---
abstract: 'The colored Tverberg theorem of Blagojević, Matschke, and Ziegler (Theorem \[A\]) provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices is a prime number. We extend this result to an optimal colored Tverberg theorem for multisets of colored points, which is valid for each prime power $r=p^k$, and includes Theorem \[A\] as a special case for $k=1$. One of the principal new ideas is to replace the ambient simplex $\Delta^N$, used in the original Tverberg theorem, by an “abridged simplex” of smaller dimension, and to compensate for this reduction by allowing vertices to repeatedly appear a controlled number of times in different rainbow simplices. Configuration spaces, used in the proof, are combinatorial pseudomanifolds which can be represented as multiple chessboard complexes. Our main topological tool is the Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions.'
author:
- |
    [Duško Jojić]{}\
    [Faculty of Science]{}\
    [University of Banja Luka]{}
- |
    Gaiane Panina\
    [St. Petersburg State University]{}\
    [St. Petersburg Department of]{}\
    [Steklov Mathematical Institute]{}
- |
    Rade T. Živaljević\
    [Mathematical Institute]{}\
    [SASA, Belgrade]{}\
date: 'May 25, 2020'
title: |
    Optimal colored Tverberg theorems\
    for multisets of points
---

Introduction {#sec:intro}
============

The following result is known as [*the topological Tverberg theorem*]{}, [@BSS; @oz87; @Vol96-1; @M].

\[thm:TT\] [([@BSS; @oz87])]{} Let $r = p^k$ be a prime power, $d\geq 1$, and $N = (r-1)(d+1)$. Then for every continuous map $f : \Delta^N \rightarrow \mathbb{R}^d$, defined on an $N$-dimensional simplex, there exist disjoint faces $\Delta_1,\dots, \Delta_r$ of $\Delta^{N}$ such that $$f(\Delta_1)\cap\dots\cap f(\Delta_r) \neq \emptyset \, .$$

It is known [@bfz2] that the condition on $r$ is essential, and that if $r$ is not a prime power the topological Tverberg theorem may fail in general.

The following relative of Theorem \[thm:TT\] is sometimes referred to as the [*Optimal colored Tverberg theorem*]{} [@BMZ], see also the review paper [@Z17] where it is classified as a [*type C colored Tverberg theorem*]{}.

[([@BMZ])]{}\[A\] Let $r\geq 2$ be a prime, $d\geq 1$, and $N:=(r-1)(d+1)$. Let $\Delta^N$ be an $N$-dimensional simplex with a partition (coloring) of its vertex set into $d+2$ parts, $$V  = [N+1] = C_0\uplus \dots \uplus C_d\uplus C_{d+1}\, ,$$ with $\vert C_i\vert = r-1$ for $i\leq d$ and $\vert C_{d+1}\vert =1$. Then for every continuous map $f : \Delta^N \rightarrow \mathbb{R}^d$, there are $r$ disjoint “rainbow simplices” $\Delta_1,\dots, \Delta_r$ of $\Delta^N$ satisfying $$f(\Delta_1) \cap \dots \cap f(\Delta_r) \neq\emptyset \,$$ where by definition a face $\Delta$ of $\Delta^N$ is a rainbow simplex if and only if $\vert \Delta\cap C_j\vert \leq 1$ for each $j=0,\dots, d+1$.

Note that Theorem \[A\] does not include Theorem \[thm:TT\] as a special case. Indeed, we need a stronger condition in the colored Tverberg theorem, where $r$ is a prime rather than a prime power. It remains an interesting question if this condition on $r$ can be relaxed.

Our (first) main new result (Theorem \[thm:main-p-power-mapping-red\]) is valid for each prime power $r=p^k$, and includes Theorem \[A\] as a special case for $k=1$. One of the guiding ideas in the proof is to replace the simplex $\Delta^N$ (used in both Theorems \[thm:TT\] and \[A\]) by a simplex of smaller dimension, and to compensate this by allowing its vertices to appear a controlled number of times in different faces $\Delta_i$.

\[thm:main-p-power-mapping-red\] Let $r=p^k$ be a prime power, $d\geq 1$, and $N:=k(p-1)(d+1)$. Let $\Delta^N$ be an $N$-dimensional simplex whose vertices are colored by $d+2$ colors, meaning that there is a partition $Vert(\Delta^N) = C_0\sqcup C_1\sqcup\dots\sqcup C_{d}\sqcup C_{d+1}$ into $d+2$ monochromatic subsets. We also assume that:

1.  Each of the colored sets $C_0,\dots, C_d$ has $(p-1)k$ vertices. The vertices in each $C_i$ are assigned multiplicities, as prescribed by the vector $(1,p,\dots,p^{k-1})^{\times (p-1)}\in \mathbb{N}^{k(p-1)}$.

2.  The (exceptional) color class $C_{d+1}$ contains a single vertex with multiplicity one.

(Altogether there are $N+1=k(p-1)(d+1)+1$ vertices. Counted with multiplicities, their total number is $(r-1)(d+1)+1$.)

We claim that under these conditions for any continuous map $f:\Delta^N \rightarrow  \mathbb{R}^d$ there exist $r$ (not necessarily disjoint or even different) faces $\Delta_1, \dots, \Delta_r$ of $\Delta^N$ such that:

1.  $f(\Delta_1) \cap ... \cap f(\Delta_r)\neq \emptyset$.

2.  The number of occurrences of each vertex of $\Delta^N$ in all faces $\Delta_i$, does not exceed the prescribed multiplicity of that vertex.

3.  All faces $\Delta_i$ are multicolored or [*rainbow simplices*]{}, in the sense that their vertices have different colors, $(\forall i)(\forall j) \, \vert Vert(\Delta_i)\cap C_j\vert\leq 1$.

Theorem \[thm:main-p-power-mapping-red\] has an alternative formulation where the rainbow simplices $\Delta_i$ are faces of the original simplex $\Delta^{(r-1)(d+1)}$, rather than the faces of the abridged simplex $\Delta^{k(p-1)(d+1)}$ (used in Theorem \[thm:main-p-power-mapping-red\]). While Theorem \[thm:main-p-power-mapping-red\] is more intuitive and has a clear geometrical meaning, an advantage of Theorem \[thm:main-p-power-mapping\] is that all simplices $\Delta_i$ are distinct and the result is closer in form to Theorems \[thm:TT\] and \[A\].

\[thm:main-p-power-mapping\] Assuming that $r=p^k$ is a prime power and $r' = r-1$ let $$K = K_{r',r',\dots,r',1}\cong [r']\ast [r']\ast\dots\ast [r'] \ast [1] = [r']^{\ast(d+1)}\ast [1]$$ be a $(d+1)$-dimensional simplicial complex on a vertex set $V = V_0\sqcup V_1\sqcup\dots\sqcup V_{d+1}$, divided into $(d+2)$ color classes where $\vert V_0\vert =\vert V_1\vert = \dots = \vert V_{d}\vert =r'=r-1$ and $\vert V_{d+1}\vert  =1$. Assume that $f : K\rightarrow \mathbb{R}^d$ is an $\mathbb{L}$-collapse map (Definition \[def:collapse\]), meaning that $f = \widehat{f} \circ \alpha$ for some map $\widehat{f} : K_{k(p-1),k(p-1),\dots, k(p-1),1} \longrightarrow \mathbb{R}^d$ where $$\alpha : K_{r',r',\dots,r',1} \longrightarrow K_{k(p-1),k(p-1),\dots, k(p-1),1}$$ is the simplicial map arising from a choice of a $\mathbb{L}$-collapse map $\theta : [r'] \rightarrow [k(p-1)]$. Then there exist $r$ pairwise vertex disjoint simplices ($r$ vertex-disjoint rainbow simplices) $\Delta_1,\dots, \Delta_r$ in $K$ such that $$\label{eqn:image-intersection}
          f(\Delta_1)\cap \dots \cap f(\Delta_r)\neq\emptyset \, .$$

The following extension of Theorem \[A\] is the main result of [@BMZ].

