mathbb{R}}[V]/E_{T})$ over $P({\mathbb{R}}[V])$ given by the projection $s_{T}([t])=\{[t]\hookrightarrow{\mathbb{R}}[V]\to{\mathbb{R}}[V]/E_{T}\}$, for $t\in S({\mathbb{R}}[V])$, has the property that $\pi_{V}({\rm Zero}(s_{T}))=\Delta_{T}$. Choose some isomorphism ${\mathbb{R}}[V]/E_{T}\cong{\mathbb{R}}^{d}$ to get the required section $s$. ∎

Theorem 3.3**.**

Let $(V_{l})_{l=1}^{m}$ be a family of $m$ finite sets with $\#V_{l}=d_{l}n_{l}+1$, where $d_{l}\geqslant 1$ and $n_{l}\geqslant 1$, $l=1,\ldots,m$, are positive integers. Write $n=n_{1}+\ldots+n_{m}$. Suppose that $(B_{i})_{i\in I}$ is a finite closed cover of

[TABLE]

such that any point of $Y$ lies in at most $n$ of the sets $B_{i}$.

Then for some $i\in I$ and $l\in\{1,\ldots,m\}$ the projection of $B_{i}$ to the $l$th factor $\Delta(V_{l})$ meets each of the codimension $d_{l}$ simplices $\Delta_{T}$ for $T\in\SS_{d_{l}}(V_{l})$.

The Lebesgue theorem [7, Theorem 4.1] is the special case $n_{l}=1$, $d_{l}=1$, for all $l$; the case $m=1$, $d_{1}=1$ is a result of Knaster, Kuratowski, Mazurkiewicz [7, Remark 2.2].

We follow the method of Karasev [7, Theorems 2.1 and 4.1].

Proof.

This is an immediate consequence of Proposition 3.1. Take $p$ to be the product

[TABLE]

and $\xi_{k}$ to be the multiple $d_{l}H_{l}$ of the Hopf bundle $H_{l}$ over $P({\mathbb{R}}[V_{l}])$ if $n_{1}+\ldots+n_{l-1}<k\leqslant n_{1}+\ldots+n_{l}$. The Euler class $e(\xi)=e(H_{1})^{d_{1}n_{1}}\ldots e(H_{m})^{d_{m}n_{m}}$ is non-zero. ∎

The next lemma gives a way of checking that $e(\xi)$ in the application of Proposition 3.1 is non-zero.

Lemma 3.4**.**

Suppose that $X$ is a closed subspace of a compact Hausdorff space $\hat{X}$ and that $\xi$ is the restriction of a vector bundle $\hat{\xi}$ over $\hat{X}$. Let $\eta$ be a real vector bundle over $\hat{X}$ with the property that the restriction of $\eta$ to each connected component of the complement $\hat{X}-X$ admits a nowhere zero section.

If the mod $2$ cohomology Euler class $e(\hat{\xi}\oplus\eta)$ is non-zero, then $e(\xi)$ is non-zero.

Proof.

Fix a Euclidean metric on $\eta$. For each component $C$ of $\hat{X}-X$ choose a continuous section $s_{C}$ of the sphere bundle $S(\eta\,|\,C)$, and choose a continuous function $\rho:\hat{X}\to[0,1]$ such that $\rho^{-1}(0)=X$. Then one can define a continuous section $s$ of $\eta$ with zero-set ${\rm Zero}(s)=X$ by $s(x)=\rho(x)s_{C}(x)$ if $x\in C$, $s(x)=0$ if $x\in X$.

Since $e(\hat{\xi}\oplus\eta)=e(\hat{\xi})\cdot e(\eta)$ is non-zero, the restriction $e(\xi)$ of $e(\hat{\xi})$ to ${\rm Zero}(s)=X$ is non-zero. See, for example, [4, Proposition 2.7]. ∎

This allows us to deduce Karasev’s strengthened KKM theorem [7, Theorem 2.1]).

Proposition 3.5**.**

Let $(B_{i})_{i\in I}$ be closed subsets of a simplex $\Delta(V)$ with vertex set $V$ of cardinality $dn+r+1$, where $d\geqslant 1$ and $r\geqslant 0$ are integers, such that any point of $Y=\bigcup_{i\in I}B_{i}$ lies in at most $n$ of the sets $B_{i}$. Then, either for some $i\in I$ the subset $B_{i}$ meets each codimension $d$ simplex $\Delta_{T}$ for $T\in\SS_{d}(V)$, or some connected component of $\Delta(V)-Y$ intersects every $dn$-dimensional simplex $\Delta_{T}$ for $T\in\SS_{r}(V)$.

