\begin{document}

  \title{van Kampen-Flores theorem for cell complexes}

  \baselineskip.525cm

  \parskip .05in
  \parindent .0pt

  \begin{abstract}
    The van Kampen-Flores theorem states that the $n$-skeleton of a $(2n+2)$-simplex does not embed into $\mathbb{R}^{2n}$. We give two proofs for its generalization to a continuous map from a skeleton of a certain regular CW complex into a Euclidean space. We will also generalize Frick and Harrison's result on the chirality of embeddings of the $n$-skeleton of a $(2n+2)$-simplex into $\mathbb{R}^{2n+1}$.
  \end{abstract}

  \baselineskip.525cm

  \section{Introduction}

  Recall that the van Kampen-Flores theorem \cite{F, K} states that the $n$-skeleton of a $(2n+2)$-simplex does not embed into $\mathbb{R}^{2n}$. Later, Volovikov generalized this theorem as follows: for any continuous map $f\colon\Delta^{rn+2r-2}_n\to\mathbb{R}^d$, there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $\Delta^{rn+2r-2}_n$ such that $f(\sigma_1)\cap\cdots\cap f(\sigma_r)\ne\emptyset$ whenever $(r-1)d\le rn$ and $r$ is a prime power \cite{V} (see also \cite{BZ, S}), where $X_n$ denotes the $n$-skeleton of a CW complex $X$.

  As in \cite{BFZ, BZ}, the generalized van Kampen-Flores theorem can be deduced from the topological Tverberg theorem by the so-called constraint method. Recently, the topological Tverberg theorem was generalized in \cite{HKTT} to a continuous map from a certain CW complex into a Euclidean space. We recall the definition of such a CW complex. To this end, we set terminology. Let $X$ be a regular CW complex. A \emph{face} of $X$ will mean its closed cell. For faces $\sigma_1,\ldots,\sigma_r$ of $X$, let $X(\sigma_1,\ldots,\sigma_r)$ denote the subspace of $X$ consisting of faces $\tau$ such that $\sigma_i\cap\tau=\emptyset$ for all $i$. Clearly, $X(\sigma_1,\ldots,\sigma_r)$ is a subcomplex of $X$. We say that $X$ is \emph{$n$-acyclic over $A$} if $\widetilde{H}_*(X;A)=0$ for $*\le n$, where $A$ is an abelian group and $(-1)$-acyclic over $A$ means non-empty.

  \begin{definition}[see \cite{HKTT}]
    Let $X$ be a regular CW complex. We say that $X$ is \emph{$k$-complementary $n$-acyclic over $A$} if $X(\sigma_1,\ldots,\sigma_i)$ is non-empty and $(n-\dim\sigma_1-\cdots-\dim\sigma_i)$-acyclic over $A$ for any pairwise disjoint faces $\sigma_1,\ldots,\sigma_i$ of $X$ with $\dim\sigma_1+\cdots+\dim\sigma_i\le n+1$ where $i=0,1,\ldots,k$.
  \end{definition}

  We give examples of $k$-complementary $n$-acyclic complexes.

  \begin{example}
    \label{simplex}
    An $n$-simplex is an $(r-1)$-complementary $(n-r)$-acyclic complex over any abelian group.
  \end{example}

  \begin{example}
    \label{simplicial sphere}
    Every simplicial $n$-sphere is 1-complementary $(n-1)$-acyclic over any abelian group. See \cite[Corollary 2.3]{HKTT}. An example of a non-polyhedral sphere is also available in \cite{HKTT}. More generally, every triangulation of a mod $p$ homology $n$-sphere is 1-complementary $(n-1)$-acyclic over $\mathbb{Z}/p$.
  \end{example}

  \begin{example}
    \label{saturated simplicial sphere}
    We say that a regular CW complex $X$ is {\it $r$-saturated} if $(X-X(\sigma_1))\cap\cdots\cap(X-X(\sigma_k))$ is contractible or empty for any faces $\sigma_1,\ldots,\sigma_k$ of $X$ with $k\le r$. In \cite[Corollary 2.5]{HKTT}, it is proved that every $r$-saturated simplicial $d$-sphere is $r$-complementary $(d-r)$-acyclic over any abelian group by using the Alexander duality. Since the mod $p$ Alexander duality holds for a triangulation of a mod $p$ homology sphere, we can see that every $r$-saturated triangulation of a mod $p$ homology $d$-sphere is $r$-complementary $(d-r)$-acyclic over $\mathbb{Z}/p$.
  \end{example}

  As mentioned above, the topological Tverberg theorem is generalized in \cite{HKTT} to a continuous map from a regular CW complex which is $(r-1)$-complementary $((r-1)d-1)$-acyclic over $\mathbb{Z}$ into $\mathbb{R}^d$, where $r$ is a prime power. In this paper, we will prove the following generalization of the van Kampen-Flores theorem.