[([@BMZ])]{}\[AA\] Let $r\geq 2$ be a prime, $d\geq 1$, and $N:=(r-1)(d+1)$. Let $\Delta^N$ be an $N$-dimensional simplex with a partition of its vertex set into $m+1$ parts, $$V  = [N+1] = C_0\uplus \dots \uplus C_{m}\, ,$$ with $\vert C_i\vert \leq r-1$ for $i=0,...,m$. Then for every continuous map $f : \Delta^N \rightarrow \mathbb{R}^d$, there are $r$ disjoint rainbow simplices $\Delta_1,\dots, \Delta_r$ in $\Delta^N$ satisfying $$f(\Delta_1) \cap \dots \cap f(\Delta_r) \neq\emptyset \,$$

Theorem \[AA\] is deduced from Theorem \[A\] by a direct combinatorial argument. By a similar reduction procedure we prove our (second) main result (Theorem \[thm:grand\] in Section \[Sec:Gen\]) which extends Theorem \[thm:main-p-power-mapping-red\], and unifies all known type C (and type A) colored Tverberg theorems in a single statement.

Here is an example, illustrating the optimality of Theorem \[thm:main-p-power-mapping-red\] and illuminating its relationship with other statements of Tverberg type. Let $\mathcal{S} = \{A_i^{\alpha_i}\}_{i=0}^{d+1}$ be a [*multiset*]{} of points in $\mathbb{R}^d$ where $\{A_i\}_{i=0}^d$ are the vertices of a non-degenerate simplex and $A_{d+1}$ its barycenter. Each of the points $A_i\, (i=0,\dots, d+1)$ is assigned a different color and a multiplicity $\alpha_i \in \{1, r-1\}$, where $\alpha_i=1$ only for $i=d+1$.

If we allow a scattering of points, where each $\{A_i^{\alpha_i}\}$ is replaced by a collection $\{A_i^j\}_{j=1}^{\alpha_i}$ of (possibly distinct) points in $\mathbb{R}^d$ then:

1.  The classical (affine) Tverberg theorem guarantees the existence of a partition $\mathcal{S} = S_1\sqcup\dots \sqcup S_r$ such that ${\rm Conv}(S_1)\cap \dots \cap {\rm Conv}(S_r)\neq \emptyset$;

2.  Theorem \[A\] says that, under assumption that $r$ is a prime, such a partition exists with an additional property that each set $S_k$ has at most one point in each color;

3.  Theorem \[thm:main-p-power-mapping-red\] claims that the conclusion of Theorem \[A\] is true under assumption that $r$ is a prime power, if we allow only a partial scattering where in each color the points create clusters (partition) of the type $1^{p-1}p^{p-1}\ldots (p^{k-1})^{p-1}$. (The clusters themselves can be scattered in an arbitrary fashion.)

The organizations of the paper is following. The proof of Theorem \[thm:main-p-power-mapping-red\] (and its equivalent form, Theorem \[thm:main-p-power-mapping\]) is given in Section \[sec:proof-red\]. The role of chessboard complexes and their generalizations, as configuration spaces for theorems of Tverberg type, is briefly reviewed in \[sec:conf-spaces\]. In Section \[sec:conf-spaces\] we formulate our main topological result of Borsuk-Ulam type (Theorem \[thm:main-p-power\]), used in the proof of Theorem \[thm:main-p-power-mapping-red\], and its companion (Theorem \[thm:main-p-power-degree\]), about degrees of maps from multiple chessboard complexes. The proof of Theorems \[thm:main-p-power-degree\] and \[thm:main-p-power\] is postponed for Section \[sec:proof-degree\].

In Section \[sec:chess-multiple\] we develop the theory of multiple chessboard complexes in the generality needed for applications in the proofs of Theorems \[thm:main-p-power-mapping-red\], \[thm:main-p-power-mapping\] and \[thm:main-p-power-degree\]. The focus is on multiple chessboard complexes which turn out to be pseudomanifolds (Sections \[sec:hierarchy\] and \[sec:degree\]).

In Section \[Sec:Gen\] we formulate and prove extensions of Theorem \[thm:main-p-power-mapping-red\] (Theorems \[thm:main-p-power-generalization\] and \[thm:grand\]).

In Section \[sec:comparison\] we outline the proof of [@KB Theorem 2.1] (slightly extended to the case of pseudomanifolds), as one of the central result illustrating the Eilenberg-Krasnoselskii comparison principle for degrees of equivariant maps, in the case of non-free group actions.

Chessboard complexes and equivariant maps {#sec:conf-spaces}
=========================================

The central role of [*chessboard complexes*]{}, as proper configuration spaces for [*colored Tverberg problem and its relatives*]{}, was recognized in [@ZV92] almost thirty years ago. To the present day these complexes remain, together with their generalizations (the multiple chessboard complexes) in the focus of research in this area of geometric combinatorics.

Recall that the (standard) chessboard complex $\Delta_{p,q}$ is the complex of all non-attacking placements of rooks in a $(p\times
q)$-chessboard (a placement is non-attacking if it is not allowed to have more than one rook in the same row or in the same column). More generally, the multiple chessboard complex $\Delta_{p,q}^{{\mathbbm{A}}, {\mathbbm{B}}}$ (see Section \[sec:chess-multiple\]), where $\mathbbm{A}\in \mathbb{N}^p$ and $\mathbbm{B}\in \mathbb{N}^q$, arises if we allow more than one rook in each row (each column), where their precise number is determined by vectors $\mathbbm{A}$ and $\mathbbm{B}$.

Central results in this area are the [*Topological type A colored Tverberg theorem*]{} (Theorem 2.2 in [@BMZ], reproduced here as Theorem \[A\]) and the [*Topological type B colored Tverberg theorem*]{} [@ZV92; @VZ94]. Both of these results are obtained by applications of the [*Configuration Space/Test Map scheme*]{} involving chessboard complexes.

The associated [*test maps*]{} are respectively (\[eqn:nova\]) (in the Type A case) and (\[eqn:nasa\]) (for the Type B result), $$\label{eqn:nova}
f : (\Delta_{r,r-1})^{\ast d}\ast [r]
\stackrel{\mathbb{Z}/r}{\longrightarrow} W_r^{\oplus d}$$ $$\label{eqn:nasa}
f : (\Delta_{r,2r-1})^{\ast
(k+1)}\stackrel{\mathbb{Z}/r}{\longrightarrow} W_r^{\oplus d}$$ where $W_r\subset \mathbb{R}^r$ is the standard $(r-1)$-dimensional representation of the group $S_r$.

Both theorems are consequences of the corresponding Borsuk-Ulam-type statements claiming that in either case the $S_r$-equivariant map $f$ must have a zero, where in (\[eqn:nova\]) $r$ is a prime number, and in (\[eqn:nasa\])) $r$ is a power of a prime.