Proof.

Considering $\pi_{V}:\hat{X}=P({\mathbb{R}}[V])\to\hat{Y}=\Delta(V)$, take $X\subseteq\hat{X}$ to be $\pi_{V}^{-1}(Y)$ and $p:X\to Y$ to be the restriction of $\pi_{V}$. Let each $\xi_{k}$ be the restriction of $dH$, so that $\xi$ is the restriction of $\hat{\xi}=dnH$ to $X$. Take $\eta=rH$, so that $e(\hat{\xi}\oplus\eta)=e(H)^{dn+r}$ is non-zero.

If, for each component of $\Delta(V)-Y$, written as the image $\pi_{V}(C)$ of a component $C$ of $\hat{X}-X$, there is some simplex $\Delta_{T}$, where $T\in\SS_{r}(V)$, such that $\pi_{V}(C)\cap\Delta_{T}=\emptyset$, then Lemma 3.2 provides a nowhere zero section of $rH=\eta$ over $C$. By Lemma 3.4, $e(\xi)$ is then non-zero, and we can apply Proposition 3.1 to deduce the existence of an $i\in I$ such that $B_{i}$ meets each codimension $d$ simplex. ∎

4. Karasev’s topological central point theorem

An early result of the following type appears in [11, Lemma 3.1].

Proposition 4.1**.**

Let $f:X\to Z$ be a continuous map from $X$ to a compact Hausdorff space $Z$ with covering dimension less than $n$. Suppose that the mod $2$ cohomology class $e(\xi)$ is non-zero.

Then there exists a point $z\in Z$ and $k\in\{1,\ldots,n\}$ such that $z\in f({\rm Zero}(s))$ for each section $s$ of $\xi_{k}$.

Proof.

Suppose that for each point $z\in Z$ there exist sections $s_{k}^{z}$ of $\xi_{k}$, $1\leqslant k\leqslant n$, such that $z\notin f({\rm Zero}(s_{k}^{z}))$ for each $k$. Then the open sets $(Z-f({\rm Zero}(s_{1}^{z}))\cap\cdots\cap(Z-f({\rm Zero}(s_{n}^{z}))$, $z\in Z$, cover $X$. Since $Z$ is compact with covering dimension $<n$, this open cover may be refined by a finite open cover $(W_{i})_{i\in I}$ such that each point of $Z$ lies in at most $n$ of the sets $W_{i}$.

Set $U_{i}=f^{-1}(W_{i})$. Then we may apply Theorem 2.1 to the open cover $(U_{i})_{i\in I}$ of $X$ to conclude that there exist $i$ and $k$ such that for every section $s$ of $\xi_{k}$ the zero-set ${\rm Zero}(s)$ meets $U_{i}$. So $W_{i}\not\subseteq Z-f({\rm Zero}(s))$ for every section $s$ of $\xi_{k}$. But $W_{i}\subseteq Z-f({\rm Zero}(s_{k}^{z}))$ for some $z\in Z$. This contradiction completes the proof. ∎

Theorem 4.2**.**

Let $(V_{l})_{l=1}^{m}$ be a family of $m$ finite sets with $\#V_{l}=d_{l}n_{l}+1$, where $d_{l},\,n_{l}\geqslant 1$, $l=1,\ldots,m$ are positive integers. Write $n=n_{1}+\ldots+n_{m}$. Suppose that

[TABLE]

is a continuous map to a compact Hausdorff space $Z$ with covering dimension less than $n$.

Then for some $l\in\{1,\ldots,m\}$

[TABLE]

Karasev’s topological central point theorem, as in [6, Theorem 1.1] and [7, Theorem 5.1], is the special case $m=1$.

Proof.

We apply Proposition 4.1 with $X$ and $\xi_{k}$ as in Theorem 3.3 and with $f$ equal to the composition of

[TABLE]

with $g:Y\to Z$. We recall that $\xi_{k}=d_{l}H_{l}$ if $n_{1}+\ldots+n_{l-1}<k\leqslant n_{1}+\ldots+n_{l}$, so that the Euler class $e(\xi)=e(H_{1})^{d_{1}n_{1}}\cdots e(H_{m})^{d_{m}n_{m}}$ is non-zero. ∎

As an application we prove a result of Helly-Lovász [2, Theorem 3.1].

