Higher Homotopic Distance

TL;DR

This paper explores the properties of higher homotopic distance, its relation to topological invariants like $ ext{cat}$, $ ext{secat}$, and higher topological complexity, providing new proofs and insights.

Contribution

It introduces important properties of higher homotopic distance and establishes conditions linking it to classical topological invariants, offering alternative proofs for related theorems.

Findings

Higher homotopic distance properties are characterized.

Conditions for equality between homotopic distance and invariants like $ ext{cat}$, $ ext{secat}$, and topological complexity are identified.

Alternative proofs for $ ext{TC}_n$-related theorems using higher homotopic distance are provided.

Abstract

The concept of homotopic distance and its higher analog are introduced in [6]. In this paper we introduce some important properties of higher homotopic distance, investigate the conditions under which $\cat$, $\secat$ and higher dimensional topological complexity are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distance, to some $\TC_n$-related theorems.

Equations

$$\mathrm{D}(f_{1},f_{2},\cdots,f_{n})\leq\sum_{i=0}^{k}\mathrm{D}(f_{1}\big{|}_{U_{i}},f_{2}\big{|}_{U_{i}},\cdots,f_{n}\big{|}_{U_{i}})+k.$$

$$\mathrm{D}(h_1\circ f_1,h_2\circ f_2,\cdots ,h_n\circ f_n)\le \mathrm{D}(f_1,f_2,\cdots ,f_n).$$

$$\Big{(}h_{\ell}\circ f_{\ell}\Big{)}\Big{|}_{U_{j}}\simeq h_{\ell}\circ f_{\ell}\Big{|}_{U_{j}}\simeq h_{m}\circ f_{m}\Big{|}_{U_{j}}\simeq\Big{(}h_{m}\circ f_{m}\Big{)}\Big{|}_{U_{j}}.$$

$$\mathrm{D}(f_1\circ h_1,f_2\circ h_2,\cdots ,f_n\circ h_n)\le \mathrm{D}(f_1,f_2,\cdots ,f_n).$$

$$\displaystyle\Big{(}f_{\ell}\circ h_{\ell}\Big{)}\Big{|}_{U_{j}}$$

$$\displaystyle\simeq{f_{m}}\big{|}_{U_{j}}\circ h^{\prime}_{m,j}=f_{m}\circ\iota_{j}\circ h^{\prime}_{m,j}$$

$$\displaystyle=f_{m}\circ{h_{m}}\big{|}_{U_{j}}=\Big{(}f_{m}\circ h_{m}\Big{)}\Big{|}_{U_{j}}.$$

$$\mathrm{D}(f_1,\cdots ,f_n,h_1,\cdots ,h_m)\le \mathrm{D}(f_1,\cdots ,f_n,g_{\sigma (1)},\cdots ,g_{\sigma (s)})+\mathrm{D}(g_{\beta (1)},\cdots ,g_{\beta (s^{\prime })},h_1,\cdots ,h_m)$$

$$F:U\times I\to X^{n-1},\text{ satisfying }F(x,0)=x\text{ and }F(x,1)=(x_0,\cdots ,x_0).$$

$$H(x,t)=\{\begin{array}{ll}{j}_i(F(x,2t)),& 0\le t\le \frac{1}{2}\\ j_k(F(x,2-2t)),& \frac{1}{2}\le t\le 1\end{array}$$

$$H\Big{(}x,\frac{1}{2}\Big{)}=j_{i}(F(x,1))=j_{i}(x_{0},\cdots,x_{0})=(x_{0},\cdots,x_{0})=j_{k}(F(x,1)),$$

$$F(x_{1},\cdots,x_{n-1},t)=\big{(}\mathrm{pr}_{2}(H_{1}(x,t)),\cdots,\mathrm{pr}_{i}(H_{i-1}(x,t)),\cdots,\mathrm{pr}_{n}(H_{n-1}(x,t))\big{)}$$

$$F(x,0)$$

$$\displaystyle=\big{(}\mathrm{pr}_{2}(j_{1}(x)),\cdots,\mathrm{pr}_{i}(j_{i-1}(x)),\cdots,\mathrm{pr}_{n}(j_{n-1}(x))\big{)}$$

$$=(x_1,\cdots ,x_n)$$

$$F(x,1)$$

$$=(x_0,\cdots ,x_0).$$

$$P$$

$$\displaystyle=\{(x,\gamma)\in X\times Y^{J_{n}}\ \big{|}\ \gamma_{i}(1_{i})=f_{i}(x)\text{ for all }i=1,2,\cdots,n\}.$$