The following theorem extends (\[eqn:nova\]) and serves as a basis for a new Type A topological Tverberg theorem, which extends (in a natural way) the result of Blagojevi' c, Matschke and Ziegler to the prime power case.

\[thm:main-p-power\] Let $G = (\mathbb{Z}_p)^k $ be a $p$-toral group of order $r = p^k$. Let $\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}}$ be the multiple chessboard complex (based on a $k(p-1)\times p^k$ chessboard), where $ \mathbbm{1} = (1,\dots, 1)\in \mathbb{R}^{p^k}$ and $\mathbb{L} =  (1,p,\dots, p^{k-1})^{\times (p-1)}\in \mathbb{R}^{k(p-1)}$. Let $\partial\Delta_{[p^k]}\cong S^{p^k-2}$ be the boundary of a simplex with $p^k$ vertices. Then there does not exist a $G$-equivariant map $$f :   (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)}\ast [p^k] \longrightarrow (\partial\Delta_{[p^k]})^{\ast (d+1)}\cong (S^{p^k-2})^{\ast (d+1)} \cong S^{(p^k -1)(d+1)-1} \, .$$

Theorem \[thm:main-p-power\] is a consequence of the following theorem about degrees of equivariant maps.

\[thm:main-p-power-degree\] Let $G = (\mathbb{Z}_p)^k $ be a $p$-toral group of order $r = p^k$. Let $\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}}$ be the multiple chessboard complex (based on a $k(p-1)\times p^k$ chessboard), where $ \mathbbm{1} = (1,\dots, 1)\in \mathbb{R}^{p^k}$ and $\mathbb{L} =  (1,p,\dots, p^{k-1})^{\times (p-1)}\in \mathbb{R}^{k(p-1)}$. Let $\partial\Delta_{[p^k]}\cong S^{p^k-2}$ be the boundary of a simplex with $p^k$ vertices. Then ${\rm deg}(f) \neq 0 \, (\mbox{{\rm mod} } p)$ for any $G$-equivariant map $$f :   (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)} \longrightarrow (\partial\Delta_{[p^k]})^{\ast (d+1)}\cong (S^{p^k-2})^{\ast (d+1)} \cong S^{(p^k -1)(d+1)-1} \, .$$

Chessboard pseudomanifolds {#sec:chess-multiple}
==========================

Following [@jvz; @jvz2], a multiple chessboard complex $ \Delta_{m,n}^{\mathbb{K};\mathbb{L}}  = \Delta_{m,n}^{k_1,\dots,
k_n; l_1,\dots, l_m}$ is an abstract simplicial complex with vertices in $[m]\times [n]$, where the simplices have at most $k_i$ elements in the row $[m]\times \{i\}$ and at most $l_j$ elements in each column $\{j\}\times [n]$).

We shall be mainly interested in complexes $ \Delta_{m,n}^{\mathbbm{1};\mathbb{L}}  = \Delta_{m,n}^{1,\dots, 1; l_1,\dots, l_m}$ where at most one rook is permitted in each of the rows of the chessboard $[m]\times [n]$.

\[prop:pseudomanifold\] The multiple chessboard complex $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ is a pseudomanifold if $$\label{eqn:uslov_za_mnogostrukost}
n = l_1+l_2+\dots+ l_m +1 \, .$$ More precisely, the links of simplices of codimension $1$ and $2$ are spheres of dimensions $0$ and $1$, while in codimension $3$ may appear both $2$-spheres and $2$-dimensional tori $T^2$.

[**Proof:**]{} Let $S\in \Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ and let $s_i := \vert S\cap (\{i\}\times [n])\vert$. The link ${\rm Link}(S)$ is clearly isomorphic to the multiple chessboard complex $\Delta_{m,n}^{\mathbbm{1};\mathbb{T}}$ where $\mathbb{T} = (t_1,\dots, t_m)$ and $t_i := l_i-s_i$. (Here we allow that $t_j = 0$ for some $j\in [m]$.) The proof is completed by an explicit description of all multiple chessboard complexes that arise as links of simplices in codimension $\leq 3$.

If ${\rm codim}(S) = 1$ then there exists $j_0$ such that $l_{j_0} = s_{j_0}+1$ and $l_j = s_j$ for each $j\neq j_0$. The condition (\[eqn:uslov\_za\_mnogostrukost\]) guarantees that $t_{j_0} =2$, which together with $t_j=0$ for $j\neq j_0$ implies ${\rm Link}(S)\cong  \Delta_{1,2}\cong S^0$.

If ${\rm codim}(S) = 2$ then there are two possibilities. Either (I) there exists $j_0$ such that $l_{j_0} = s_{j_0}+2$ and $l_j = s_j$ for each $j\neq j_0$, or (II) there exists $j_0\neq j_1$ such that both $l_{j_0} = s_{j_0}+1, l_{j_1} = s_{j_1}+1$ and $l_j = s_j$ for each $j\neq j_0, j_1$. In the first case ${\rm Link}(S) \cong \partial \Delta_{[3]}\cong S^1$, while in the second ${\rm Link}(S) \cong \Delta_{2,3} \cong S^1$.

If ${\rm codim}(S) = 3$ then the number of non-zero entries in the vector $\mathbb{T} = (t_1,\dots, t_m)$ is $1, 2$ or $3$. In the first case ${\rm Link}(S) \cong \partial \Delta_{[4]} \cong S^2$. In the second case ${\rm Link}(S) \cong \Delta_{2,4}^{\mathbbm{1};\mathbb{A}}$, where $\mathbb{A} = (2,1)$, hence $\Delta_{2,4}^{\mathbbm{1};\mathbb{A}}\cong S^2$ .

Finally, in the third case ${\rm Link}(S) \cong \Delta_{3,4} \cong T^2$. $\square$

Hierarchy of pseudomanifolds $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ {#sec:hierarchy}
--------------------------------------------------------------------

From here on we tacitly assume that the chessboard complex $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ satisfies the condition $n = l_1+\ldots + l_m+1$. If $\mathbb{L} = \mathbbm{1}\in \mathbb{N}^m$ then $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}} = \Delta_{n-1,n}$ is a standard chessboard complex [@BLVZ], while in the case $m=1$ the complex $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}} \cong \partial \Delta_{[n]}$ is the boundary sphere $\partial \Delta_{[n]} \cong S^{n-2}$ of the simplex $\Delta_{[n]}:= 2^{[n]}$.

The pseudomanifolds $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ form a poset category where the complexes $\Delta_{n-1,n}$ and $\partial \Delta_{[n]}$ play the role of the initial and terminal object. The morphisms in this category are the [*$\theta$-collapse maps*]{} $\Omega_\theta$, described in the following definition.

\[def:collapse\] Assuming $m'\geq m$, choose an epimorphism $\theta :
[m']\rightarrow [m]$. Let $\widehat{\theta} : [m']\times
[n]\rightarrow [m]\times [n]$ be the associated map of chessboards where $\widehat{\theta}(i,j) = (\theta(i), j)$. We say that a sequence $\mathbb{B} = (b_1,\ldots, b_{m})$ is obtained by a $\theta$-collapse from a sequence $\mathbb{A} = (a_1,\ldots, a_{m'})$ if $b_i = \sum_{\theta(j) =i}~a_j$. Define $\Omega_\theta : \Delta_{m',n}^{\mathbbm{1};\mathbb{A}} \rightarrow
\Delta_{m,n}^{\mathbbm{1};\mathbb{B}}$ as the induced map of multiple chessboard complexes where $\Omega_\theta(S) :=
\widehat{\theta}(S)$, for each simplex $S\in  \Delta_{m,n}^{\mathbbm{1};\mathbb{A}})$.

The special cases $\Omega_\theta : \Delta_{n-1,n} \rightarrow
\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ and $\Omega_\theta : \Delta_{m,n}^{\mathbbm{1};\mathbb{L}} \rightarrow \partial\Delta_{[n]}$ are of particular importance. These maps are completely determined by the vector $\mathbb{L}$ and in this case the corresponding $\Omega_\theta$ (and other closely related maps) are referred to as $\mathbb{L}$-collapse maps.

The group $S_n$, permuting the rows of the chessboard $[m]\times [n]$, acts on the multiple chessboard complex $\Delta_{m,n}^{\mathbbm{1};\mathbbm{L}}$. The simplicial map $\Omega_\theta : \Delta_{m',n}^{\mathbbm{1};\mathbb{L}'} \rightarrow \Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$, associated to a collapse map $\theta : [m'] \rightarrow [m]$, is clearly $S_n$-equivariant.

Orientability of pseudomanifolds $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ 
-------------------------------------------------------------------------

\[prop:orientation\] The pseudomanifold $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ is always orientable. It has a fundamental class $\tau \in H_d(\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}; \mathbb{Z})\cong \mathbb{Z}$ where $d={\rm dim}(\Delta_{m,n}^{\mathbbm{1};\mathbb{L}})= m-1$. A permutation $g\in S_n$ reverses the orientation (changes the sign of $\tau$) if and only if $g$ is odd.