Corollary 4.3**.**

Suppose that $C_{l,v}$, $l=1,\ldots,m$, $v\in V_{l}$, $\#V_{l}=d_{l}+1$, are convex subsets of a real vector space $E$ with the property that the intersection $C_{1,v_{1}}\cap\cdots\cap C_{m,v_{m}}$ is non-empty for each $(v_{1},\ldots,v_{m})\in V_{1}\times\cdots\times V_{m}$.

If $\dim E<m$, then, for some $l$, the intersection $\bigcap_{v\in V_{l}}C_{l,v}$ is non-empty.

Proof.

We apply Theorem 4.2 with $n=m$, $n_{l}=1$, $Z=E$. For each $(v_{1},\ldots,v_{m})\in V_{1}\times\cdots\times V_{m}$ choose $z_{v_{1},\ldots,v_{m}}\in C_{1,v_{1}}\cap\cdots C_{m,v_{m}}\subseteq Z$. Take $g$ to be the piecewise linear map

[TABLE]

We conclude from Theorem 4.2, noting that a codimension $d_{l}$ simplex in $\Delta(V_{l})$ is a point, that there is some $l$ and $z\in Z$ such that, for each $v\in V_{l}$ the vector $z$ can be written as

[TABLE]

where $\sum_{v_{j}\in V_{j}}t_{j}(v_{j})=1$ for each $j$ and so $\sum_{(v_{1},\ldots,v_{m})}t_{1}(v_{1})\cdots t_{m}(v_{m})=1$. Since each $z_{v_{1},\ldots,v_{m}}$ with $v_{l}=v$ lies in the convex set $C_{l,v}$, we see that $z\in C_{l,v}$, as required. ∎

Bárány’s dual result [2, Theorem 2.1] (as formulated in [5, Theorem 3.1] and [8, Theorem 3]) can be obtained in a similar fashion.

Corollary 4.4**.**

Let $K\subseteq E$ be a non-empty compact convex subspace of a finite-dimensional real vector space $E$. Suppose that $V_{1},\ldots,V_{m}$ are finite sets with $\#V_{l}=d_{l}+1$, $l=1,\ldots,m$, and that $\varphi_{l}:V_{l}\to E$ are maps with the property that for each $(v_{1},\ldots,v_{m})\in V_{1}\times\cdots\times V_{m}$ the convex hull of $\{\varphi_{1}(v_{1}),\ldots,\varphi_{m}(v_{m})\}$ in $E$ is disjoint from $K$.

If $\dim E<m$, then, for some $l\in\{1,\ldots,m\}$, the convex hull of $\varphi_{l}(V_{l})$ in $E$ is disjoint from $K$.

Proof.

Choose a basepoint $*\in K$. We again apply Theorem 4.2 with $n=m$, $n_{l}=1$, and take $Z$ to be the affine space of affine linear maps $z:E\to{\mathbb{R}}$ such that $z(*)=-1$. Notice that the covering dimension of $Z$ is equal to $\dim E$.

For each $(v_{1},\ldots,v_{m})\in V_{1}\times\cdots\times V_{m}$ choose an affine linear map $z_{v_{1},\ldots,v_{m}}\in Z$ taking strictly positive values on $\{\varphi_{1}(v_{1}),\ldots,\varphi_{m}(v_{m})\}$ and strictly negative values on $K$. As in the proof of Helly’s theorem, take $g$ to be the piecewise linear map

[TABLE]

Theorem 4.2 provides some $l$ and $z\in Z$ such that, for each $v\in V_{l}$ the affine linear map $z$ can be written as

[TABLE]

where $\sum_{(v_{1},\ldots,v_{m})}t_{1}(v_{1})\cdots t_{m}(v_{m})=1$, so that $z$ takes a strictly positive value at each $\varphi_{l}(v)\in\varphi_{l}(V_{l})$ and strictly negative values on $K$. ∎

As observed by Sarkaria [10] (and expounded in [3]), Tverberg’s theorem is an easy consequence of Corollary 4.4. The following generalization, discussed in [3, Theorem 3.8] and due to Arocha, Bárány, Bracho, Fabila and Montejano [1], can be viewed as a coincidence theorem.

