ESTIMATING THE SIZE OF THE $(H, G)$-COINCIDENCES SET IN REPRESENTATION SPHERES

Abstract

Let W be a real vector space and let V be an orthogonal representation of a group G such that $V^{G} = \{0\}$ (for the set of fixed points of G). Let $S(V)$ be the sphere of V and suppose that $f: S(V) \to W$ is a continuous map. We estimate the size of the $(H, G)$ -coincidences set if G is a cyclic group of prime power order $\mathbb {Z}_{p^k}$ or a p-torus $\mathbb {Z}_p^k$ .

1 Introduction

Let G be a finite group which acts on a space X and let $f: X \to Y$ be a continuous map from X into another space Y. If H is a subgroup of G, then H acts on the right on each orbit $Gx$ of G as follows: if $y \in Gx$ and $y = gx$ , with $g \in G$ , then $h \cdot y = g h^{-1} x$ . A point $x\in X$ is said to be an $(H, G)$ -coincidence point of f (as introduced by Gonçalves et al. in [6]) if f sends every orbit of the action of H on the G-orbit of x to a single point. Of course, if H is the trivial subgroup, then every point of X is an $(H, G)$ -coincidence. If $H = G$ , this is the usual definition of a G-coincidence point, that is, $f(x) = f(g x )$ for all $g\in G$ . Let us denote by $A(f, H, G)$ the set of all $(H, G)$ -coincidence points. Borsuk–Ulam theorems estimate the size of the set $A(f, H, G)$ . For the case when the target space Y is a CW-complex, this problem was considered by Gonçalves et al. [6] (for the subgroup $H= \mathbb {Z}_p$ of a finite group G, X a homotopy sphere and Y a CW-complex) and Gonçalves et al. [7] (for the subgroup $H= \mathbb {Z}_p$ of a finite group G, X under certain (co)homological assumptions and Y a CW-complex). In [5], by considering the target space $Y=M$ a manifold and H a proper nontrivial subgroup of G, we proved a formulation of the Borsuk–Ulam theorem for manifolds in terms of $(H,G)$ -coincidences which has applications to the famous topological Tverberg problem (see for example, [1]).

Let W be a real vector space and let V be an orthogonal representation of a group G with $V^{G} = \{0\}$ . Let $S(V)$ be the sphere of V and suppose that $f: S(V) \to W$ is a continuous map. We estimate the size of $A(f, H, G)$ if G is a cyclic group of prime power order $\mathbb {Z}_{p^k}$ or a p-torus $\mathbb {Z}_p^k$ (Theorems 3.1, 3.2 and 3.5).

2 Bourgin–Yang versions of the Borsuk–Ulam theorem for $\mathbb {Z}_{p^k}$ and $\mathbb {Z}_p^k$

Let $G = \mathbb {Z}_{p^k}$ be a cyclic group of prime power order, $k\geq 1$ . Given two powers $p^m, p^n$ of p with $ 1 \leq m\leq n\leq k-1$ , we set

$$ \begin{align*} \mathcal{A}_{m,n}:= \{ G/H \mid H \subset G, p^m \le \vert H\vert \le p^n\}, \end{align*} $$

where $\vert H\vert $ is the cardinality of H. We write $\mathcal {A}_X$ for a set of all the G-orbits of a space X (up to a homeomorphism and thus up to an isomorphism of finite G-sets).

Let V be an orthogonal representation of $G = \mathbb {Z}_{p^k}$ , p prime, $k\geq 1$ , such that $V^G=\{0\}$ (for the set of fixed points of G). For $G=\mathbb {Z}_{p^k}$ , with p odd, every nontrivial irreducible orthogonal representation is even dimensional and admits a complex structure [10], so V also admits such a structure. We write $d(V)=\dim _{\mathbb {C}} V =\tfrac 12 \dim _{\mathbb {R}} V $ , an integral numerical invariant of V.

The following Bourgin–Yang versions of the Borsuk–Ulam theorem for complex orthogonal representations of $G=\mathbb {Z}_{p^{k}}$ , p prime, $k\geq 1$ and for real orthogonal representations of $G=\mathbb {Z}_{2^{k}}$ , $k\geq 1$ are from [8].