As a consequence the $S_n$-pseudomanifolds $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ and $\Delta_{m',n}^{\mathbbm{1};\mathbb{L}'}$ are concordant in the sense that each $g\in S_n$ either changes the orientation of both of the complexes if none of them.

[**Proof:**]{} Let $\Omega_\theta :  \Delta_{m,n}^{\mathbbm{1};\mathbb{L}} \rightarrow \partial \Delta_{[n]}$ be the collapse map associate to the constant map $\theta : [m] \rightarrow [1]$ (Definition \[def:collapse\]). In other words $\Omega_\theta$ is the map induced by the projection $[m]\times [n] \rightarrow [1]\times [n]$ of chessboards, where a simplex $S\in \Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ is mapped to a simplex $S' \in \partial \Delta_{[n]}$ if and only if $$(\forall i\in [n])  \, (\{i\}\times [m])\cap S \neq\emptyset \,  \Leftrightarrow \, i\in S')  \, .$$ Let $\widehat{S}$ be the simplex $S \in \Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ oriented by listing its vertices in the increasing order of rows. Note that if $\Omega_\theta(S) = S'\in \partial \Delta_{[n]}$ then $\Omega_\theta(\widehat{S})= \widehat{S'}$.

Choose an orientation $\mathcal{O}'$ on the sphere $\partial \Delta_{[n]}$ and use this orientation to define, via the collapse map $\Omega_\theta$, an orientation $\mathcal{O}$ on $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$. More explicitly, an ordered simplex $\widehat{S}$ is positively oriented with respect to $\mathcal{O}$ if and only if $\widehat{S'}$ is positively oriented with respect to the orientation $\mathcal{O}'$. It is not difficult to check that $\mathcal{O}$ is indeed and orientation on the pseudomanifold $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$ which has all the properties listed in Proposition \[prop:orientation\]. $\square$

Degree of the collapse map $\Omega_\theta$ {#sec:degree}
------------------------------------------

In the following proposition we calculate the degree of the map $\Omega_\theta$.

\[prop:degree\] The degree of the map $\Omega_\theta : \Delta_{m,n}^{\mathbbm{1};\mathbb{A}} \rightarrow
\Delta_{m',n}^{\mathbbm{1};\mathbb{B}}$ is, $$\label{eqn:degree-1}
\mbox{\rm deg} (\Omega_\theta) =
\genfrac(){0pt}{0}{\mathbb{B}}{\mathbb{A}} =
\frac{b_1!\, b_2!\ldots b_{m'}!}{a_1!\, a_2!\ldots a_{m}!}.$$ In the special case when $m'=1$ we obtain that the degree of the map $\Omega_\theta$ is the multinomial coefficient, $$\label{eqn:degree-2}
\mbox{\rm deg} (\Omega_\theta) = \frac{(a_1+ a_2+\ldots + a_m)!}{a_1!\,
a_2!\ldots a_m!}$$ and in the special case $a_1=a_2=\ldots=a_m=1$ [(\[eqn:degree-1\])]{} reduces to the formula, $$\label{eqn:degree-3}
\mbox{\rm deg} (\Omega_\theta) =  b_1!\, b_2!\ldots b_{m'}!.$$

[**Proof:**]{} Each simplicial map $\Omega_\theta : \Delta_{m,n}^{\mathbbm{1};\mathbb{A}} \rightarrow \Delta_{m',n}^{\mathbbm{1};\mathbb{B}}$ is non-degenerate in the sense that it maps bijectively the top dimensional simplices of $\Delta_{m,n}^{\mathbbm{1};\mathbb{A}}$ to top dimensional simplices of $\Delta_{m,n}^{\mathbbm{1};\mathbb{B}}$. Moreover, it is an orientation preserving map so in order to calculate the degree of $\Omega_\theta$ it is sufficient to calculate the cardinality of the preimage $\Omega_\theta^{-1}(c_0)$ of the barycenter $c_0$ of a chosen top dimensional simplex of $\Delta_{m,n}^{\mathbbm{1};\mathbb{B}}$.

Since the degree is multiplicative it is sufficient to establish formula (\[eqn:degree-3\]). A simple calculation shows that the cardinality of the set $\Omega_\theta^{-1}(c_0)$ is, in the case of a map $\Omega_\theta : \Delta_{n-1,n}  \rightarrow \Delta_{m',n}^{\mathbbm{1};\mathbb{B}}$, indeed given by the formula (\[eqn:degree-3\]). $\square$

Proof of Theorem \[thm:main-p-power-mapping-red\] {#sec:proof-red}
=================================================

By convention $\Delta = \Delta_C$ is a simplex spanned by a set $C$, in particular $\Delta^N \cong \Delta_C$ where $C = Vert(\Delta^N) =  C_0\sqcup C_1\sqcup\dots\sqcup C_{d}\sqcup C_{d+1}$.

Recall that a set $S\subset C$ (and the corresponding face $\Delta_S\subseteq \Delta_C$) is called a [*rainbow set*]{} (rainbow face) if $\vert S\cap C_i \vert \leq 1$ for all $ i=0,1,\dots, d+1$. It follows that the set of all rainbow simplices is a subcomplex of $\Delta_C$ which has a representation as a join of $0$-dimensional simplicial complexes: $$\label{eqn:duga-1}
       \mathfrak{R} = \mathfrak{Rainbow} := C_0\ast C_1\ast \dots \ast C_d \ast C_{d+1} \subset \Delta_C \, .$$ By assumption $\vert C_i \vert = m:= k(p-1)$ for $i=0,1,\dots, d$ and $\vert C_{d+1}\vert = 1$, or more explicitly $C_i = \{c^i_{\alpha, \beta}\} \,
  (0\leqslant \alpha \leqslant k-1;  1\leqslant \beta \leqslant  p-1)$ for all $0\leq i\leq d$, and $C_{d+1} = \{c_0\}$. Theorem \[thm:main-p-power-mapping-red\] claims that for each continuous map $f : \Delta_C \rightarrow \mathbb{R}^d$ there exist rainbow faces $\Delta_1,\dots, \Delta_r \in \mathfrak{R}$ such that:

1.  Vertex $c_0$ appears in at most one of the faces $\Delta_i$;

2.  For all $i,\alpha,\beta$ the vertex $c^i_{\alpha,\beta}$ may appear in not more than $p^\alpha$ faces $\Delta_1, \dots, \Delta_r$;

3.  $f(\Delta_1)\cap\dots\cap f(\Delta_r) \neq \emptyset$.

An $r$-tuple $(\Delta_1, \dots, \Delta_r)$ of rainbow simplices is naturally associated to the join $\Delta_1 \ast  \dots \ast  \Delta_r\in \mathfrak{R}^{\ast r}$. Our immediate objective is to identify the subcomplex $\mathfrak{R}^{\ast r}_\Lambda\subset  \mathfrak{R}^{\ast r}$ which collects all $r$-tuples $(\Delta_1, \dots, \Delta_r)$ satisfying conditions (1) and (2).