$$\widetilde{H}_{1}:f_{1}\big{|}_{U_{j}}\simeq f_{2}\big{|}_{U_{j}}$$

$$\cdots$$

$$\widetilde{H}_{n-1}:f_{n-1}\big{|}_{U_{j}}\simeq f_{n}\big{|}_{U_{j}}.$$

$${H}_{1}:g\big{|}_{U_{j}}\simeq f_{1}\big{|}_{U_{j}}$$

$$\cdots$$

$${H}_{n}:g\big{|}_{U_{j}}\simeq f_{n}\big{|}_{U_{j}}.$$

$$s_j:U_j$$

$$x$$

$${\beta }_x(1_i)=H_i(x,1_i)=f_i(x)\text{ for each }i=1,2,\cdots ,n,$$

$${\gamma }_x(1_1)=f_1(x),{\gamma }_x(1_2)=f_2(x),\cdots ,{\gamma }_x(1_n)=f_n(x)$$

$${\gamma }_x(0_1)={\gamma }_x(0_2)=\cdots ={\gamma }_x(0_n).$$

$$\mathrm{D}(f_1,f_2,\cdots ,f_n)\le \mathrm{cat}(X).$$

$$P$$

$$\mathrm{pr}_{m}\big{|}_{U_{j}}=\overline{\mathrm{pr}}_{m}\big{|}_{U_{j}\cap X^{n}}\simeq\overline{\mathrm{pr}}_{\ell}\big{|}_{U_{j}\cap X^{n}}=\mathrm{pr}_{\ell}\big{|}_{U_{j}}$$

Full text

1. Introduction

The notion of homotopic distance is first introduced by Macias-Virgos and Mosquera-Lois in [6], which is a homotopy invariant whose special cases are topological complexity ($\mathrm{TC}$) and Lusternik-Schnirelmann category ($\mathrm{cat}$), and is defined as follows.

Definition 1.1**.**

[6] Given two maps $f,g:X\rightarrow Y$, the homotopic distance between $f$ and $g$ is the least non-negative integer $k$ such that one can find an open cover $\{U_{0},\cdots,U_{k}\}$ for $X$ satisfying f\big{|}_{U_{i}}\simeq g\big{|}_{U_{i}} for all $i=0,1,\cdots,k$. It is denoted by $\mathrm{D}(f,g)$. If there is no such a covering, we write $\mathrm{D}(f,g)=\infty$.

The organization of the paper is as follows:

In Section 2, we will recall the higher homotopic distance and introduce some propositions and lemmas. These lemmas will be mainly used in Section 3 and Section 4 to prove main theorems of this paper.

Motivated from the fact that homotopic distance has a relation between topological complexity and Lusternik-Schnirelmann category, we will show that $n$-th homotopic distance of some specific maps is equal to $n$-th topological complexity and Lusternik-Schnirelmann category. Moreover, we will give the relation between $n$-th homotopic distance and sectional category. Later in the same section, we will give alternative proofs of the well-known theorems about $\mathrm{TC}_{n}$.

In the last section, we will prove the homotopy invariance of higher homotopic distance and deduce that $\mathrm{TC}_{n}$ is homotopy invariant.

For a further reading about the variances of homotopic distance, we refer the interested readers to see [7] in which categorical version of homotopic distance between functors is introduced.

2. Higher Homotopic Distance and Some of Its Properties

Higher homotopic distance, as well as usual homotopic distance, was first introduced in [6] by Macias-Virgos and Mosquera-Lois. In this section, we will recall its definition and introduce some of its properties which are of importance to prove some main theorems.

Definition 2.1**.**

[6] Given $f_{i}:X\rightarrow Y$ for $i\in\{1,2,\cdots,n\}$, the $n$-th homotopic distance $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})$ is the least non-negative integer $k$ such that there exist open subsets $U_{0},U_{1},\cdots,U_{k}$ which covers $X$ and satisfy $f^{j}_{1}|_{U_{j}}\simeq f^{j}_{2}|_{U_{j}}\simeq\cdots\simeq f^{j}_{n}|_{U_{j}}$ for all $j\in\{0,1,\cdots,k\}$.

If there is no such a covering, we define $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=\infty$.

The following four propositions are direct consequences of the definition.

Proposition 2.2**.**

$\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=\mathrm{D}(f_{\sigma(1)},f_{\sigma(2)},\cdots,f_{\sigma(n)})$* holds for any permutation $\sigma$ of $\{1,2,\cdots,n\}$.*

∎

Proposition 2.3**.**

$\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=0$* iff $f_{i}\simeq f_{i+1}$ for each $i\in\{1,2,\cdots,n-1\}$.*

∎

Proposition 2.4**.**

Given maps $f_{i}:X\rightarrow Y$ and $g_{i}:X\rightarrow Y$ for $i\in\{1,2,\cdots,n\}$. If $f_{i}\simeq g_{i}$ for each $i$, then $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=\mathrm{D}(g_{1},g_{2},\cdots,g_{n})$.