  \begin{theorem}
    \label{main}
    Let $X$ be a regular CW complex which is $(r-1)$-complementary $(r(n+1)-2)$-acyclic over $\mathbb{Z}/p$, where $r=p^k$ for a prime number $p$. If $(r-1)d\le rn$, then for any continuous map $f\colon X_n\to\mathbb{R}^d$, there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $X_n$ such that
    \[
      f(\sigma_1)\cap\cdots\cap f(\sigma_r)\ne\emptyset.
    \]
  \end{theorem}

  By Example \ref{simplex}, $\Delta^{rn+2r-2}$ is $(r-1)$-complementary $(r(n+1)-2)$-acyclic over any abelian group. So Theorem \ref{main} recovers Volovikov's generalized van Kampen-Flores theorem stated above. By Example \ref{simplicial sphere}, Theorem \ref{main} specializes to the following corollaries which are of particular interest.

  \begin{corollary}
    The $n$-skeleton of a $(r-1)$-saturated triangulation of a mod $p$ homology $(r(n+2)-3)$-sphere does not embed into $\mathbb{R}^d$ whenever $r=p^k$ and $(r-1)d\le rn$.
  \end{corollary}

  \begin{corollary}
    The $n$-skeleton of a triangulation of a mod $2$ homology $(2n+1)$-sphere does not embed into $\mathbb{R}^{2n}$.
  \end{corollary}

  We will give two proofs for Theorem \ref{main}. One proof applies the constraint method to a generalized topological Tverberg theorem in \cite{HKTT} (Section \ref{Constraint method}). The other proof is topological and includes results on a group action on the nerve of a covering (Proposition \ref{G-map nerve}) and a skeleton of a discretized configuration (Lemma \ref{Conf weight}), which are of independent interest. Moreover, the topological proof applies to generalize the result on chirality recently obtained by Frick and Harrison \cite{FH}. An embedding $f\colon X\to\mathbb{R}^{n+1}$ is called \emph{chiral} if $f$ is not isotopic to $\alpha\circ f$ for each orientation reversing homeomorphism $\alpha\colon\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$. Here, an isotopy means a homotopy $h\colon X\times[0,1]\to Y$ such that $h(-,t)\colon X\to Y$ is an embedding for each $t\in[0,1]$. Since every orientation reversing homeomorphism $\alpha\colon\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ is isotopic to a reflection, we only need to consider a specific reflection in the definition of chirality. There are other definitions of chirality, and for the backgroud of this subject, we refer to Flapan's book \cite{Flapan} and Section 2.1 of \cite{FH}. In \cite{FH}, the following theorem is proved.

  \begin{theorem}
    \label{Frick-Harrison}
    Each embedding $\Delta^{2n+2}_n\to\mathbb{R}^{2n+1}$ is chiral.
  \end{theorem}

  Note that if all embeddings of $X$ into $\mathbb{R}^{n+1}$ are chiral, then there is no embedding of $X$ into $\mathbb{R}^n$. Then Theorem \ref{Frick-Harrison} can be thought of as a generalization of the van Kampen-Flores theorem. As mentioned above, we can apply the results in the topological proof for Theorem \ref{main} to generalize Theorem \ref{Frick-Harrison} as follows.

  \begin{theorem}
    \label{main 2}
    Let $X$ be a regular CW complex which is 1-complementary $2n$-acyclic over $\mathbb{Z}/2$. Then each embedding $X_n\to\mathbb{R}^{2n+1}$ is chiral.
  \end{theorem}

  By Example \ref{simplicial sphere}, we get:

  \begin{corollary}
    Every embedding of the $n$-skeleton of a triangulation of a mod $2$ homology $(2n+1)$-sphere into $\mathbb{R}^{2n+1}$ is chiral.
  \end{corollary}

  
\section{Group action on a homotopy colimit}\label{Group action on a homotopy colimit}

  This section considers a group action on a homotopy colimit induced from a group action on an index category. We will consider a homotopy colimit of a functor over a poset for simplicity, but results are easily generalized to a functor over a small category.

  We begin with a group action on a simplicial complex. For a simplicial complex $K$, let $P(K)$ denote the face poset of $K$. Then the barycentric subdivision of $K$ is $\Delta(P(K))$. We record the following basic property, which we will freely use.

  \begin{lemma}
    \label{action order complex}
    For a simplicial $G$-complex $K$, $\Delta(P(K))$ is a simplicial $G$-complex such that $\Delta(P(K))^G$ is its subcomplex.
  \end{lemma}

  \begin{proof}
    It is obvious that $\Delta(P(K))$ is a simplicial $G$-complex. If a simplex $\{x_0<\cdots<x_n\}\in\Delta(P(K))$ is fixed by $G$, then each of its face $\{x_{i_0}<\cdots<x_{i_k}\}$ is fixed by $G$ too. Thus $\Delta(P(K))^G$ is a subcomplex of $\Delta(P(K))$, completing the proof.
  \end{proof}

