A note on covers defining relative and sectional categories

TL;DR

This paper characterizes the relative category using open covers and extends the definitions of relative and sectional categories to arbitrary covers in ANR spaces, broadening their applicability.

Contribution

It provides a new characterization of relative category via open covers and generalizes the definitions to arbitrary covers for ANR spaces.

Findings

Relative category characterized by open covers

Extension of sectional and relative categories to arbitrary covers in ANR spaces

Broader applicability of categorical concepts in topological spaces

Abstract

We give a characterization of relative category in the sense of Doeraene-El Haouari by using open covers. We also prove that relative category and sectional category can be defined by taking arbitrary covers (not necessarily open) when dealing with ANR spaces.

Equations

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$F_{n-1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$T^n(i_X)=\{(x_0,x_1,...,x_n)\in X^{n+1}:x_i\in A\text{mbox}forsomei\}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$L_i(x,t)=\{\begin{array}{ll}x,& x\in X\setminus B_i\\ H_i(x,t{h}_i(x)),& x\in U_i\end{array}$$

$${\phi }_i:=g_i\circ f_i:X\to X$$

$$G_i(x,t)=\{\begin{array}{ll}{L}_i^{\prime }(x,2t),& 0\le t\le \frac{1}{2}\\ H({\phi }_i(x),2t-1),& \frac{1}{2}\le t\le 1\end{array}$$

$$\Omega :=U_0\cap (X\setminus \overline{V_1})\cap ...\cap (X\setminus \overline{V_n}))$$

$$H_i^{\prime }(x,t)=\{\begin{array}{ll}H(x,t),& x\in \Omega \\ H_i(x,t),& x\in V_i\end{array}$$

$$...⟶H^{n_i}(X,U_i)\stackrel{q_i^{\ast }}{⟶}{H}^{n_i}(X,A)\stackrel{k_i^{\ast }}{⟶}{H}^{n_i}(U_i,A)⟶...$$

$$(U_i,A)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$x_0\cup x_1\cup ...\cup x_n=q_0^{\ast }({\bar{x}}_0)\cup q_1^{\ast }({\bar{x}}_1)\cup ...\cup q_n^{\ast }({\bar{x}}_n))=q^{\ast }({\bar{x}}_0\cup {\bar{x}}_1\cup ...\cup {\bar{x}}_n)$$

$$\text{mbox}nil{H}^{\ast }(X,A)=\text{mbox}nil{H}^{\ast }(X/A)=\text{mbox}cuplength(X/A)$$

$$A_i\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\text{mbox}{cat}_g(X)=\text{mbox}{secat}_g(\ast \stackrel{x_0}{⟶}X)$$

$$\text{mbox}{cat}_g(X)=\text{mbox}{secat}_g(p:P(X)\to X)$$

$$E\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\ast \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\text{mbox}{TC}_g(X):=\text{mbox}{secat}_g({\pi }_X:X^I\to X\times X)$$

$$\text{mbox}{cat}_g(X)\le \text{mbox}{TC}_g(X)\le \text{mbox}{cat}_g(X\times X)$$

$$\mathcal{V}=\{B(y,\epsilon (y)):y\in Y\}$$

$$\max \{d(g(u),f(a_u)),d(u,a_u))\}<\epsilon (f(a_u))$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$p(s^{\prime }(u))\in p(B(s(a_u),\delta (s(a_u))))\subseteq B(a_u,\epsilon (a_u))$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$F(x,t)=\{\begin{array}{ll}G(x,2t),& 0\le t\le \frac{1}{2}\\ {\beta }^{\prime }(H(\beta (x),2t-1)),& \frac{1}{2}\le t\le 1\end{array}$$

Full text

2010 Mathematics Subject Classification : 55M30, 55M15, 55P99.

Keywords : Relative category, sectional category, Lusternik-Schnirelmann category, topological complexity

Introduction

The aim in this paper is twofold. On the one hand we establish a characterization of relative category by means of open covers. Relative category of a map $f$, written $\mbox{relcat}(f),$ was introduced in [2] by J.P. Doeraene and M. El Haouari in order to give a relative version of sectional category [13] (also called Schwarz genus). However, unlike sectional category, relative category is not defined by means of open covers but, instead, directly by the join construction, that is, the homotopy pullback of the homotopy pushout of two maps having the same target. However, considering in this paper covers by relatively sectional open subsets in a given cofibration $i_{X}:A\hookrightarrow X$ we can establish another numerical homotopy invariant, denoted as $\mbox{relcat}^{op}(i_{X})$ in such a way it agrees with $\mbox{relcat}(i_{X})$ under non very restrictive conditions. In this sense, in the first section of this paper we prove the following theorem:

Theorem. Let $i_{X}:A\hookrightarrow X$ be a cofibration where $X$ is a normal space. Then $\mbox{relcat}(i_{X})=\mbox{relcat}^{op}(i_{X}).$

As a consequence of this result we are able to prove that Fadell-Huseini’s relative category [3] is an upper bound of Doeraene-El Haouari’s relative category. We can also obtain a certain cohomological lower bound.