∎

Proposition 2.5**.**

If $1<m<n$ and $f_{1},f_{2},\cdots,f_{m},\cdots,f_{n}:X\rightarrow Y$ are maps, then $\mathrm{D}(f_{1},f_{2},\cdots,f_{m})\leq\mathrm{D}(f_{1},f_{2},\cdots,f_{n})$.

∎

Proposition 2.6**.**

If $f_{1},f_{2},\cdots,f_{n}:X\rightarrow Y$ are maps and if $\{U_{0},U_{1},\cdots,U_{k}\}$ is any open covering of $X$, then we have

[TABLE]

Proof.

Let \mathrm{D}(f_{1}\big{|}_{U_{i}},f_{2}\big{|}_{U_{i}},\cdots,f_{n}\big{|}_{U_{i}})=m_{i} for all $i\in\{0,1,\cdots,k\}$. Then there exists an open covering $\{U^{0}_{i},U^{1}_{i},\cdots,U^{m_{i}}_{i}\}$ of $U_{i}$ such that f_{1}\big{|}_{U^{j}_{i}}\simeq f_{2}\big{|}_{U^{j}_{i}}\simeq f_{n}\big{|}_{U^{j}_{i}} for all $j\in\{0,1,\cdots,m_{i}\}$.

Notice that the collection $\mathcal{U}=\{U^{0}_{0},U^{1}_{0},\cdots,U^{m_{0}}_{0},U^{0}_{1},U^{1}_{1},\cdots,U^{m_{1}}_{1},\cdots,U^{0}_{k},U^{1}_{k},\cdots,U^{m_{k}}_{k}\}$ is an open cover for $X$ such that f_{1}\big{|}_{V}\simeq f_{2}\big{|}_{V}\simeq\cdots\simeq f_{n}\big{|}_{V} for all $V\in\mathcal{U}$. The required inequality follows from the cardinality of $\mathcal{U}$ is $(m_{0}+m_{1}+\cdots+m_{k})+k+1$. ∎

The following propositions will be used to give the main results in the third and the fourth sections.

Proposition 2.7**.**

Given maps $f_{i}:X\rightarrow Y$ and $h_{i}:Y\rightarrow Z$ for $i\in\{1,2,\cdots,n\}$. If $h_{i}\simeq h_{i+1}$ for every $i\in\{1,2,\cdots,n-1\}$, then

[TABLE]

Proof.

Suppose $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=k$. Then there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $X$ such that $f_{1}|_{U_{j}}\simeq f_{2}|_{U_{j}}\simeq\cdots\simeq f_{n}|_{U_{j}}$ for each $j\in\{0,1,\cdots,k\}$.

For each $j\in\{0,1,\cdots,k\}$ and for any distinct $\ell,m\in\{1,2,\cdots,n\}$, we have

[TABLE]

Therefore $\mathrm{D}(h_{1}\circ f_{1},h_{2}\circ f_{2},\cdots,h_{n}\circ f_{n})\leq k$.

∎

Proposition 2.8**.**

Given maps $f_{i}:X\rightarrow Y$ and $h_{i}:Z\rightarrow X$ for $i\in\{1,2,\cdots,n\}$. If $h_{i}\simeq h_{i+1}$ for every $i\in\{1,2,\cdots,n-1\}$, then

[TABLE]

Proof.

Suppose $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=k$. Then there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $X$ such that $f_{1}|_{U_{j}}\simeq f_{2}|_{U_{j}}\simeq\cdots\simeq f_{n}|_{U_{j}}$ for each $j\in\{0,1,\cdots,k\}$.

Let $V_{j}:=h^{-1}_{\ell}(U_{j})\subseteq Z$. Notice that open subsets $V_{j}$ cover $Z$. Denote by $h^{\prime}_{i,j}:V_{j}\rightarrow U_{j}$ the restriction on both domain and range, and denote by $\iota_{j}:U_{j}\xhookrightarrow{}X$ the inclusion.

Then for each $j\in\{0,1,\cdots,k\}$ and for any distinct $\ell,m\in\{1,2,\cdots,n\}$, we have

[TABLE]

So $\mathrm{D}(f_{1}\circ h_{1},f_{2}\circ h_{2},\cdots,f_{n}\circ h_{n})\leq k$. ∎

Lemma 2.9**.**

[8]** Let $\mathcal{U}=\{U_{0},U_{1},\cdots,U_{m}\}$ and $\mathcal{V}=\{V_{0},V_{1},\cdots,V_{n}\}$ be two open coverings of a normal space $X$ such that each set of $\mathcal{U}$ satisfies Property (A) and each set of $\mathcal{V}$ satisfies Property (B). If Property (A) and Property (B) are inherited by open subsets and disjoint unions, then $X$ has an open covering $\mathcal{W}=\{W_{0},W_{1},\cdots,W_{m+n}\}$ which satisfies both Property (A) and Property (B).