  Hereafter, let $P$ denote a poset. We understand $P$ as a category by regarding $x\ge y$ as a unique morphism $x\to y$. Let $F\colon P\to\mathbf{Top}$ be a functor. Recall that the homotopy colimit of $F$, denoted by $\operatorname{hocolim} F$, is defined as the coequalizer of two maps
  \[
    \alpha,\beta\colon\coprod_{x\ge y\in P}F(x)\times\Delta(P_{\le y})\to\coprod_{x\in P}F(x)\times\Delta(P_{\le x})
  \]
  such that $\alpha$ restricts to $F(x\ge y)\times 1\colon F(x)\times\Delta(P_{\le y})\to F(y)\times\Delta(P_{\le y})$ and $\beta$ restricts to the inclusion $F(x)\times\Delta(P_{\le y})\to F(x)\times\Delta(P_{\le x})$, where $P_{\le x}=\{y\in P\mid y\le x\}$.

  We recall from \cite{JS} a condition for a $G$-action on $P$ inducing a $G$-action on $\operatorname{hocolim} F$.

  \begin{definition}
    We call a functor $F\colon P\to\mathbf{Top}$ a \emph{$G$-functor} if it is equipped with a natural transformation $\eta_g\colon F\circ g\to F$ for each $g\in G$ such that $\eta_{gh} = \eta_g \circ \eta_h$ for all $g,h\in G$.

  \end{definition}

  It is immediate from the definition that if $F\colon P\to\mathbf{Top}$ is a $G$-functor, then there is a natural $G$-action on $\operatorname{colim} F$. Now we show:

  \begin{proposition}
    \label{action hocolim}
    If $F\colon P\to\mathbf{Top}$ is a $G$-functor, then there is a natural $G$-action on $\operatorname{hocolim} F$ such that the projections
    \[
      \operatorname{hocolim} F\to\Delta(P)\quad\text{and}\quad\operatorname{hocolim} F\to\operatorname{colim} F
    \]
    are $G$-maps.
  \end{proposition}

  \begin{proof}
    Except for the statement about the projection $\operatorname{hocolim} F\to\operatorname{colim} F$, the proposition is proved in\cite[Proposition 2.4]{JS}, and it is obvious that the projection $\operatorname{hocolim} F\to\operatorname{colim} F$ is a $G$-map.
  \end{proof}

  We consider the fixed point set of $\operatorname{hocolim} F$ for a $G$-functor $F$.

  \begin{proposition}
    \label{subdivision}
    Let $K$ be a simplicial $G$-complex, and let $F\colon P(K)\to\mathbf{Top}$ be a $G$-functor. Then there is a $G$-functor $\widehat{F}\colon P(\Delta(P(K)))\to\mathbf{Top}$ satisfying :
    \begin{enumerate}
      \item $\operatorname{colim} F\cong\operatorname{colim}\widehat{F}$;

      \item $(\operatorname{colim}\widehat{F})^G=\operatorname{colim}\widehat{F}\vert_{P(\Delta(P(K))^G)}$ and $(\operatorname{hocolim}\widehat{F})^G=\operatorname{hocolim}\widehat{F}\vert_{P(\Delta(P(K))^G)}$
    \end{enumerate}
  \end{proposition}

  \begin{proof}
    Define a functor $\widehat{F}\colon P(\Delta(P(K)))\to\mathbf{Top}$ by
    \[
      \widehat{F}(\{x_0<\cdots<x_n\})=F(x)\quad\text{and}\quad\widehat{F}(\{x_0<\cdots<x_n\}\ge\{y_0<\cdots<y_k\})=F(x_n\ge y_k).
    \]
    Then it is straightforward to see it has the desired property, where (2) uses Lemma \ref{action order complex}.
  \end{proof}

  \begin{corollary}
    \label{hocolim-colim}
    Let $K$ be a simplicial $G$-complex, where $G$ is a finite group, and let $F\colon P(K)\to\mathbf{Top}$ be a $G$-functor. If $F(x>y)$ is a cofibration for each $x>y$, then the projection
    \[
      \operatorname{hocolim}\widehat{F}\to\operatorname{colim}\widehat{F}
    \]
    is a $G$-homotopy equivalence, where $\widehat{F}$ is as in Proposition \ref{subdivision}.
  \end{corollary}

  \begin{proof}
    By Proposition \ref{subdivision} and \cite[Projection Lemma 1.6]{ZZ} (cf. the proof of \cite[Proposition 5.6]{KL}), the projection
    \[
      (\operatorname{hocolim}\widehat{F})^H\to(\operatorname{colim}\widehat{F})^H
    \]
    is a homotopy equivalence for any subgroup $H$ of $G$. Then the proof is done.
  \end{proof}

  \section{Proof of Theorem \ref{main}}\label{Proof of Theorem}

  This section proves Theorem \ref{main}. Recall that the \emph{discretized configuration space} $\mathbb{C}onf_r(X)$, or the \emph{$r$-fold deleted product}, of a regular CW complex $X$ is defined as the subcomplex of $X^r$ consisting of faces $\sigma_1\times\cdots\times \sigma_r$ such that $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint faces of $X$. The symmetric group $\Sigma_r$, hence its subgroups, act on $\mathbb{C}onf_r(X)$ by coordinate permutation.