Theorem 2.1 [8, Theorem 3.6]

Let V, W be two complex orthogonal representations of the cyclic group $G=\mathbb {Z}_{p^k}$ , $p> 2 $ prime, $k\geq 1$ , such that $V^G=W^G=\{0\}$ . Let $f: S(V)\,\,{\overset {G}\to }\,\, W$ be an equivariant map and $Z_f :=f^{-1}(0)=\{v\in S(V) \mid f(v)=0\}$ . Suppose $\mathcal {A}_{S(V)} \subset \mathcal {A}_{m,n}$ and $\mathcal {A}_{S(W)} \subset \mathcal {A}_{{m},{n}}$ . Then

$$ \begin{align*} \displaystyle\dim \,Z_f \geq 2\bigg(\bigg\lceil\frac{(d(V)-1)m}{n}\bigg \rceil - d(W)\bigg). \end{align*} $$

Theorem 2.2 [8, Theorem 3.9]

Let V, W be two real orthogonal representations of the cyclic group $G=\mathbb {Z}_{2^k}$ , $k\geq 1$ , such that $V^G=W^G=\{0\}$ . Let $f: S(V)\,\,{\overset {G}\to }\,\, W$ be an equivariant map and $Z_f=f^{-1}(0)$ . Suppose that $\mathcal {A}_{S(V)} \subset \mathcal {A}_{m,n}$ and $\mathcal {A}_{S(W)} \subset \mathcal {A}_{{m},{n}}$ . Then

$$ \begin{align*} \dim(Z_f) \geq \bigg \lceil\frac{(d(V)-1)m}{n}\bigg \rceil -d(W). \end{align*} $$

The next result is the classical version of the Bourgin–Yang theorem for a p-torus $\mathbb {Z}_p^k = \mathbb {Z}_p \times \mathbb {Z}_p\times \cdots \times \mathbb {Z}_p$ (k times).

Theorem 2.3 [9, Theorem 2.1]

Let V and W be two orthogonal representations of the group $G = \mathbb {Z}_p^k$ such that $V^{G} = W^{G} = \{ 0 \}$ . Let $f: S(V) \to W$ be a continuous map. Then

$$ \begin{align*} \dim \, Z_f \geq \dim_{\mathbb{R}} V - \dim_{\mathbb{R}} W - 1. \end{align*} $$

For further recent extensions of the Bourgin–Yang theorem, see [2, 3].

3 Estimating the size of the $(H, G)$ -coincidences set

Let $W^{\prime }$ be a real vector space and $f:S(V)\to W^{\prime }$ a continuous map. In this section, we estimate the size of the set $A(f,\mathbb {Z}_{p^i}, \mathbb {Z}_{p^k})$ under various assumptions.

Theorem 3.1. Let V be a complex orthogonal representation of the cyclic group $G=\mathbb {Z}_{p^k}$ , $p \geq 3$ prime and $k\geq 1$ , such that $V^G=\{0\}$ and let $W^{\prime }$ be a real vector space. Let $f:S(V)\to W^{\prime }$ be a continuous map.

1. (1) If $\mathcal {A}_{S(V)} \subset \mathcal {A}_{1,p^{k-1}}$ , then for all i with $1\leq i \leq k$ ,

   $$ \begin{align*} \dim A(f,\mathbb{Z}_{p^i}, \mathbb{Z}_{p^k})\geq 2\bigg \lceil\frac{d(V)-1}{p^{k-1}}\bigg \rceil-(p^{k} - p^{k-i})\,dW^{\prime}. \end{align*} $$

2. (2) If $\mathcal {A}_{S(V)} \subset \mathcal {A}_{1,p^{i-1}}$ for some i with $1\leq i \leq k$ , then

   $$ \begin{align*} \dim A(f,\mathbb{Z}_{p^i}, \mathbb{Z}_{p^k})\geq 2\bigg \lceil\frac{d(V)-1}{p^{i-1}}\bigg \rceil-(p^{k} - p^{k-i})\,dW^{\prime}. \end{align*} $$