Theorem 2.10**.**

Let $X$ be a normal spaceand $n,m\in\mathbb{Z}^{+}$ with $n\leq m$. If $f_{1},\cdots,f_{n},g_{1},\cdots,g_{m},h_{1},\cdots,h_{m}:X\rightarrow Y$ are maps, then

[TABLE]

where $\sigma$ and $\beta$ is a permutation of $\{1,2,\cdots,m\}$ and $\{1,2,\cdots,n\}$ respectively, such that $g_{\sigma(i_{0})}\simeq g_{\beta(j_{0})}$ for some $i_{0}\in\{1,2,\cdots,m\}$ and $j_{0}\in\{1,2,\cdots,n\}$.

Proof.

Let $\mathrm{D}(f_{1},\cdots,f_{n},g_{\sigma(1)},\cdots,g_{\sigma(s)})=k_{1}$ and $\mathrm{D}(g_{\beta(1)},\cdots,g_{\beta(s^{\prime})},h_{1},\cdots,h_{m})=k_{2}$. So there exists an open covering $\mathcal{U}=\{U_{0},\cdots,U_{k_{1}}\}$ of $X$ such that f_{1}\big{|}_{U_{i}}\simeq\cdots\simeq f_{n}\big{|}_{U_{i}}\simeq g_{\sigma(1)}\big{|}_{U_{i}}\simeq\cdots g_{\sigma(s)}\big{|}_{U_{i}} for all $i=0,1,\cdots,k_{1}$. Similarly there exists an open covering $\mathcal{V}=\{V_{0},\cdots,V_{k_{2}}\}$ such that g_{\beta(1)}\big{|}_{V_{j}}\simeq\cdots g_{\sigma(s^{\prime})}\big{|}_{V_{j}}h_{1}\big{|}_{V_{j}}\simeq h_{m}\big{|}_{V_{j}} for all $j=0,1,\cdots,k_{2}$.

From the assumption, $g_{\sigma(i_{0})}\simeq g_{\beta(j_{0})}$ for some $i_{0}$ and $j_{0}$, and by Lemma 2.9, we have an open covering $\mathcal{W}=\{W_{0},\cdots,W_{k_{1}+k_{2}}\}$ of $X$ such that f_{1}\big{|}_{W_{k}}\simeq\cdots\simeq f_{n}\big{|}_{W_{k}}\simeq g_{1}\big{|}_{W_{k}}\simeq\cdots\simeq g_{m}\big{|}_{W_{k}}\simeq h_{1}\big{|}_{W_{k}}\simeq\cdots\simeq h_{m}\big{|}_{W_{k}} for all $k=0,1,\cdots,k_{1}+k_{2}$. ∎

3. $\mathrm{TC}_{n}$, $\mathrm{cat}$, $\mathrm{secat}$ of a fibration and $n$-th homotopic distance

In the first half of this section we will introduce the relation between $n$-th homotopic distance with $\mathrm{cat}$ and with $\mathrm{TC}_{n}$ and with $\mathrm{secat}$ of a fibration. In the second half, we will prove some properties of $\mathrm{TC}_{n}$ such as its relation with $\mathrm{cat}$, via higher homotopic distance.

Let us begin this section by recalling the definitions of $\mathrm{cat}$, $\mathrm{secat}$ and $\mathrm{TC}_{n}$.

Definition 3.1**.**

[3] Lusternik Schnirelmann category $\mathrm{cat}(X)$ of a space $X$ is the least integer $k\geq 0$ such that there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $X$ with the property that the inclusion on each $U_{i}$ is null-homotopic. Such an open $U_{i}$ is usually called categorical.

Definition 3.2**.**

[10] Sectional category $\mathrm{secat}(q)$ of a fibration $q:E\rightarrow B$ is the least integer $k\geq 0$ such that there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $B$ such that there is a section $s_{i}$ over $U_{i}$ for every $i=0,1,\cdots,k$.

A special case of the sectional category, $\mathrm{TC}_{n}$, is defined as follows.