  In \cite{HKTT}, a homotopy decomposition of $\mathbb{C}onf_r(X)$ in terms of a homotopy colimit is given, and by computing the Bousfield-Kan spectral sequence for this homotopy colimit, the following is proved in the case $A=\mathbb{Z}$. We can easily see that the computation works for any abelian group $A$ too.

  \begin{lemma}
    \label{Conf acyc}
    If $X$ is $(r-1)$-complementary $n$-acyclic over $A$, then $\mathbb{C}onf_r(X)$ is $n$-acyclic over $A$.
  \end{lemma}

  For the rest of this section, let $r=p^k$ for a prime number $k$, and let $G=(\mathbb{Z}/p)^k$. Note that $G$ acts on $\mathbb{C}onf_r(X)$ since $G$ is a subgroup of $\Sigma_r$. We consider the weight of a $G$-space. The \emph{weight} of a $G$-space $X$, denoted by $\operatorname{wgt}_G(X)$, is defined as the greatest integer $n$ such that the map
  \[
    \pi^*\colon H^*(BG;\mathbb{Z}/p)\to H^*(X\times_GEG;\mathbb{Z}/p)
  \]
  is injective for $*\le n$, where $\pi\colon X\times_GEG\to BG$ denotes the projection. The weight of a $G$-space is studied in \cite{HK,V} (cf. \cite{R}). It is called the \emph{index} in \cite{V}, but this is confusing because we already have the index of a free $G$-space. So we use the name ``weight" as in \cite{HK}. The following properties of a weight are proved in \cite{V}. If $X$ and $Y$ are $G$-spaces, then we consider $X*Y$ as a $G$-space by the diagonal $G$-action.

  \begin{proposition}
    \label{weight}
    \begin{enumerate}
      \item If there is a $G$-map $X\to Y$ between $G$-spaces $X$ and $Y$, then
      \[
        \operatorname{wgt}_G(X)\le\operatorname{wgt}_G(Y).
      \]

      \item If a $G$-space $X$ is $n$-acyclic over $\mathbb{Z}/p$, then
      \[
        \operatorname{wgt}_G(X)\ge n+1.
      \]

      \item Let $S$ be a finite $G$-complex such that $S^G=\emptyset$ and $H^*(S;\mathbb{Z}/p)\cong H^*(S^n;\mathbb{Z}/p)$. Then
      \[
        \operatorname{wgt}_G(S^n)=n.
      \]
      \item Let $S$ be as above. Then for a $G$-space $X$,
      \[
        \operatorname{wgt}_G(X*S)\le\operatorname{wgt}_G(X)+n+1.
      \]
    \end{enumerate}
  \end{proposition}

  We consider a group action on the nerve of a covering. Let $K$ be a simplicial complex having a covering
  \begin{equation}
    \label{covering}
    K=K_1\cup\cdots\cup K_r.
  \end{equation}
  Suppose that a symmetric group $\Sigma_r$ acts on $K$ by permuting $K_1,\ldots,K_r$. Let $N(K)$ denote the nerve of the covering \eqref{covering}. Then $\Sigma_r$ acts on $N(K)$ too. Define a $\Sigma_r$-functor $F\colon N(K)\to\mathbf{Top}$ by $F(K_{i_1}\cap\cdots\cap K_{i_k})=K_{i_1}\cap\cdots\cap K_{i_k}$ and the canonical inclusion.

  \begin{lemma}
    \label{hocolim-colim special}
    Let $\widehat{F}\colon P(\Delta(NK))\to\mathbf{Top}$ be the $\Sigma_r$-functor constructed from $F\colon NK\to\mathbf{Top}$ in Proposition \ref{subdivision}. Then the projection $\operatorname{hocolim}\widehat{F}\to\operatorname{colim}\widehat{F}$ is a $\Sigma_r$-homotopy equivalence.
  \end{lemma}

  \begin{proof}
    This lemma immediately follows from Corollary \ref{hocolim-colim}.
  \end{proof}

  \begin{proposition}
    \label{G-map nerve}
    There is a $\Sigma_r$-map
    \[
      K\to\Delta(N(K)).
    \]
  \end{proposition}

  \begin{proof}
    By Proposition \ref{action hocolim} and Lemma \ref{hocolim-colim special}, there is a string of $\Sigma_r$-map
    \[
      K=\operatorname{colim} F\xrightarrow{\cong}\operatorname{colim}\widehat{F}\to\operatorname{hocolim}\widehat{F}\to\Delta(P(\Delta(NK)))\xrightarrow{\cong}\Delta(N(K)).
    \]
    Thus the proof is finished.
  \end{proof}