Proof. Let i be fixed with $1\leq i\leq k$ . Consider the real vector space $\oplus _{j=1}^{p^k} W^{\prime }$ , which is the direct sum of $p^{k}$ copies of $W^{\prime }$ . The space $\oplus _{j=1}^{p^k} W^{\prime }$ admits an action of the cyclic group $G=\mathbb {Z}_{p^{k}}$ , given by

$$ \begin{align*} g(w_{1},w_{2},\ldots,w_{p^{k}})=(w_{2},\ldots,w_{p^{k}},w_{1}) \end{align*} $$

for a fixed generator $g \in G$ and for each $(w_{1},\ldots ,w_{p^{k}})\in \oplus _{j=1}^{p^k} W^{\prime }$ .

Denote by $\Delta ({W^{\prime }}^{p^{k-i}})$ the diagonal of $ \oplus _{j=1}^{p^k} W^{\prime } = {W^{\prime }}^{p^{k-i}} \oplus \cdots \oplus {W^{\prime }}^{p^{k-i}}$ . Then

$$ \begin{align*} {\underset{j=1}{\overset{p^k}\bigoplus}} W^{\prime}= \Delta ({W^{\prime}}^{p^{k-i}})\oplus (\Delta ({W^{\prime}}^{p^{k-i}}))^{\bot}, \end{align*} $$

where $\Delta ({W^{\prime }}^{p^{k-i}})^{\bot }$ is the orthogonal complement of $\Delta ({W^{\prime }}^{p^{k-i}})$ . Now $\Delta ({W^{\prime }}^{p^{k-i}})$ is a G-subspace of $\oplus _{j=1}^{p^k} W^{\prime }$ of dimension $p^{k-i}\dim W^{\prime } $ , so $\Delta (W^{p^{k-i}})^{\bot }$ is a G-subrepresentation of $\oplus _{j=1}^{p^k} W^{\prime }$ of dimension $(p^{k}- p^{k-i})\dim W^{\prime }$ for which $({\Delta ({W^{\prime }}^{p^{k-i}})^{\bot }})^G=\{0\}$ .

Denote by $a_1, \ldots , a_r$ a set of representatives of the left lateral classes of $G/\mathbb {Z}_{p^i}$ , where $r = p^{k-i}$ . Consider the map

$$ \begin{align*} F:S(V)\to \Delta ({W^{\prime}}^{p^{k-i}})\oplus \Delta ({W^{\prime}}^{p^{k-i}})^{\bot} \end{align*} $$

defined by

$$ \begin{align*} F(x) = (F_0 (x), F_1(x), \ldots, F_{p^i - 1} (x)), \end{align*} $$

where $F_{j} (x) = (f(a_1 h^j x), \ldots , f(a_r h^j x))$ , $j = 0, 1, \ldots , p^i - 1$ , for a fixed generator $h \in \mathbb {Z}_{p^i}.$ The linear orthogonal projection along the diagonal $\Delta ({W^{\prime }}^{p^{k-i}})$ defines a G-equivariant map $\rho :\Delta ({W^{\prime }}^{p^{k-i}})\oplus \Delta ({W^{\prime }}^{p^{k-i}})^{\bot }\to \Delta ({W^{\prime }}^{p^{k-i}})^{\bot }$ . Let us denote by l the composition

$$ \begin{align*} S(V) \stackrel{F}\rightarrow \Delta ({W^{\prime}}^{p^{k-i}})\oplus \Delta ({W^{\prime}}^{p^{k-i}})^{\bot} \stackrel{\rho}\rightarrow \Delta ({W^{\prime}}^{p^{k-i}})^{\bot}, \end{align*} $$

with

$$ \begin{align*} Z_{l}=l^{-1}(0)=(\rho \circ F)^{-1}(0)=F^{-1}(\Delta ({W^{\prime}}^{p^{k-i}}))= A(f,\mathbb{Z}_{p^i}, \mathbb{Z}_{p^k}). \end{align*} $$