Definition 3.3**.**

[9] For $n\in\mathbb{N}$, let $J_{n}$ be the wedge sum of $n$ closed intervals $[0,1]_{i}$ for $i=1,2,\cdots,n$ where the zeros $0_{i}$’s are identified. For a path-connected space $X$, denote by $X^{J_{n}}$ the space of paths with $n$-legs. Then there is a fibration $e_{n}:X^{J_{n}}\to X^{n}$ defined by $e_{n}(f)=(f(1_{1}),f(1_{2}),\cdots,f(1_{n}))$, and the sectional category (or Schwarz genus) of this fibration is called $n$-dimensional topological complexity of $X$, denoted by $\mathrm{TC}_{n}(X)$.

If there is no such a covering, we write $\mathrm{TC}_{n}(X)=\infty$.

Theorem 3.4**.**

For a fixed $x_{0}\in X$, consider the inclusions $j_{i}:X^{n-1}\hookrightarrow X^{n}$ given by $j_{i}(x_{1},\cdots,x_{n-1})=(x_{1},\cdots,x_{i-1},x_{0},x_{i},x_{i+1},\cdots,x_{n-1})$ for $i\in\{1,2,\cdots,n\}$. Then $D(j_{1},\cdots,j_{n})=\mathrm{cat}(X^{n-1})$.

Proof.

Let $U\subset X^{n-1}$ be categorical.

Our aim is to show that each $j_{i}$ restricted on $U$ is homotopic with each other for $i\in\{1,2,\cdots,n\}$. It suffices to show that for any distinct $i,k\in\{1,\cdots,n\}$, we have $j_{i}|_{U}\simeq j_{k}|_{U}$.

For convenience, we will write $x=(x_{1},\cdots,x_{n})$. Consider the homotopy

[TABLE]

Define the map $H:U\times I\rightarrow X^{n}$ by

[TABLE]

so that $H(x,0)=j_{i}(x)$, $H(x,1)=j_{k}(x)$ are satisfied. Since

[TABLE]

$H$ is continuous. Hence $\mathrm{D}(j_{1},\cdots,j_{n})\leq\mathrm{cat}(X^{n-1})$.

For the other way around, assume that we have ${j_{1}}|_{U}\simeq\cdots\simeq{j_{n}}|_{U}$, i.e., there exists a homotopy $H_{i}:U\times I\rightarrow X^{n}$ such that $H_{i}(x,0)=j_{i}(x)$ and $H_{i}(x,1)=j_{i+1}(x)$ for each $i\in\{1,\cdots,n-1\}$.

Define $F:U\times I\rightarrow X^{n-1}$ by

[TABLE]

where $\mathrm{pr}_{i}:X^{n}\rightarrow X$ is the projection maps into the $i$-th factor and which satisfies

[TABLE]

and

[TABLE]

Thus $\mathrm{cat}(X^{n-1})\leq\mathrm{D}(j_{1},\cdots,j_{n})$. ∎

Theorem 3.5**.**

Let $f_{1},f_{2},\cdots,f_{n}:X\rightarrow Y$ be maps. Consider the fibration $e_{n}:Y^{J_{n}}\rightarrow Y^{n}$ as defined in Definition 3.3. If $q:P\rightarrow X$ is the pullback of the fibration $e_{n}$ by the map $F:=(f_{1},f_{2},\cdots,f_{n}):X\rightarrow Y^{n}$, then $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=\mathrm{secat}(q)$.

[TABLE]

Proof.

Before we start proving the theorem, observe that

[TABLE]

Suppose $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=k$. Then there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $X$ such that f_{1}\big{|}_{U_{j}}\simeq f_{2}\big{|}_{U_{j}}\simeq\cdots\simeq f_{n}\big{|}_{U_{j}} for all $j\in\{0,1,\cdots,k\}$.

For each $U_{j}$, we have the homotopies

[TABLE]

[TABLE]

[TABLE]

Fix one of the homotopies, say $\widetilde{H}_{1}$. Write the function $\widetilde{H}_{1}(x,\frac{1}{2})=:g(x)$. So there are new homotopies

[TABLE]

[TABLE]

[TABLE]

Define a continuous map

[TABLE]

where $\beta_{x}:J_{n}\rightarrow Y$ defined by \beta_{x}\big{|}_{I_{i}}=H_{i} with $\beta_{x}(0_{1})=\cdots=\beta_{x}(0_{n})=g(x)$ are glued.

Since

[TABLE]

each $s_{j}$ is a section of $q$ over $U_{j}$. Hence $\mathrm{secat}(q)\leq k$.

On the other way around, if $\mathrm{secat}(q)=k$ then we have an open covering $\{U_{0},\cdots,U_{k}\}$ of $X$ such that there is a section $s_{j}:U_{j}\rightarrow P$ (i.e., $q\circ s_{j}=\mathrm{Id}_{U_{j}}$) for each $j=0,1,\cdots,k$.