  We apply Proopsition \ref{G-map nerve} to our situation. Let
  \[
    P=P(\mathbb{C}onf_r(X)_{r(n+1)-1})-P(\mathbb{C}onf_r(X_n))
  \]
  and let $P_i=\{\sigma_1\times\cdots\times\sigma_r\in P\mid\dim\sigma_i>n\}$ for $i=1,\ldots,r$. Then $P=P_1\cup\cdots\cup P_r$ and each $P_i$ is an upper ideal of $P$, implying
  \[
    \Delta(P)=\Delta(P_1)\cup\cdots\cup\Delta(P_r).
  \]

  \begin{lemma}
    \label{Conf G-map}
    Let $X$ be a regular CW complex which is $(r-1)$-complementary $(r(n+1)-2)$-acyclic over $\mathbb{Z}/p$. Then there is a $G$-map
    \[
      \mathbb{C}onf_r(X)_{r(n+1)-1}\to\mathbb{C}onf_r(X_n)*S^{r-2}
    \]
    where $G$ acts on $S^{r-2}$ such that $(S^{r-2})^G=\emptyset$.
  \end{lemma}

  \begin{proof}
    The map
    \[
      \Delta(P(\mathbb{C}onf_r(X)_{r(n+1)-1}))\to\Delta(P(\mathbb{C}onf_r(X_n)))*\Delta(P),\quad\sigma\mapsto(\sigma-\sigma\cap\Delta(P))\sqcup(\sigma\cap\Delta(P))
    \]
    is a simplicial $\Sigma_r$-map. Then we get a $G$-map
    \[
      \mathbb{C}onf_r(X)_{r(n+1)-1}\to\mathbb{C}onf_r(X_n)*\Delta(P).
    \]
    Since $P_1\cap\cdots\cap P_r=\emptyset$, $\Delta(N(\Delta(P)))$ is a $\Sigma_r$-subcomplex of $\Delta(P(\partial\Delta^{r-1}))$, we get a composition of $\Sigma_r$-maps
    \[
      \Delta(P)=\operatorname{colim}\widehat{F}\cong\operatorname{colim}\widehat{F}\to\operatorname{hocolim}\widehat{F}\to\Delta(N\Delta(P))\to\Delta(P(\partial\Delta^{r-1}))\cong\partial\Delta^{r-1}.
    \]
    Thus we get a $\Sigma_r$-map
    \[
      \mathbb{C}onf_r(X)_{r(n+1)-1}\to\mathbb{C}onf_r(X_n)*\partial\Delta^{r-1}
    \]
    which is also a $G$-map because $G$ is a subgroup of $\Sigma_r$. Clearly, $(\partial\Delta^{r-1})^G=\emptyset$, completing the proof.
  \end{proof}

  \begin{lemma}
    \label{Conf weight}
    If $X$ is a regular CW complex which is $(r-1)$-complementary $(r(n+1)-2)$-acyclic over $\mathbb{Z}/p$, then
    \[
      \operatorname{wgt}_G(\mathbb{C}onf_r(X_n))\ge rn.
    \]
  \end{lemma}

  \begin{proof}
    By Lemma \ref{Conf acyc}, $\mathbb{C}onf_r(X)$ is $(r(n+1)-2)$-acyclic over $\mathbb{Z}/p$. Then 
    $\mathbb{C}onf_r(X)_{r(n+1)-1}$ is $(r(n+1)-2)$-acyclic over $\mathbb{Z}/p$ too, and so by Proposition \ref{weight},
    \[
      \operatorname{wgt}_G(\mathbb{C}onf_r(X)_{r(n+1)-1})\ge r(n+1)-1.
    \]
    Thus by Proposition \ref{weight} and Lemma \ref{Conf G-map},
    \begin{align*}
      r(n+1)-1&\le\operatorname{wgt}_G(\mathbb{C}onf_r(X)_{r(n+1)-1})\\
      &\le\operatorname{wgt}_G(\mathbb{C}onf_r(X_n)*S^{r-2})\\
      &\le\operatorname{wgt}_G(\mathbb{C}onf_r(X_n))+r-1.
    \end{align*}
    Therefore we get $\operatorname{wgt}_G(\mathbb{C}onf_r(X_n))\ge rn$.
  \end{proof}

  Now we are ready to prove Theorem \ref{main}.