By assumption $\Delta_{i, \nu} : = \Delta_i\cap C_\nu$ is either empty or a singleton, for each rainbow simplex $\Delta_i$. A moment’s reflection reveals that the union $\cup \{\Delta_{i, \nu}\}_{i = 1}^{r}$ is a simplex in $ \Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}}$, for $0\leq\nu\leq d$ and a simplex in $[r] =  [p^k]$ if $\nu = d+1$. It immediately follows that $$\mathfrak{R}^{\ast r}_\Lambda  \cong  (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)}\ast [r] \, .$$

Let $\hat{f} : \mathfrak{R} \rightarrow \mathbb{R}^d$ be the restriction of the map $f : \Delta_C \rightarrow \mathbb{R}^d$. The corresponding map defined on the $r$-tuples of rainbow simplices, satisfying conditions (1) and (2) is the map $$\hat{F} : (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)}\ast [r] \longrightarrow (\mathbb{R}^d)^{\ast r} \, .$$ By composing with the projection $ (\mathbb{R}^d)^{\ast r} \rightarrow  (\mathbb{R}^d)^{\ast r}/D$ (where $D\cong \mathbb{R}^d$ is the diagonal) and the embedding $(\mathbb{R}^d)^{\ast r}/D \hookrightarrow  (W_r)^{\oplus (d+1)}$, where $W_r\cong \mathbb{R}^r/\mathbb{R}$ is the standard $(r-1)$-dimensional representation of $S_r$, we obtain a map $$\hat{F} : (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)}\ast [r] \longrightarrow (W_r)^{\oplus (d+1)}$$ which has a zero in a simplex $(\Delta_1, \dots, \Delta_r)$ if and only if $f(\Delta_1)\cap \dots\cap f(\Delta_r)\neq \emptyset$. Since the sphere $S((W_r)^{\oplus (d+1)})\cong (S(W_r))^{\ast (d+1)}$ is equivariantly homeomorphic to $(\partial\Delta_{[r]})^{\ast (d+1)}$ a zero exists by Theorem \[thm:main-p-power\], which concludes the proof of Theorem \[thm:main-p-power-mapping-red\]. $\square$

Proof of Theorem \[thm:main-p-power-degree\] {#sec:proof-degree}
============================================

We are supposed to show that the degree ${\rm deg}(f)$ of each $G$-equivariant map $$\label{eqn:p-power-degree}
      f :   (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)} \longrightarrow (\partial\Delta_{[p^k]})^{\ast (d+1)}\cong (S^{p^k-2})^{\ast (d+1)} \cong S^{(p^k -1)(d+1)-1}$$ where $G = (\mathbb{Z}_p)^k$ is a $p$-toral group, is non-zero modulo $p$. Following the [*Comparison principle for equivariant maps*]{} (Section \[sec:comparison\]) we should:

1.  Exhibit a particular map (\[eqn:p-power-degree\]) such that ${\rm deg}(f) \neq 0$ modulo $p$;

2.  Check if the conditions of Theorem \[thm:K-B-2.1-sing\] are satisfied.

The following proposition provides the needed example for the first part of the proof.

\[prop:map-example\] The $\theta$-collapse map $$\label{eqn:map-example}
\Omega_\theta : \Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}} \longrightarrow \partial\Delta_{[p^k]} \cong \Delta_{1,r}^{1,r'},$$ where $\theta: [k(p-1)] \rightarrow [1]$ is a constant map, has a non-zero degree modulo $p$.

[**Proof:**]{} We calculate the degree of the map (\[eqn:map-example\]) by applying the formula (\[eqn:degree-2\]). Recall that $\mathbb{L} =  (1,p,\dots, p^{k-1})^{\times (p-1)}\in \mathbb{R}^{k(p-1)}$ so in this case $$\label{eqn:degree-4}
\mbox{\rm deg} (\Omega_\theta) = \frac{(p^k-1)!}{[(p^{k-1})!\, (p^{k-2})! \ldots p!\, 1!]^{p-1}} \, .$$ The well-known formula for the highest power of $p$ dividing $m!$ is $${\rm ord}_p(m!)  = \left\lfloor \frac{m}{p}\right\rfloor + \left\lfloor \frac{m}{p^2}\right\rfloor + \dots  \, .$$ By applying this formula we obtain $${\rm ord}_p((p^k-1)!) =  {\rm ord}_p((p^k)!) - k = p^{k-1} + p^{k-2} +\dots + 1 - k$$ and by applying the same formula to the denominator of (\[eqn:degree-4\]) we obtain exactly the same quantity. $\square$

In light of Proposition \[prop:map-example\] the map $$\label{eqn:collapse-(d+1)}
       (\Omega_\theta)^{\ast (d+1)}:   (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)} \longrightarrow (\partial\Delta_{[p^k]})^{\ast (d+1)}\cong (S^{p^k-2})^{\ast (d+1)} \cong S^{(p^k -1)(d+1)-1}$$ has a non-zero degree ${\rm deg}((\Omega_\theta)^{\ast (d+1)}) = ({\rm deg}(\Omega_\theta))^{ d+1}$ modulo $p$, which completes part (A) of the proof.

For the second part we begin with the observation that $\partial\Delta_{[r]}\, (r=p^k)$ is $S_r$-equivariantly homeomorphic to the unit sphere $S(W_r)$ in the standard $S_r$-representation $W_r := \{x\in \mathbb{R}^r \mid \, x_1+\dots+ x_r = 0\}$. As a consequence, for each subgroup $H\subseteq S_r$ the corresponding fixed point set $\partial\Delta_{[r]}^H \cong S(W_r)^H = S(W_r^H)$ is also a sphere.

The action of $H$ decomposes $[r]$ into orbits $[r] = O_1\sqcup \dots \sqcup O_t$. From here easily follows a combinatorial description of the fixed point set $\partial\Delta_{[r]}^H$. A point $x\in \partial\Delta_{[r]}$, with barycentric coordinates $\{\lambda_i\}_{i=1}^r$, is fixed by $H$ if and only if the barycentric coordinates are constant in each of the orbits. Summarising, $\partial\Delta_{[r]}^H$ is precisely the boundary of the simplex with vertices $\{o_i\}_{i=1}^t$, where $o_i$ is the barycenter of the face $\Delta_{O_i}\subset \Delta_{[r]}$.

Let $\Delta_{m,r}^{\mathbbm{1};\mathbb{L}}$ be a multiple chessboard complex, where $\mathbbm{L} = (l_1,\dots, l_m)$ and $m = k(p-1)$. It is not difficult to see that the barycenter $b_{i,j}$ of (geometric realization of) the simplex $\{i\}\times O_j$ is in the fixed point set $(\Delta_{m,r}^{\mathbbm{1};\mathbb{L}})^H$ if and only if $\vert O_j\vert \leq l_i $.

More generally, a point $x$ is in $(\Delta_{m,r}^{\mathbbm{1};\mathbb{L}})^H$ if and only if it can be expressed as a convex combination $$x = \sum_{(i,j)\in S} \lambda_{i,j} b_{i,j}$$ where $S$ is a subset of $[m]\times [t]$ satisfying

1.  If $(i,j), (i',j)\in S$ then $i=i'$;

2.  $(\forall i\in [m]) \, \sum \{\vert O_j\vert   \mid \, (i,j)\in S\}\leq l_i$.

The $\theta$-collapse map $\Omega_\theta$, where $\theta: [m]\rightarrow [1]$ is the constant map, maps $(\Delta_{m,r}^{\mathbbm{1};\mathbb{L}})^H$ to $\partial\Delta_{[r]}^H$. Moreover $\Omega_\theta(b_{i,j}) = o_j$ and, in light of (1) and (2), the simplex with vertices $\{b_{i,j}\}_{(i,j)\in S}$ is mapped bijectively to a face of $\partial\Delta_{[r]}^H$. The following inequality is an immediate consequence, $${\rm dim}((\Delta_{m,r}^{\mathbbm{1};\mathbb{L}})^H) \leq {\rm dim}(\partial\Delta_{[r]}^H) \, .$$ From here and (\[eqn:collapse-(d+1)\]) we obtain the inequality $${\rm dim}  [(\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)}]^H \leq {\rm dim}[(\partial\Delta_{[p^k]})^{\ast (d+1)}]^H$$ which finishes the proof of part (B) and concludes the proof of the theorem. $\square$

It follows from the part (B) of the proof of Theorem \[thm:main-p-power-degree\] that $\Delta = (\Delta_{m,r}^{\mathbbm{1};\mathbb{L}})^H$ is also a “chessboard complex”. Indeed, $S\subseteq [m]\times [t]$ is a simplex in $\Delta$ if and only if $S$ has at most one rook in each row $[m]\times\{j\}$ and the total weight of the set $(\{i\}\times  [t])\cap S$ is at most $l_i$, where the weight of each element $(i,j)$ is $\vert O_j\vert$.