Since $q$ maps to the first factor, $s_{j}:U_{j}\rightarrow P$ is defined by $s(x)=(x,\gamma_{x})$ such that $\gamma_{x}$ satisfies

[TABLE]

and

[TABLE]

Let say $\gamma_{x}(0_{i})=y_{0}=g(x)$ for some map $g:X\rightarrow Y$, for some (and all) $i$.

Each $i$-th leg gives a path, denote it by $H^{i}_{x}:I_{i}\rightarrow Y$ satisfying $H^{i}_{x}(0)=g(x)$ and $H^{j}_{x}(1_{i})=f_{i}(x)$. So over each $U_{j}$, all $f_{i}$’s are homotopic to each other. Thus $\mathrm{D}(f_{1},\cdots,f_{n})\leq k$.

∎

Corollary 3.6**.**

If $f_{1},f_{2},\cdots,f_{n}:X\rightarrow Y$ are maps with path connected spaces $X$ and $Y$, then

[TABLE]

Proof.

Suppose $\mathrm{cat}(X)=k$. Then there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $X$ such that \mathrm{Id}\big{|}_{U_{i}}\simeq c\big{|}_{U_{i}} for all $i\in\{0,1,\cdots,k\}$ where $c$ is a constant map. Let say $H_{i}$ is the homotopy between these maps for each $i$ with H_{i}(x,0)=\mathrm{Id}\big{|}_{U_{i}}(x).

[TABLE]

By the homotopy lifting property, there exists $\widetilde{H}_{i}$ such that $q\circ\widetilde{H}_{i}=H_{i}$ and H_{i}(x,0)=\mathrm{Id}\big{|}_{U_{i}} is lifted to some $\widetilde{f}_{i}$. If we choose the sections as $\widetilde{f}_{i}$ for each $i$, then we have q\circ\widetilde{f}_{i}=\mathrm{Id}\big{|}_{U_{i}} as required. Hence this shows that $\mathrm{secat}(q)\leq k$. Summarizing, we have $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=\mathrm{secat}(q)\leq\mathrm{cat}(X)$ where the equality follows from Theorem 3.5. ∎

Remark 3.7**.**

Notice that the necessary condition in Theorem 3.4 also follows from Corollary 3.6, if we take $f_{i}=j_{i}:X^{n-1}\rightarrow X^{n}$.

Corollary 3.8**.**

If $f_{1},f_{2},\cdots,f_{n}:X\rightarrow Y$ are maps, then $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})\leq\mathrm{TC}_{n}(Y)$.

Proof.

If the pullback of $e_{n}:Y^{J_{n}}\rightarrow Y^{n}$ is the fibration $q:P\rightarrow X$, then as mentioned in [6], $\mathrm{secat}(q)\leq\mathrm{secat}(e_{n})$. Hence by Theorem 3.5, $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=\mathrm{secat}(q)\leq\mathrm{TC}_{n}(Y)$. ∎

Lemma 3.9**.**

[2]** Let $X$ be a path-connected space and consider the fibration $e_{n}$ as described in Definition 3.3. Let $U\subseteq X$ be an open. Then there is a section $s:U\to X^{J_{n}}$ of $e_{n}$ if and only if each composition $f_{i}:U\xhookrightarrow{\iota}X^{n}\xrightarrow{\mathrm{pr}_{i}}X$ is homotopic to some map $g\colon U\to X$, where $\mathrm{pr}_{i}:X^{n}\rightarrow X$ is the projection to the $i$-th factor.

The following theorem whose proof follows directly from Lemma 3.9 is introduced in [6].

Theorem 3.10**.**

[6*]** If $X$ is path-connected space and each map $\mathrm{pr}_{i}:X^{n}\rightarrow X$ projects to the $i$-th factor for $i\in\{1,2,\cdots,n\}$, then $\mathrm{D}(\mathrm{pr}_{1},\mathrm{pr}_{2},\cdots,\mathrm{pr}_{n})=\mathrm{TC}_{n}(X)$.

Although Corollary 3.11 and Theorem 3.12 are already known, here we will give new alternative proofs using higher homotopic distance.

Corollary 3.11**.**

[9]** $\mathrm{TC}_{n}(X)\leq\mathrm{TC}_{n+1}(X)$.

Proof.

For convenience, let us denote the projection maps by two different notations depending on their domains, that is, $\mathrm{pr}_{i}:X^{n}\rightarrow X$ and $\overline{pr}_{i}:X^{n+1}\rightarrow X$ which project onto the $i$-th factor.

Suppose $\mathrm{TC}_{n+1}(X)=k$. By Theorem 3.10, there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $X^{n+1}$ such that \overline{\mathrm{pr}}_{1}\big{|}_{U_{j}}\simeq\overline{\mathrm{pr}}_{2}\big{|}_{U_{j}}\simeq\cdots\simeq\overline{\mathrm{pr}}_{n+1}\big{|}_{U_{j}} for all $j\in\{0,1,\cdots,k\}$.