  \begin{proof}
    [Proof of Theorem \ref{main}]
    Suppose there is a map $X_n\to\mathbb{R}^d$ such that for each pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $X_n$, $f(\sigma_1)\cap\cdots\cap f(\sigma_r)=\emptyset$. Then as in \cite[The proof of Theorem 3.10]{BZ}, there is a $G$-map
    \[
      \mathbb{C}onf_r(X_n)\to(\mathbb{R}^d)^r-\Delta,\quad(x_1,\ldots,x_r)\mapsto(f(x_1),\ldots,f(x_r))
    \]
    where $\Delta=\{(x_1,\ldots,x_r)\in(\mathbb{R}^d)^r\mid x_1=\cdots=x_r\}$. It is obvious that $(\mathbb{R}^d)^r-\Delta$ is $G$-homotopy equivalent to the unit sphere $S$ of the orthogonal complement of $\Delta\subset(\mathbb{R}^d)^r$. So there is a $G$-map $\mathbb{C}onf_r(X)\to S$. Since $S^G=\emptyset$, it follows from Proposition \ref{weight} and Lemma \ref{Conf weight} that
    \[
      rn\le\operatorname{wgt}_G(\mathbb{C}onf_r(X_n))\le\operatorname{wgt}_G(S)=(r-1)d-1.
    \]
    Thus we must have $(r-1)d > rn$, completing the proof.
  \end{proof}

  \section{Chirality}\label{Chirality}

  This section proves Theorem \ref{main 2} by applying results in the previous sections. We will use the following two lemmas.

  \begin{lemma}
    \label{weight product}
     For $i=1,2$, let $X_i$ be a $\mathbb{Z}/2$-space with $n_i=\operatorname{wgt}_{\mathbb{Z}/2}(X_i)$, and let $(\mathbb{Z}/2)^2$ act on $\mathbb{R}^{n_1+n_2}$ by
     \[
       (x_1,\ldots,x_{n_1},x_{n_1+1},\ldots,x_{n_1+n_2})\cdot g_i=
       \begin{cases}
         (-x_1,\ldots,-x_{n_1},-x_{n_1+1},\ldots,-x_{n_1+n_2})&i=1\\
         (x_1,\ldots,x_{n_1},-x_{n_1+1},\ldots,-x_{n_1+n_2})&i=2
       \end{cases}
     \]
     where $g_1=(1,0),\,g_2=(0,1)\in(\mathbb{Z}/2)^2$. Then every $(\mathbb{Z}/2)^2$-map
    \[
      X_1\times X_2\to\mathbb{R}^{n_1+n_2}
    \]
    has a zero.
  \end{lemma}

  \begin{proof}
    Let $G=(\mathbb{Z}/2)^2$ and $G_i$ be the $i$-th $\mathbb{Z}/2$ in $G$ for $i=1,2$. Recall that the mod 2 cohomology of $BG$ is given by
    \[
      H^*(BG;\mathbb{Z}/2)=\mathbb{Z}/2[x_1,x_2],\quad|x_i|=1
    \]
    where $x_i$ corresponds to a generator of  $H^1(BG_i;\mathbb{Z}/2)\cong\mathbb{Z}/2$ for $i=1,2$. Let $K_1$ denote the kernel of the natural map $H^*(BG;\mathbb{Z}/2)\to H^*((X_1\times X_2)\times_GEG;\mathbb{Z}/2)$. Since $(X_1\times X_2)\times_GEG\simeq(X_1\times_{G_1}EG_1)\times(X_2\times_{G_2}EG_2)$, we have
    \[
      K_1=(x_1^{n_1+1},x_2^{n_2+1}).
    \]
    Let $K_1$ denote the kernel of the natural map $H^*(BG;\mathbb{Z}/2)\to H^*(S^{n_1+n_2-1}\times_GEG;\mathbb{Z}/2)$, where the $G$-action on $S^{n_1+n_2-1}$ is a restriction of that on $\mathbb{R}^{n_1+n_2}$. By considering the Mayer-Vietoris exact sequence for a covering $S^{n_1+n_2-1}\times_GEG=((S^{n_1-1}\times D^{n_2})\times_GEG)\cup((D^{n_1}\times S^{n_2-1})\times_GEG)$, we can see that
    \[
      (x_1+x_2)^{n_1}x_2^{n_2}\in K_2.
    \]
    Now we suppose that a $G$-map $X_1\times X_2\to\mathbb{R}^{n_1+n_2}$ does not have a zero. Then by normalizing, we get a $G$-map $X_1\times X_2\to S^{n_1+n_2-1}$. This implies $K_2\subset K_1$, which contradicts to the above computation. Thus every $G$-map $X_1\times X_2\to\mathbb{R}^{n_1+n_2}$ must have a zero, completing the proof.
  \end{proof}