It follows that $\Delta$ can be classified as a complex of the type $\Delta_{m,t}^{\mathbbm{1}, \mathcal{L}}$ (cf. [@jvz Definition 2.3])), where $\mathcal{L}$ is family of threshold (simplicial) complexes.

Proof of Theorem \[thm:main-p-power\]
-------------------------------------

If such a map $f$ exists, then there is a commutative diagram $$\label{CD:3-to-2}
\begin{CD}
 (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)}\ast [p^k] @>{f}>> (\partial\Delta_{[p^k]})^{\ast (d+1)}\\
@A\pi AA @A\cong AA\\
 (\Delta_{k(p-1),p^k}^{\mathbbm{1};\mathbb{L}})^{\ast (d+1)} @>\widehat{f}>> (\partial\Delta_{[p^k]})^{\ast (d+1)}
\end{CD}$$ where $\pi$ is the inclusion map and $\widehat{f} = f\circ\pi$. The map $\pi$ is homotopic to a constant map (since ${\rm Image}(\pi)$ is contained in ${\rm Star}(v)$ for any vertex $v\in [p^k]$). It follows that ${\rm deg}(f) = 0$ which contradicts Theorem \[thm:main-p-power-degree\].

Colored Tverberg theorems for multisets of points {#Sec:Gen}
=================================================

In this section we prove (as our most general results) Theorems \[thm:main-p-power-generalization\] and \[thm:grand\], which unify the optimal colored Tverberg theorem (Theorem \[AA\]), and the (primary) colored Tverberg theorem for multisets of points (Theorem \[thm:main-p-power-mapping-red\]).

These results are deduced from Theorem \[thm:main-p-power-mapping-red\] by a combinatorial reduction procedure, and they extend Theorem \[thm:main-p-power-mapping-red\] in a similar vein Theorem \[AA\] generalizes Theorem \[A\]. Following [@Z17] they can be also classified as Type C colored Tverberg theorems (characterized by the condition that the number of colors is at least $d+2$, where $d$ is the dimension of the ambient euclidean space).

One of the central assumptions in Theorem \[AA\] is the requirement that $\vert  C_i \vert \leq r-1$ for each color class $C_i$. Here we need a similar condition expressed in the language of multisets (Definition \[def:subordinate\]).

The collection of multiplicities $\mathbbm{L} = (1, p, \dots, p^{k-1})^{\times (p-1)}$, which arises in Theorem \[thm:main-p-power-mapping-red\], is called the *standard multiplicity passport*. More generally, the multiplicity passport of an arbitrary multiset $A = \{a_1^{\alpha_1}, a_2^{\alpha_2},\dots,  a_s^{\alpha_s}\}$ is the vector $pass(A) = (\alpha_1, \alpha_2,\dots, \alpha_s)\in \mathbb{N}^s $.

Strictly speaking the order of entries in the vector $pass(A)$ is of no importance and the entries equal to zero can be ignored. This means that the passport $pass(A)$ can be often identified as a partition $\lambda = 1^{m_1}2^{m_2}\ldots q^{m_q}\ldots$, where the exponents indicate the number of times each multiplicity occurs in the multiset $A$. From this point of view $pass(\mathbbm{L}) = 1^{p-1}p^{p-1}\ldots (p^{k-1})^{p-1}$.

\[def:subordinate\] We say that a multiset $A = \{a_1^{\alpha_1}, a_2^{\alpha_2},\dots,  a_s^{\alpha_s}\}$ is $\mathbbm{L}$-small if its multiplicity passport is [*subordinated*]{} to $\mathbbm{L}$ in the sense that the vector $(\alpha_1, \dots, \alpha_s)$ (with some reordering) is obtained from the standard multiplicity passport $\mathbbm{L}$ by removing some entries.

Equivalently, $A$ is $\mathbbm{L}$-small if its passport is a “divisor” of  $\mathbbm{L}$ in the sense that $pass(A) = 1^{m_0}p^{m_1}\ldots (p^{k-1})^{m_{k-1}}$ for some integers $0\leq m_i \leq p-1$.

The following result is both a consequence and a natural extension of Theorem \[thm:main-p-power-mapping-red\]. The number of colors and the total number of vertices (counted with multiplicities) is the same as in Theorem \[thm:main-p-power-mapping-red\]. The main difference is that the color classes are treated equally (there are no “exceptional” colors).

\[thm:main-p-power-generalization\] Let $r=p^k$ be a prime power, $d\geq 1$, and $N = k(p-1)(d+1)$. Let $\Delta = \Delta^N$ be a simplex whose vertices are colored by $d+2$ colors, meaning that there is a partition $Vert(\Delta) = C_0\sqcup C_1\sqcup\dots\sqcup C_{d}\sqcup C_{d+1}$ into $d+2$ monochromatic subsets. We also assume that:

1.  In each of the sets $C_i$ the vertices are assigned multiplicities in such a way that the multiplicity passport of $C_i$ is $\mathbbm{L}$-small in the sense of Definition \[def:subordinate\].

2.  The total sum of all multiplicities over all the vertices is $(r-1)(d+1)+1$.

Under these conditions for any continuous map $f:\Delta \rightarrow  \mathbb{R}^d$ there exist $r$ faces $\Delta_1, \dots, \Delta_r$ of $\Delta$ such that:

1.  $f(\Delta_1) \cap ... \cap f(\Delta_r)\neq \emptyset$.

2.  The number of occurrences of each vertex of $\Delta^N$ in all faces $\Delta_i$, does not exceed the prescribed multiplicity of that vertex.

3.  All faces $\Delta_i$ are rainbow simplices.

[**Proof:**]{} The proof mimics the construction used in [@BMZ] for reducing Theorem \[AA\] to Theorem \[A\] , see also [@sar-2] and [@L] for earlier appearance of similar ideas.

Enlarge the simplex $\Delta$ to a bigger simplex $\overline{\Delta} = \Delta \ast \Delta'$, where $\Delta'$ is the face spanned by added (new) vertices. Initially we add a vertex $v_0$ (of multiplicity one) and place it in a new color class $C_{d+2} = \{v_0\}$. We continue by adding (where necessary) new vertices to the old color classes (with proper multiplicities) in such a way that the passports of all old color classes become standard. (Here we use the assumption that the multiplicity passport of each $C_i$ is $\mathbbm{L}$-small.) Note that the total multiplicity of all added vertices is $r-1$.

Let $\bar{f }:\overline{\Delta} \rightarrow  \mathbb{R}^d\times \mathbb{R}$ be an extension of the map $f:\Delta \rightarrow  \mathbb{R}^d$ defined by $$\bar{f}((1-t)x + ty)  = (1-t)f(x) + te_{d+1}   \quad \mbox{for each $x\in \Delta$ and $y\in \Delta'$} \, .$$ By Theorem \[thm:main-p-power-mapping-red\], applied to the map $\bar{f}$, there exist $r$ rainbow simplices $F_i\subset \overline{\Delta}$ such that $\bigcap_{i=1}^r \bar{f}(F_i)$ is non-empty, where the total number of occurrences of each vertex of $\overline{\Delta}$ in rainbow simplices $F_i$ does not exceed the multiplicity of that vertex.

Since the total multiplicity of all new vertices (= vertices of $\Delta'$) is $r-1$, there must exist i such that $F_i\cap \Delta' = \emptyset$. It follows that $$\bar{f}(F_1) \cap \dots \cap \bar{f}(F_r)= {f}(F_1\cap \Delta) \cap \dots \cap f(F_r\cap \Delta)\subset \mathbb{R}^d$$ which means that $\Delta_i := F_i\cap \Delta \, (i=1,\dots, r)$ is a collection of rainbow simplices satisfying the conclusion of the theorem.