Notice that \overline{\mathrm{pr}}_{i}\big{|}_{X^{n}}=\mathrm{pr}_{i} for any $i\in\{1,2,\cdots,n\}$. So

[TABLE]

for any distinct $m,\ell\in\{1,2,\cdots,n\}$ and for all $j\in\{0,1,\cdots,k\}$. Hence $\mathrm{D}(\mathrm{pr}_{1},\mathrm{pr}_{2},\cdots,\mathrm{pr}_{n})=\mathrm{TC}_{n}(X)\leq k$.

∎

Theorem 3.12**.**

[1]** $\mathrm{TC}_{n}(X)\leq\mathrm{cat}(X^{n})$.

Proof.

For a fixed $x_{0}\in X$, consider the inclusions

[TABLE]

for $\ell=1,2,\cdots,n$.

Let suppose $\mathrm{cat}(X^{n})=k$. Then there exists an open covering $\{U_{0},U_{1},\cdots,U_{k}\}$ of $X^{n}$ such that j_{\ell}\big{|}_{U_{i}}\simeq j_{\ell+1}\big{|}_{U_{i}} for all $i$ and for all $\ell$. Let $x=(x_{1},x_{2},\cdots,x_{n})$ and let say $F_{i}^{\ell}:U_{i}\times I\rightarrow X^{n+1}$ be the homotopy such that $F_{i}^{\ell}(x,0)=j_{\ell}(x)$ and $F_{i}^{\ell}(x,1)=j_{\ell+1}(x)$ for every $\ell$.

On each $U_{i}$, define a homotopy $H_{i}^{\ell}:U_{i}\times I\rightarrow X$ by $H_{i}^{\ell}(x,t)=(\mathrm{pr}_{\ell}\circ F_{i}^{\ell})(x,t)$ where $\mathrm{pr}_{\ell}:X^{n}\rightarrow X$ is the projection that maps onto the $\ell$-th factor. Since

[TABLE]

and

[TABLE]

for each $\ell=1,2,\cdots,n$, we have \mathrm{pr}_{1}\big{|}_{U_{i}}\simeq\mathrm{pr}_{2}\big{|}_{U_{i}}\simeq\cdots\simeq\mathrm{pr}_{n}\big{|}_{U_{i}} for all $i=0,1,\cdots,k$. Thus $\mathrm{TC}_{n}(X)=\mathrm{D}(\mathrm{pr}_{1},\mathrm{pr}_{2},\cdots,\mathrm{pr}_{n})\leq k$ by Lemma 3.10.

∎

Theorem 3.13**.**

Let $f_{1},f_{2},\cdots,f_{n}:X\rightarrow Y$ and $g_{1},g_{2},\cdots,g_{n}:\widetilde{X}\rightarrow\widetilde{Y}$ be maps. If $X\times\widetilde{X}$ is normal space, then we have

[TABLE]

Proof.

Let $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=k_{1}$ and $\mathrm{D}(g_{1},g_{2},\cdots,g_{n})=k_{2}$. So there exists an open covering $\{U_{0},\cdots,U_{k_{1}}\}$ of $X$ such that f_{1}\big{|}_{U_{i}}\simeq f_{2}\big{|}_{U_{i}}\simeq\cdots\simeq f_{n}\big{|}_{U_{i}} for all $i$ and there exists an open covering $\{V_{0},\cdots,V_{k_{2}}\}$ of $\widetilde{X}$ such that g_{1}\big{|}_{V_{j}}\simeq g_{n}\big{|}_{V_{j}}\simeq\cdots\simeq g_{n}\big{|}_{V_{j}} for all $j$. Then we have homotopies $\widetilde{F_{s}}:f_{s}\simeq f_{s+1}$ for each $s=1,2,\cdots,{n-1}$. By the argument in Theorem 3.5, we can find new homotopies between $f_{s}$ and a fixed map $f$ for all $s=1,2,\cdots,n$. Similarly, we can find new homotopies between $g_{s}$ and a fixed map $g$.

Let say that Property (A) is the following: On each $U_{i}$, there exists homotopies $f_{s}\times\mathrm{Id}_{\widetilde{X}}\simeq f\times\mathrm{Id}_{\widetilde{X}}$ for all $s=1,2,\cdots,n$. Similarly, let Property (B) be the following: On each $V_{i}$, there exists homotopies $\mathrm{Id}_{X}\times g_{s}\simeq\mathrm{Id}_{X}\times g$ for all $s$.