  \begin{lemma} \label{lemma psi}
    \label{Psi}
    For a regular CW complex $X$, there is a continuous map $\Psi\colon\mathbb{C}onf_2(X)_{2n+1}\to\mathbb{R}^d$ such that for each $(x,y)\in\mathbb{C}onf_2(X)_{2n+1}$,
    \[
      \Psi(x,y)=-\Psi(y,x)\quad\text{and}\quad\Psi^{-1}(0)=\mathbb{C}onf_2(X_n).
    \]
  \end{lemma}

  \begin{proof}
    Let $P=P(\mathbb{C}onf_2(X)_{2n+1})-P(\mathbb{C}onf_2(X_n))$ as in the proof of Lemma \ref{Conf G-map}. Then $\mathbb{C}onf_2(X)_{2n+1}$ is a $\mathbb{Z}/2$-subspace of $\mathbb{C}onf_2(X_n)*\Delta(P)$ and there is a $\mathbb{Z}/2$-map $f\colon\Delta(P)\to S^0=\{-1,+1\}$ as in the proof of Lemma \ref{Conf G-map}. Let $r\colon\mathbb{C}onf_2(X_d)* \Delta(P)-\mathbb{C}onf_2(X_n)\to S^0$ be the natural projection, and let $q\colon\mathbb{C}onf_2(X_n)* \Delta(P)\to[0,1]$ be the projection onto the join parameter such that $q^{-1}(0)=\mathbb{C}onf_2(X_n)$ and $q^{-1}(1)=\Delta(P)$. Now we define
    \[
      \Psi\colon\mathbb{C}onf_2(X)_{2n+1}\to\mathbb{R},\quad x\mapsto\begin{cases}0&x\in\mathbb{C}onf_2(X_n)\\q(x)\cdot f(r(x))&x\not\in\mathbb{C}onf_2(X_n).\end{cases}
    \]
    It is straightforward to check that this map satisfies the conditions in the statement.
  \end{proof}

  \begin{proof}
    [Proof of Theorem \ref{main 2}]
    Let $\mathrm{Emb}(A,B)$ denote the space of embeddings of $A$ into $B$ with compact-open topology. Let $\mathbb{Z}/2$ act on $S^1$ by the antipodal map and on $\mathrm{Emb}(A,\mathbb{R}^N)$ by the induced map from a reflection on $\mathbb{R}^N$. By \cite[Lemma 2.1]{FH}, every embedding $A\to\mathbb{R}^N$ is chiral if and only if there is a $\mathbb{Z}/2$-map $S^1\to\mathrm{Emb}(A,\mathbb{R}^N)$. Let $F\colon X\times S^1\to\mathbb{R}^{2n+1}$ be a $\mathbb{Z}/2$-map, where $\mathbb{Z}/2$ actions on $X\times S^1$ and $\mathbb{R}^{2n+1}$ are given by
    \[
      X\times S^1\to X\times S^1,\quad(x,y)\mapsto(x,-y)
    \]
    and
    \[
      \mathbb{R}^{2n+1}\to\mathbb{R}^{2n+1},\quad(x_1,\ldots,x_{2n},x_{2n+1})\mapsto(x_1,\ldots,x_{2n},-x_{2n+1}).
    \]
    Then since any $\mathbb{Z}/2$-map $X_n\times S^1\to\mathbb{R}^{2n+1}$ extends to $\mathbb{Z}/2$-map $X\times S^1\to\mathbb{R}^{2n+1}$, it suffices to prove that there is $y\in S^1$ such that $F(-,y)\vert_{X_n}\colon X_n\to\mathbb{R}^{2n+1}$ is not an embedding. Define a map
    \[
      \widehat{F}\colon\mathbb{C}onf_2(X)_{2n+1}\times S^1\to\mathbb{R}^{2n+2},\quad((x_1,x_2),y)\mapsto(\Psi(x_1,x_2),F(x_1,y)-F(x_2,y)),
    \]
    where $\Psi \colon \mathbb{C}onf_2(X)_{2n+1} \to \mathbb{R}$ is a function described in Lemma \ref{lemma psi}. Let $(\mathbb{Z}/2)^2$ act on $\mathbb{R}^{2n+2}$ by
    \[
      (x_0,\ldots,x_{2n},x_{2n+1})\cdot g_i=
      \begin{cases}
        (-x_0,\ldots,-x_{2n},-x_{2n+1})&i=1\\
        (x_0,\ldots,x_{2n},-x_{2n+1})&i=2
      \end{cases}
    \]
    for $(x_0,\ldots,x_{2n},x_{2n+1})\in\mathbb{R}^{2n+2}$, where $g_1=(1,0),\,g_2=(0,1)\in(\mathbb{Z}/2)^2$. Then $\widehat{F}$ is a $(\mathbb{Z}/2)^2$-map. If $\widehat{F}$ does not has a zero, then by normalizing, we get a $(\mathbb{Z}/2)^2$-map $\mathbb{C}onf_2(X)\times S^1\to S^{2n+1}$. By Lemma \ref{Conf acyc}, $\mathbb{C}onf_2(X)_{2n+1}$ is $2n$-acyclic over $\mathbb{Z}/2$, and so by Proposition \ref{weight}, $\operatorname{wgt}_{\mathbb{Z}/2}(\mathbb{C}onf_2(X)_{2n+1})\ge 2n+1$. By Proposition \ref{weight}, we also have $\operatorname{wgt}_{\mathbb{Z}/2}(S^1)=1$. Then by Lemma \ref{weight product}, $\widehat{F}((x_1,x_2),y)=0$ for some $(x_1,x_2)\in\mathbb{C}onf_2(X)_{2n+1}$ and $y\in S^1$. Namely, $\Psi(x_1,x_2)=0$ and $F(x_1,y)=F(x_2,y)$. By the definition of $\Psi$, we have $x_1,x_2\in X_n$, and therefore $F(-,y)\colon X_n\to\mathbb{R}^{2n+1}$ is not injective, completing the proof.
  \end{proof}