For a fixed generator $g\in G$ , we can take $h = g^{p^{k-i}}, a_1\, {=}\ e, a_2 = g, \ldots , a_r\, {=}\ g^{p^{k-i} - 1},$ and then F is a G-equivariant map. Moreover,

$$ \begin{align*} \mathcal{A}_{S(\Delta({W^{\prime}}^{p^{k-i}})^\bot)}\subset \mathcal{A}_{1,p^{i-1}}\subset \mathcal{A}_{1,p^{k-1}}. \end{align*} $$

To check the validity of the inclusion $\mathcal {A}_{S(\Delta ({W^{\prime }}^{p^{k-i}})^\bot )}\subset \mathcal {A}_{1,p^{i-1}}$ , it suffices to prove that the cardinality of the orbit $\mathbb {Z}_{p^k} w$ belongs to the set $\{p^k, p^{k-1}, \ldots , p^{k-i+1} \}$ for any $w=(w_1, \ldots , w_{p^k}) \in S(\Delta ({W^{\prime }}^{p^{k-i}})^\bot )$ . From [4, Ch. 1, Proposition 4.1], the cardinality of the orbit $\mathbb {Z}_{p^k} w$ belongs to the set $\{p^{k}, p^{k-1}, \ldots , p, p^{0}=1 \}$ . Let $w=(w_1, \ldots , w_{p^k})$ be an element in $S(\Delta ({W^{\prime }}^{p^{k-i}})^\bot )$ and suppose that $|\mathbb {Z}_{p^k} w|\in \{p^{k-i}, p^{k-i-1}, \ldots , p^{0}=1 \},$ that is, $|\mathbb {Z}_{p^k} w|=p^{\,j}$ for some j with $0\leq j\leq k-i$ .

Assertion. We have $\mathbb {Z}_{p^k} w =\{w, gw,\ldots ,g^{p^{\,j} -1} w\}$ , for a fixed generator g of $\mathbb {Z}_{p^{k}}$ .

In fact, consider a cyclic group G, $g\in G$ a fixed generator and $\{w, gw,\ldots ,g^{s-1}w\}$ the maximum set of the first s elements of the orbit $Gw$ that are distinct from each other. From this definition, $g^{s}w\in \{w, gw,\ldots ,g^{s-1}w\}.$ Suppose that

$$ \begin{align*} g^{s}w=g^{i}w \quad\text{for some}\ i\ \text{with } 1\leq i\leq s-1. \end{align*} $$

Then

$$ \begin{align*} g^{s-i}w=w \quad\mbox{where }1\leq s-i\leq s-1. \end{align*} $$

However, this contradicts the definition of the set $\{w, gw,\ldots ,g^{s-1}w\}$ .

Now, if $g^{t}w \in Gw$ , for some $t\in \mathbb {N}$ , we have $t=ns+r$ with $0\leq r\leq s-1$ . Therefore,

$$ \begin{align*} g^{t}w=g^{ns+r}w=g^{r}(g^{ns})w=g^{r}w\in \{w,gw,\ldots,g^{s-1}w\}, \end{align*} $$

since $g^{ns} w = (g^{s}\cdots g^{s})w=w$ and $0\leq r\leq s-1$ .

Thus, for a fixed generator g of $\mathbb {Z}_{p^{k}}$ ,

$$ \begin{align*} w = g^{p^{\,j}}w & = g^{p^{\,j}}(w_{1},\ldots,w_{p^{\,j}},\ldots,w_{(p^{k-j}-1) p^{\,j}+1},\ldots, w_{p^{k}}) \\ & = (w_{p^{\,j}+1},\ldots, w_{2p^{\,j}}, \ldots,w_{(p^{k-j}-1) p^{\,j}+1},\ldots, w_{p^{k}},w_{1},\ldots,w_{p^{\,j}}) \end{align*} $$

and so $w\in \Delta ({W^{\prime }}^{p^{\,j}})$ . Since

$$ \begin{align*} \Delta({W^{\prime}})\subset \Delta({W^{\prime}}^{p})\subset \cdots \subset\Delta({W^{\prime}}^{p^{k-i-1}})\subset \Delta({W^{\prime}}^{p^{k-i}}) \end{align*} $$

and $j\in \{0,1,\ldots ,k-i\}$ , we conclude that $w\in \Delta ({W^{\prime }}^{p^{\,j}})\subset \Delta ({W^{\prime }}^{p^{k-i}}),$ which is a contradiction since $\Delta ({W^{\prime }}^{p^{k-i}})\cap S(\Delta ({W^{\prime }}^{p^{k-i}})^{\bot })=\emptyset .$

This proves the assertion and the theorem follows from Theorem 2.1.