Corollary 4.5**.**

Let $r\geqslant 0$ and $m\geqslant 1$ be integers. For $l=1,\ldots,m$, $s=0,\ldots,r$, let $V_{l,s}$ be non-empty finite sets and $\varphi_{l,s}:V_{l,s}\to F$ be maps to a finite-dimensional real vector space $F$ satisfying the two conditions:

(i)* for each $l\in\{1,\ldots,m\}$, there is a non-zero vector in $F$ that can be expressed, for each $s=0,\ldots,r$, as a linear combination with non-negative coefficients of the elements of $\varphi_{l,s}(V_{l,s})$;*

(ii)* for each $s$ and $v_{l}\in V_{l,s}$, $l=1,\ldots,m$, the convex hull of $\{\varphi_{1,s}(v_{1}),\ldots,\varphi_{m,s}(v_{m})\}$ is disjoint from $\{0\}$.*

Then, if $r\cdot\dim F<m$, there is a partition $\{1,\ldots,m\}=\bigsqcup_{s=0}^{r}I_{s}$ into $r+1$ non-empty subsets $I_{s}$ and a non-zero vector $c\in F$ such that

[TABLE]

for some $\lambda_{i}\geqslant 0$ and $v_{i}\in V_{i,s}$ for $i\in I_{s}$.

If, further, there is some affine hyperplane $H$ in $F$ that contains all the subsets $\varphi_{l,s}(V_{l,s})$ but does not contain [math], then $c$ may be chosen in the hyperplane $H$ and then $\sum_{i\in I_{s}}\lambda_{i}=1$ for each $s$.

Proof.

Let $L_{r}$ be the quotient of ${\mathbb{R}}^{r+1}=\bigoplus_{s=0}^{r}{\mathbb{R}}e_{s}$ by the subspace generated by $e_{0}+\ldots+e_{r}$ and write $[e_{s}]$ for the coset of $e_{s}$.

We take $V_{l}=\bigsqcup_{s=0}^{r}V_{l,s}$, $E=L_{r}\otimes F$, $\varphi_{l}:V_{l}\to E$ defined by $\varphi_{l}(v)=[e_{s}]\otimes\varphi_{l,s}(v)$ for $v\in V_{l,s}$, and apply Corollary 4.4 with $K=\{0\}$. By assumption, the convex hull of each $\varphi_{l}(V_{l})$ in $E$ contains [math]. (Notice that, if $a_{1},\ldots,a_{r}\in F$, then $\sum_{s}[e_{s}]\otimes a_{s}=0$ if and only if $a_{1}=a_{2}=\cdots=a_{r}$.)

So there exist $v_{i}\in V_{i}$, $i=1,\ldots,m$, $\lambda_{i}\geqslant 0$, and a non-zero $c\in F$ such that $\sum_{i\,:\,v_{i}\in V_{i,s}}\lambda_{i}\varphi_{i,s}(v_{i})=c$. Take $I_{s}=\{i\mid v_{i}\in V_{i,s}\}$.

If there is a linear form $\alpha:F\to{\mathbb{R}}$ taking the value $1$ on all $\varphi_{l,s}(V_{l,s})$, we can scale to arrange that $\alpha(c)=1$, and then $\sum_{i\in I_{s}}\lambda_{i}=1$. ∎

The original Tverberg theorem is the case in which $V_{l,s}=\{*\}$ is a single point for all $l,\,s$ and $\varphi_{l,s}(*)$ is independent of $s$.

Appendix A Cohomology

It is a classical result that, if $(A_{k})_{k=1}^{n}$ is a closed cover of a compact Hausdorff space $X$ and, for each $k$, $e_{k}$ is a mod $2$ cohomology class of $X$ that restricts to zero on $A_{k}$, then the product $e_{1}\cdots e_{n}$ is zero. Here is the corresponding version of Corollary 2.2, which was used by Karasev in the form [7, Lemma 3.2].

Theorem A.1**.**

Let $X$ be a compact Hausdorff space and let $e_{1},\ldots,e_{n}$ by classes in the mod $2$ cohomology of $X$.

Suppose that $(A_{i})_{i\in I}$ is a finite closed cover of $X$ such that each point of $X$ lies in $A_{i}$ for at most $n$ indices $i\in I$ and that for each $i\in I$ and $k\in\{1,\ldots,n\}$ the restriction of $e_{k}$ to $A_{i}$ is zero.

Then $e_{1}\cdots e_{n}=0$.

Proof.

By the argument used in the proof of Corollary 2.2 one can manufacture an open cover $(U_{i})$ such that each cohomology class is represented by a map which is null (not just null-homotopic) on $U_{i}$. The construction in the proof of Theorem 2.1 gives an open cover $(U_{J})$ indexed by the non-empty subsets $J$ of $I$ with $\#J\leqslant n$. The map representing $e_{k}$ is null on the disjoint union of the sets $U_{J}$ with $\#J=k$. Since these $n$ open sets cover $X$, the product $e_{1}\cdots e_{n}$ is represented by the null map, and so the cohomology class $e_{1}\cdots e_{n}$ is zero. ∎