  \section{Constraint method}\label{Constraint method}

  This section gives another proof of Theorem \ref{main} by using the constraint method (see \cite{BFZ, BZ}). Let $r=p^k$ for a prime number $p$.

  \begin{lemma}
    \label{dim}
    Let $X$ be a regular CW complex which is $(r-1)$-complementary $(d(r-1)-1)$-acyclic over $\mathbb{Z}/p$. Then for any continuous map $f\colon X\to\mathbb{R}^d$, there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $X$ such that
    \[
      f(\sigma_1)\cap\cdots\cap f(\sigma_r)\ne\emptyset\quad\text{and}\quad\dim\sigma_1+\cdots+\dim\sigma_r\le d(r-1).
    \]
  \end{lemma}

  \begin{proof}
    Suppose that $f(\sigma_1) \cap \cdots \cap f(\sigma_r) = \emptyset$ for every $r$-tuple $\sigma_1, \cdots, \sigma_r$ of pairwise disjoint faces of $X$ satisfying $\dim \sigma_1 + \cdots + \dim \sigma_r \le(r-1)d$.

    Then the natural $(\mathbb{Z}/p)^k$-map $\mathbb{C}onf_r(X)\to(\mathbb{R}^d)^r$ restricts to a $(\mathbb{Z}/p)^k$-map $\mathbb{C}onf_r(X)_{d(r-1)}\to(\mathbb{R}^d)^r-\Delta$, where $\Delta$ is as in Section \ref{Proof of Theorem}. On the other hand, by Lemma \ref{Conf acyc}, $\mathbb{C}onf_r(X)$ is $((r-1)d-1)$-acyclic over $\mathbb{Z}/p$, and so $\mathbb{C}onf_r(X)_{d(r-1)}$ is $((r-1)d-1)$-acyclic over $\mathbb{Z}/p$ too. It is easy to see that the proof of \cite[Proposition 3.2]{HKTT} works in verbatim over $\mathbb{Z}/p$. Then we obtain that there is no $(\mathbb{Z}/p)^k$-map $\mathbb{C}onf_r(X)_{d(r-1)}\to(\mathbb{R}^d)^r-\Delta$, a contradiction. Thus the proof is finished.
  \end{proof}

  \begin{proof}
    [Proof of Theorem \ref{main}]
    Since $X_n$ is a subcomplex of $X$, there is a continuous map $c\colon X\to\mathbb{R}$ such that $c^{-1}(0)=X_n$. Consider a map
    \[
      g\colon X\to\mathbb{R}^{d+1},\quad x\mapsto(f(x),c(x)).
    \]
    By Lemma \ref{Conf acyc}, $\mathbb{C}onf_r(X)$ is $(r(n+1)-2)$-acyclic over $\mathbb{Z}/p$. Then since $(r-1)(d+1)-1\le r(n+1)-2$, we can apply Lemma \ref{dim} to a map $g$, so that there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $X$ such that $g(\sigma_1)\cap\cdots\cap g(\sigma_r)\ne\emptyset$ and
    \[
      \dim\sigma_1+\cdots+\dim\sigma_r\le (r-1)(d+1)\le r(n+1)-1.
    \]
    Thus for some $x_1\in\sigma_1,\ldots,x_r\in\sigma_r$, we have $g(x_1)=\cdots=g(x_r)$. By replacing $\sigma_1,\ldots,\sigma_r$ by their faces, we may assume each $x_i$ belongs to the interior of $\sigma_i$. Since the condition $\dim\sigma_1+\cdots+\dim\sigma_r\le r(n+1) - 1$ is still satisfied, $\dim\sigma_i\le n$ for some $i$, so that $g(x_1)=\cdots=g(x_r)=0$. Thus $\dim\sigma_i\le n$ for all $n$, completing the proof.
  \end{proof}

  

\end{document}