We also have the following estimate for the size of $A(f,\mathbb {Z}_{2^i}, \mathbb {Z}_{2^k})$ .

Theorem 3.2. Let V be a real orthogonal representation of the cyclic group $G=\mathbb {Z}_{2^k}$ , $k\geq 1$ , such that $V^G=\{0\}$ and let $W^{\prime }$ be a real vector space. Let $f:S(V)\to W^{\prime }$ be a continuous map.

1. (1) If $\mathcal {A}_{S(V)} \subset \mathcal {A}_{1,2^{k-1}}$ , then for all i with $1\leq i \leq k$ ,

   $$ \begin{align*} \dim \, A(f,\mathbb{Z}_{2^i}, \mathbb{Z}_{2^k})\geq \bigg\lceil\frac{d(V)-1}{2^{k-1}}\bigg\rceil-(2^{k-1} - 2^{k-i})\,dW^{\prime}. \end{align*} $$

2. (2) If $\mathcal {A}_{S(V)} \subset \mathcal {A}_{1,2^{i-1}}$ , then for some i with $1\leq i \leq k$ ,

   $$ \begin{align*} \dim A(f,\mathbb{Z}_{2^i}, \mathbb{Z}_{2^k})\geq \bigg \lceil\frac{d(V)-1}{2^{i-1}}\bigg\rceil-(2^{k-1} - 2^{k-i})\,dW^{\prime}. \end{align*} $$

Proof. For $G=\mathbb {Z}_{2^k}$ , $k\geq 1$ , using the same steps as in the proof of Theorem 3.1 and applying Theorem 2.2 gives the result.

Remark 3.3. We observe that Theorems 3.1 and 3.2 have peculiar characteristics that differentiate them from the classic results on $(H, G)$ -coincidences. The first is that the action of the group G on the sphere $S(V)$ is not necessarily free. The second is that the theorems provide an estimate for the dimension of the set of $(H, G)$ -coincidences of a continuous function $f: S(V) \to W^{\prime }$ , for all subgroups $H = \mathbb {Z}_{p^i}$ of $G = \mathbb {Z}_{p^k}$ .

Example 3.4. Let G and $W^{\prime }$ be $\mathbb {Z}_4$ and $\mathbb {R}$ , respectively. Let $\pi : S^1 \to \mathbb {R}$ be the projection on the first factor and $p: \mathbb {R} \to \mathbb {R}$ be the polynomial function $p(x)= x (x-1)(x+1)$ . Consider the action of $\mathbb {Z}_4$ on $S^1$ as the rotation of $\pi /4$ . Then $f = p \circ \pi $ is such that $A(f, \mathbb {Z}_2, \mathbb {Z}_4) = \{ (1,0), (0,1), (-1,0), (0, -1)\}$ and therefore $\dim A(f, \mathbb {Z}_2, \mathbb {Z}_4) = 0$ . In this case, we have the equality

$$ \begin{align*} \dim A(f,\mathbb{Z}_{2^i}, \mathbb{Z}_{2^k})= \bigg\lceil\frac{d(V)-1}{2^{i-1}}\bigg\rceil-(2^{k-1} - 2^{k-i})\,dW^{\prime}, \end{align*} $$

where $V = \mathbb {R}^2$ , $k=2$ and $i=1$ .

If we take $p(x) = x^2 (x - 1)(x + 1)$ and $f = p \circ \pi $ , then all points of $S^1$ are $(\mathbb {Z}_2, \mathbb {Z}_4)$ -coincidence points of f, that is, $A(f, \mathbb {Z}_2, \mathbb {Z}_4) = S^1$ and therefore, $\dim A(f, \mathbb {Z}_2, \mathbb {Z}_4) = 1$ .

The next result is an $(H, G)$ -coincidence version of the Bourgin–Yang theorem for p-torus $\mathbb {Z}_p^k$ .