In the following theorem we further relax, in comparison to Theorem \[thm:main-p-power-generalization\], the condition on the number of colors used for coloring the vertices of simplices. Recall that in the primary colored Tverberg theorem for multisets of points (Theorem \[thm:main-p-power-mapping-red\]) there were altogether $d+2$ color classes, but one of them was exceptional (one-element) color class. In Theorem \[thm:main-p-power-generalization\] the number of colors is again $d+2$, but there are no exceptional color classes. In the following theorem the number of color classes is $\geq d+2$ and (as in Theorem \[thm:main-p-power-generalization\]) all colors are treated equally.

\[thm:grand\] Suppose that $r=p^k$ is a prime power and $m\geq d\geq 1$, where $m+2$ is the number of colors and $d$ is the dimension of an euclidean space. Set $N=k(p-1)(d+1)$ and $M = k(p-1)(m+1)$.

Let $\Delta \cong \Delta^N \subseteq \Delta^M$ be a simplex with a coloring of vertices $$Vert(\Delta) = {C}_0\sqcup \dots \sqcup  {C}_m \sqcup {C}_{m+1}\, .$$ Assume that this coloring can be extended to a “primary coloring” of $\Delta^M$ $$Vert(\Delta^M) = {V}_0\sqcup \dots \sqcup  {V}_m \sqcup {V}_{m+1}\quad ({C}_j := Vert(\Delta) \cap V_j)$$ in the sense that:

1.  Each of the colored sets $V_0,\dots, V_m$ has $k(p-1)$ vertices. The vertices in each $V_i$ are assigned multiplicities, as prescribed by the vector $(1,p,\dots, p^{k-1})^{\times (p-1)}\in \mathbb{N}^{k(p-1)}$.

2.  The (exceptional) color class $V_{m+1}$ contains a single vertex with multiplicity one.

Moreover we assume that the set $W:= Vert(\Delta^M)\setminus Vert(\Delta)$ admits a partition $W = \uplus_{j=1}^{m-d} E_j$ such that each $E_j$, with multiplicities inherited from $Vert(\Delta^M)$ is $\mathbbm{L}$-small in the sense of Definition \[def:subordinate\].

We claim that under these conditions for any continuous map $f:\Delta \rightarrow  \mathbb{R}^d$ there exist $r$ (not necessarily disjoint or even different) faces $\Delta_1, \dots, \Delta_r$ of $\Delta$ such that:

1.  $f(\Delta_1) \cap ... \cap f(\Delta_r)\neq \emptyset$.

2.  The number of occurrences of each vertex of $\Delta$ in all faces $\Delta_i$, does not exceed the prescribed multiplicity of that vertex.

3.  All faces $\Delta_i$ are rainbow simplices, in the sense that their vertices have different colors, $(\forall i)(\forall j) \, \vert Vert(\Delta_i)\cap {C}_j\vert\leq 1$.

[**Proof:**]{} The proof is a straightforward generalization of the proof of Theorem \[thm:main-p-power-generalization\] so we omit the details.

Comparison principle for equivariant maps  {#sec:comparison}
==========================================

The following theorem is proved in [@KB] (Theorem 2.1 in Section 2). Note that the condition that the $H_i$-fixed point sets $S^{H_i}$ are [*locally $k$-connected*]{} for $k\leq {\rm dim}(M^{H_i})-1$ is automatically satisfied if $S$ is a representation sphere. So in this case it is sufficient to show that the sphere $S^{H_i}$ is (globally) $({\rm dim}(M^{H_i})-1)$-connected which is equivalent to the condition $${\rm dim}(M^{H_i}) \leq {\rm dim}(S^{H_i})\, (i=1,\dots, m) \, .$$

\[thm:K-B-2.1\] Let $G$ be a finite group acting on a compact topological manifold $M = M^n$ and on a sphere $S \cong S^n$ of the same dimension. Let $N\subset M$ be a closed invariant subset and let $(H_1), (H_2), \dots, (H_k)$ be the orbit types in $M\setminus N$. Assume that the set $S^{H_i}$ is both globally and locally $k$-connected for all $k=0, 1,\dots, {\rm dim}(M^{H_i})-1$, where $i = 1,\dots, k$. Then for every pair of $G$-equivariant maps $\Phi, \Psi : M\longrightarrow S$, which are equivariantly homotopic on $N$, there is the following relation $$\label{eqn:fundam-congruence}
 {\rm deg}(\Psi) \equiv {\rm deg}(\Phi)  \quad ({\rm mod}\, GCD\{\vert G/H_1\vert, \dots, \vert G/H_k\vert\}) \, .$$

The proof of the following extension of Theorem \[thm:K-B-2.1\] to manifolds with singularities doesn’t require new ideas. By a [*singular topological manifold*]{} we mean a topological manifold with a codimension $2$ singular set. In particular Theorem \[thm:K-B-2.1-sing\] applies to pseudomanifolds $\Delta_{m,n}^{\mathbbm{1};\mathbb{L}}$, introduced in Section \[sec:chess-multiple\].

\[thm:K-B-2.1-sing\] Let $G$ be a finite group acting on a compact “singular topological manifold” $M = M^n$ and on a sphere $S \cong S^n$ of the same dimension. Let $N\subset M$ be a closed invariant subset and let $(H_1), (H_2), \dots, (H_k)$ be the orbit types in $M\setminus N$. Assume that the set $S^{H_i}$ is both globally and locally $k$-connected for all $k=0, 1,\dots, {\rm dim}(M^{H_i})-1$, where $i = 1,\dots, k$. Then for every pair of $G$-equivariant maps $\Phi, \Psi : M\longrightarrow S$, which are equivariantly homotopic on $N$, there is the following relation $$\label{eqn:fundam-congruence-sing}
 {\rm deg}(\Psi) \equiv {\rm deg}(\Phi)  \quad ({\rm mod}\, GCD\{\vert G/H_1\vert, \dots, \vert G/H_k\vert\}) \, .$$

[**Proof:**]{} Following into footsteps of the proof of Theorem \[thm:K-B-2.1\] (see [@KB Theorem 2.1]) we define a $G$-equivariant map $$f_0 : (M\times \{0,1\}) \cup (N\times [0,1]) \longrightarrow B\setminus \{O\}$$ where $B = {\rm Cone}(S)$ is a cone over the sphere $S$ (with the apex $O$), $\Psi$ and $\Phi$ are restrictions of $f_0$ on $M\times \{0\}$ (respectively $M\times \{1\}$) and the restriction of $f_0$ on $N\times [0,1]$ is a homotopy between $\Psi \vert_N$ and $\Phi\vert_N$.

If $f : M\times [0,1] \rightarrow B$ is a $G$-equivariant extension of $f_0$ then ([@KB Lemma 2.1]) ${\rm deg}(f) = \pm ({\rm deg}(\Psi) - {\rm deg}(\Psi))$ and the relation (\[eqn:fundam-congruence-sing\])will follow if $$\label{eqn:a-congruence}
  {\rm deg}(f) = \sum_{i=1}^{m}  a_i \cdot\vert G/H_i\vert$$ for some integers $a_i\in \mathbb{Z}$.

The proof of the following lemma ([@KB Lemma 2.2]) is quite general, in particular it holds for “singular topological manifolds”.

\[lema:4-lema\] There exists a $G$-equivariant extension $f : M\times [0,1] \rightarrow B$ of the map $f_0$ satisfying the following conditions:

1.  $K = f^{-1}(O) = \bigcup_{j=1}^m T_j$ where $T_u\cap T_v = \emptyset$ for $u\neq v$;

2.  $T_j = G(K_j)$ for a compact set $K_j$;

3.  $K_j = H_j(K_j)$ is $H_j$-invariant;

4.  $g(K_j)\cap h(K_j)=\emptyset$ if $gh^{-1}\notin H_j \, (j=1,\dots, m) $.