$\mathcal{U}=\{U_{1}\times\widetilde{X},\cdots,U_{k_{1}}\times\widetilde{X}\}$ is a covering of $X\times\widetilde{X}$ satisfying Property (A) and $\mathcal{V}=\{X\times V_{1},\cdots,X\times V_{k_{2}}\}$ is a covering of $X\times\widetilde{X}$ satisfying Property (B). So by Lemma 2.9, there exists $\mathcal{W}=\{W_{0},W_{1},\cdots,W_{k_{1}+k_{2}}\}$ which satisfies both Property (A) and Property (B). Therefore, we have the following homotopies

[TABLE]

and

[TABLE]

where $F_{m}:W_{s}\times I\rightarrow Y\times\widetilde{Y}$ and $G_{m}:W_{s}\times I\rightarrow Y\times\widetilde{Y}$ for all $m=1,2,\cdots,n$ and for $s=1,2,\cdots,k_{1}+k_{2}$.

For $m=1,2,\cdots,n$, define H_{m}:=\big{(}\mathrm{pr}_{1}\circ F_{m},\mathrm{pr}_{2}\circ G_{m}\big{)} where $\mathrm{pr}_{1}:Y\times\widetilde{Y}\rightarrow Y$ and $\mathrm{pr}_{2}:Y\times\widetilde{Y}\rightarrow\widetilde{Y}$ projection maps onto the first and the second factor, respectively. Hence for $z\in X\times\widetilde{X}$,

[TABLE]

and

[TABLE]

So each $f_{m}\times g_{m}$ is homotopic to each other for all $m=1,2,\cdots,n$ on each $W_{s}$. Thus $\mathrm{D}(f_{1}\times g_{1},f_{2}\times g_{2},\cdots,f_{n}\times g_{n})\leq k_{1}+k_{2}$. ∎

Corollary 3.14**.**

[1]** $\mathrm{TC}_{n}(X_{1}\times X_{2})\leq\mathrm{TC}_{n}(X_{1})+\mathrm{TC}_{n}(X_{2})$.

Proof.

Take the maps $\mathrm{pr}_{1}^{i},\mathrm{pr}_{2}^{i},\cdots,\mathrm{pr}_{n}^{i}:X_{i}^{n}\rightarrow X_{i}$ for $i=1,2$ where $\mathrm{pr}_{j}^{i}$ denotes the projection onto the $j$-th factor. Then

[TABLE]

∎

4. Homotopy Invariance of the Higher Homotopic Distance

Theorem 4.1**.**

Higher homotopic distance is homotopy invariance in the sense that if $X\simeq X^{\prime}$ and $Y\simeq Y^{\prime}$ homotopy equivalent spaces connecting the maps $f_{i}:X\rightarrow Y$ and $g_{i}:X^{\prime}\rightarrow Y^{\prime}$ for all $i\in\{1,2,\cdots,n\}$, then $\mathrm{D}(f_{1},f_{2},\cdots,f_{n})=\mathrm{D}(g_{1},g_{2},\cdots,g_{n})$.

Proof.

If $X\simeq X^{\prime}$, then there exist maps $\alpha:X\rightarrow X^{\prime}$ and $\beta:X^{\prime}\rightarrow X$ such that $\alpha\circ\beta\simeq\mathrm{Id}_{X^{\prime}}$ and $\beta\circ\alpha\simeq\mathrm{Id}_{X}$.

By Proposition 2.8, $\mathrm{D}(f_{1}\circ\beta,f_{2}\circ\beta,\cdots,f_{n}\circ\beta)\leq\mathrm{D}(f_{1},f_{2},\cdots,f_{n})$. On the other hand,

[TABLE]

where the inequality follows from Proposition 2.8 and the second equality follows from Proposition 2.4. So $\mathrm{D}(f_{1}\circ\beta,f_{2}\circ\beta,\cdots,f_{n}\circ\beta)=\mathrm{D}(f_{1},f_{2},\cdots,f_{n})$.

If $Y\simeq Y^{\prime}$, then there exist maps $\gamma:Y\rightarrow Y^{\prime}$ and $\eta:Y^{\prime}\rightarrow Y$ such that $\gamma\circ\eta\simeq\mathrm{Id}_{Y^{\prime}}$ and $\eta\circ\gamma\simeq\mathrm{Id}_{Y}$. In a similar way, using Proposition 2.7, one can see that $\mathrm{D}(g_{1},g_{2},\cdots,g_{n})=\mathrm{D}(\beta_{Y}\circ g_{1},\eta\circ g_{2},\cdots,\eta\circ g_{n})$. Hence

[TABLE]

∎

Corollary 4.2**.**

[9]** $\mathrm{TC}_{n}$ is homotopy invariant.