Theorem 3.5. Let V and $W^{\prime }$ be two orthogonal representations of the group $G = \mathbb {Z}_p^k$ such that $V^{G} = {W^{\prime }}^{G} = \{ 0 \}$ . Let $f: S(V) \to W^{\prime }$ be a continuous map. Then

$$ \begin{align*} \dim A(f,\mathbb{Z}_p^i, \mathbb{Z}_p^k) \geq \dim_{\mathbb{R}} V + (p^k - p^{k - i})\dim_{\mathbb{R}} {W^{\prime}} - 1. \end{align*} $$

Proof. Let $a_1, \ldots , a_r$ be a set of representatives of the left lateral classes of $G/\mathbb {Z}_p^i$ , where $r = p^{k-i}$ . Let $\mathbb {Z}_p^i = \{h_1, \ldots , h_{p^i}\}$ be a fixed enumeration of elements of $\mathbb {Z}_p^i$ . Consider the map

$$ \begin{align*} F:S(V)\to \Delta ({W^{\prime}}^{p^{k-i}})\oplus \Delta ({W^{\prime}}^{p^{k-i}})^{\bot} \end{align*} $$

defined by

$$ \begin{align*} F(x) = (F_1 (x), F_2(x), \ldots, F_{p^i} (x)), \end{align*} $$

where $F_j (x) = (f(a_1 h_j x), \ldots , f(a_r h_j x))$ , $j = 1, \ldots , p^i$ .

For a fixed enumeration $\mathbb {Z}_p^k = \{ g_1, \ldots , g_{p^k}\}$ of the elements of $\mathbb {Z}_p^k$ , we define a $\mathbb {Z}_p^k$ -action on $\Delta ({W^{\prime }}^{p^{k-i}})\oplus \Delta ({W^{\prime }}^{p^{k-i}})^{\bot }$ as follows: for each $g_j \in \mathbb {Z}_p^k$ and for each $(y_1, \ldots , y_{p^k}) \in \Delta ({W^{\prime }}^{p^{k-i}})\oplus \Delta ({W^{\prime }}^{p^{k-i}})^{\bot }$ , set

$$ \begin{align*} g_j \, (y_1, \ldots, y_{p^k}) = (y_{\sigma_{g_j} (1)}, \ldots, y_{\sigma_{g_j} (p^k)}), \end{align*} $$

where the permutation $\sigma _{g_j}$ is defined by $\sigma _{g_j} (k) = u$ , $g_k g_j = g_u$ . Then F becomes $\mathbb {Z}_p^k$ -equivariant.

The linear orthogonal projection along the diagonal $\Delta ({W^{\prime }}^{p^{k-i}})$ defines a G-equivariant map

$$ \begin{align*} \rho:\Delta ({W^{\prime}}^{p^{k-i}})\oplus \Delta ({W^{\prime}}^{p^{k-i}})^{\bot}\to \Delta ({W^{\prime}}^{p^{k-i}})^{\bot}. \end{align*} $$

Let us denote by l the composition

$$ \begin{align*} S(V) \stackrel{F}\rightarrow \Delta ({W^{\prime}}^{p^{k-i}})\oplus \Delta ({W^{\prime}}^{p^{k-i}})^{\bot} \stackrel{\rho}\rightarrow \Delta ({W^{\prime}}^{p^{k-i}})^{\bot}, \end{align*} $$

with $Z_{l}=l^{-1}(0)=(\,\rho \circ F)^{-1}(0)=F^{-1}(\Delta ({W^{\prime }}^{p^{k-i}}))= A(f,\mathbb {Z}_p^i, \mathbb {Z}_p^k).$ From Theorem 2.3, $\dim Z_l \geq \dim _{\mathbb {R}} V + \dim _{\mathbb {R}} \Delta ({W^{\prime }}^{p^{k-i}})^{\bot } - 1$ , that is,

$$ \begin{align*} \dim A(\,f,\mathbb{Z}_p^i, \mathbb{Z}_p^k) \geq \dim_{\mathbb{R}} V + (p^k - p^{k - i})\dim_{\mathbb{R}}{W^{\prime}} - 1.\\[-38pt] \end{align*} $$