The proof of Theorem \[thm:K-B-2.1-sing\] is completed as in [@KB Section 2.1.3] by observing that “singular topological manifolds” also have absolute and relative fundamental classes.

More explicitly, if $F_j$ is the restriction $f$ to a sufficiently small neighborhood of $K_j$ then $${\rm deg}(f)  = \sum_{j=1}^{m} {\rm deg}(F_j) \, .$$ By the same argument as in [@KB] we deduce from Lemma \[lema:4-lema\] that ${\rm deg}(F_j) = a_j\cdot \vert  G/H_j \vert$ for some $a_j\in \mathbb{Z}$, and the relation (\[eqn:fundam-congruence-sing\]) is an immediate consequence. $\square$

[000000]{}

S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, Eliminating higher-multiplicity intersections. III. Codimension 2, 2017 (v1 – 2015), arXiv: 1511.03501.

I. Bárány, P.V.M. Blagojevi' c and G.M. Ziegler. Tverberg’s Theorem at 50: Extensions and Counterexamples, *Notices Amer. Math. Soc.*, vol. 63, no. 7, 2016.

I. B' ar' any and D.G. Larman. A colored version of Tverberg’s theorem. [*J. London Math. Soc.*]{}, 45:314–320, 1992.

I. B' ar' any, S.B. Shlosman, and A. Szűcs. On a topological generalization of a theorem of Tverberg. *J. London Math. Soc.*, 23:158–164, 1981.

I. Bárány, P. Soberon. Tverberg’s theorem is 50 years old: a survey, *Bull. Amer. Math. Soc.* 55 no. 4, pp. 459–492, 2018.

A. Bj" orner, L. Lov' asz, S.T. Vre' cica, and R.T. Živaljevi' c. Chessboard complexes and matching complexes. *J. London Math. Soc. (2)* 49 (1994), 25–39.

P.V.M. Blagojevi[' c]{}, F. Frick, G.M. Ziegler. Tverberg plus constraints, [*Bull. Lond. Math. Soc.*]{}, 46:953–967, 2014.

P.V.M. Blagojevi[' c]{}, F. Frick, and G. M. Ziegler, Barycenters of polytope skeleta and counterexamples to the topological Tverberg conjecture, via constraints, *J. Eur. Math. Soc. (JEMS)*, 21 (7), 2107–2116 (2019).

P.V.M. Blagojevi' c, B. Matschke, G. M. Ziegler, Optimal bounds for the colored Tverberg problem, *J. Eur. Math. Soc. (JEMS)*, 2015, 17:4, 739–754, arXiv: 0910.4987.

P.V.M. Blagojević, G. M. Ziegler, Beyond the Borsuk–Ulam theorem: the topological Tverberg story, in [*“A journey through discrete mathematics”*]{}, M. Loebl, J. Nešetřil, R. Thomas (Eds.), Springer 2017, 273–341.

F. Frick, Counterexamples to the topological Tverberg conjecture, 2015, 3 pp., arXiv: 1502.00947.

S. Hell. Tverberg’s theorem with constraints. *J. Combinatorial Theory*, Ser. A 115:1402–1406, 2008.

D. Joji' c, I. Nekrasov, G. Panina, R. Živaljevi' c. [Alexander r-tuples and Bier complexes]{}, *Publ. Inst. Math. (Beograd) (N.S.)* 104(118) (2018), 1–22.

D. Joji[ć]{}, W. Marzantowicz, S.T. Vre[ć]{}ica, R.T. [Ž]{}ivaljevi[ć]{}. Topology of unavoidable complexes, arXiv:1603.08472 \[math.AT\].

D. Joji' c, G. Panina, R. Živaljevi' c, *A Tverberg type theorem for collectively unavoidable complexes*, to appear in [*Israel J. Math.*]{} arXiv:1812.00366 \[math.CO\].

D. Joji' c, G. Panina, R. Živaljevi' c, *Splitting necklaces, with constraints*, arXiv:1907.09740 \[math.CO\].

D. Joji' c, S.T. Vre[ć]{}ica, G. Panina, R. Živaljevi' c, *Generalized chessboard complexes and discrete Morse theory*, submitted.

D. Joji[ć]{}, S.T. Vre[ć]{}ica, R.T. [Ž]{}ivaljevi[ć]{}. Multiple chessboard complexes and the colored Tverberg problem. *J. Combin. Theory Ser. A*, 145:400–425, 2017.

D. Joji[ć]{}, S.T. Vre[ć]{}ica, R.T. [Ž]{}ivaljevi[ć]{}. Symmetric multiple chessboard complexes and a new theorem of Tverberg type. *J. Algebraic Combin.*, 46:15–31, 2017.

D. Joji[ć]{}, S.T. Vre[ć]{}ica, R.T. [Ž]{}ivaljevi[ć]{}. Topology and combinatorics of unavoidable complexes, arXiv:1603.08472v1 \[math.AT\], (unpublished preprint).

A.M. Kushkuley, Z.I. Balanov. *Geometric Methods in Degree Theory for Equivariant Maps*, Lecture Notes in Math. 1632, Springer 1996.

M. de Longueville, Erratum to: “Notes on the topological Tverberg theorem”. *Discrete Math.* 247, 271–297 (2002).

I. Mabillard, U. Wagner. Eliminating Tverberg points, I. An analogue of the Whitney trick. In [*Proc. 30th Ann. Symp. on Computational Geometry*]{} (SoCG), Kyoto 2014 (ACM, 2014), pp. 171–180.

I. Mabillard, U. Wagner. Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems, arXiv:1508.02349, August 2015.

I. Mabillard, U. Wagner. Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the $r$-Metastable Range, [*32nd International symposium on computational geometry*]{} (SoCG’16), LIPIcs. Leibniz Int. Proc. Inform., 2016. arXiv:1601.00876 \[math.GT\], January 2016.

J. Matoušek. *Using the Borsuk-Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry*. Universitext, Springer-Verlag, Heidelberg, 2003 (Corrected 2nd printing 2008).

M. Ozaydin. Equivariant maps for the symmetric group. Unpublished manuscript, available online at <http://minds.wisconsin.edu/handle/1793/63829>, 1987.

K.S. Sarkaria, Tverberg partitions and Borsuk–Ulam theorems. *Pacific J. Math.* 196, 231–241 (2000).

A. B. Skopenkov. A user’s guide to the topological Tverberg conjecture, *Russ. Math. Surv.*, 2018, 73 (2), 323–353.

A.Yu. Volovikov, On a topological generalizationof the Tverberg theorem, *Math. Notes* 59 (3) (1996) 324–326.

A.Y. Volovikov. *On the van Kampen-Flores theorem*, [Math. Notes]{}, 59:477–481, 1996.

S. Vre' cica and R. Živaljevi' c. New cases of the colored Tverberg theorem. In H. Barcelo and G. Kalai, editors, *Jerusalem Combinatorics ’93*, Contemp. Math. Vol. 178, pp. 325–334, A.M.S. 1994.

S. Vre[ć]{}ica, R. [Ž]{}ivaljevi[ć]{}. Chessboard complexes indomitable. , 118(7):2157–2166, 2011.

R.T. Živaljevi' c and S.T. Vre' cica. The colored Tverberg’s problem and complexes of injective functions. *J. Combin. Theory Ser. A* 61 (1992), 309–318.

R.T. Živaljević. Topological methods in discrete geometry. Chapter 21 in *Handbook of Discrete and Computational Geometry*, third ed., J.E. Goodman, J. O’Rourke, and C.D. Tóth, CRC Press LLC, Boca Raton, FL, 2017.

R. Živaljevi' c. User’s guide to equivariant methods in combinatorics, I and II. , (I) 59(73):114–130, 1996 and (II) 64(78):107–132, 1998.