On the other hand, our second objective in this paper is to establish generalized versions to both sectional category and relative category. Indeed, in [11, 12] T. Srinivasan defined the notion of generalized Lusternik-Schnirelmann category of a space $X,$ $\mbox{cat}_{g}(X),$ as the least nonnegative integer $n$ such that $X$ admits a cover constituted by $n+1$ subsets (not necessarily open) which are contractible in $X.$ It is a numerical homotopy invariant which agrees with $\mbox{cat}(X)$, the usual notion of Lusternik-Schnirelmann category, provided $X$ is an ANR. Following the same spirit as in Srinivasan’s work, in the second section of the paper we first extend this notion to sectional category, with the so-called generalized sectional category, denoted as $\mbox{secat}_{g}(-),$ establishing that it agrees with the usual sectional category when dealing with ANR spaces:

Theorem. Let $p:E\rightarrow B$ be a fibration between ANR spaces. Then $\mbox{secat}_{g}(p)=\mbox{secat}(p).$

This result has a nice consequence involving the generalized topological complexity. In this sense, when $X$ has the homotopy type of a CW-complex, it is proved that the topological complexity of $X$ may be defined by using a general cover of $X\times X,$ not necessarily open. Another consequence is that we can recover T. Srinivasan’s result $\mbox{cat}(X)=\mbox{cat}_{g}(X)$ when $X$ is an ANR.

Finally, using the open cover characterization of relative category given in the first section we also introduce a generalized version of $\mbox{relcat}(-)$, denoted as $\mbox{relcat}_{g}(-),$ proving:

Theorem. Let $i_{X}:A\hookrightarrow X$ be a cofibration between ANR spaces. Then $\mbox{relcat}(i_{X})=\mbox{relcat}_{g}(i_{X}).$

As a consequence, we can compare the monoidal topological complexity with its generalized counterpart, establishing that they agree whenever we are considering ANR spaces.

1 Doeraene-El Haouari’s relative category defined by open covers

We begin in this section by recalling the notion of relative category of a map [2]. As we have commented in the introduction, relative category is defined by using the join construction, in particular Ganea maps. In general, the join of two maps $f:X\rightarrow Z$ y $g:Y\rightarrow Z$, $X*_{Z}Y,$ is given by the homotopy pushout of the homotopy pullback of $f$ and $g:$

[TABLE]

where the dotted arrow, called join map, is the corresponding co-whisker map, which is induced by the weak universal property of the homotopy pushout. Now, if $i_{X}:A\rightarrow X$ is a map, then $n$-th Ganea map of $i_{X}$, $g_{n}:G_{n}(X)\rightarrow X,$ is the join map inductively defined in the following join construction ($n\geq 0$):

[TABLE]

where $g_{0}:=i_{X}:A\rightarrow X.$ Then the relative category of $i_{X}$, denoted $\mbox{relcat}(i_{X})$, is defined as the least nonnegative integer $n$ such that $g_{n}:G_{n}(X)\rightarrow X$ admits a homotopy section $\sigma:X\rightarrow G_{n}(X)$ satisfying that $\sigma\circ i_{X}\simeq\alpha_{n}.$ It is important to note that relative category has a Whitehead characterization. Indeed, for all $n$ we can also consider the $n$-th sectional fat-wedge $t_{n}:T^{n}(i_{X})\rightarrow X^{n+1}$ inductively defined as follows: For $n=0$ set $T^{0}(i_{X}):=A$ and $t_{0}=i_{X}:A\rightarrow X$. If $t_{n-1}:T^{n-1}(i_{X})\rightarrow X^{n}$ is already given, then $t^{n}$ is the join map

[TABLE]

Moreover, there exists a homotopy commutative diagram of the form:

[TABLE]

J.P. Doeraene and El Haouari prove the following characterization of relative category:

Theorem 1.1**.**

[2, Prop. 26]** Let $i_{X}:A\rightarrow X$ be a map. Then $\mbox{relcat}(i_{X})\leq n$ if and only if there exists a map $f:X\rightarrow T^{n}(i_{X})$ making commutative, up to homotopy, the following diagram

[TABLE]

For our purposes of establishing an open cover characterization of relative category we are specially interested in cofibrations. Throughout this paper by a cofibration we will mean a closed map having the homotopy extension property. The main reason of using cofibrations is that, in this case, we have a nice explicit description of the $n$-th sectional fat-wedge:

Proposition 1.2**.**

[5*, Cor. 11]** *Let $i_{X}:A\hookrightarrow X$ be a cofibration. Then the $n$-th sectional fat-wedge $t_{n}:T^{n}(i_{X})\hookrightarrow X^{n+1}$ is, up to homotopy equivalence, the following subspace of $X^{n+1}$:

[TABLE]

$t_{n}$ being the canonical inclusion ($t_{n}$ is actually, a cofibration).**

Moreover, one can easily check that $\tau_{n}:A\rightarrow T^{n}(i_{X})$ is defined, up to homotopy equivalence, as $\tau_{n}(a)=(a,a,...,a).$ If $\Delta_{n+1}:X\rightarrow X^{n+1}$ denotes the $(n+1)$-th diagonal map, then there exists a strictly commutative diagram:

[TABLE]

Now we give our definition of relative category by means of open covers. As we have previously said, we will consider cofibrations.

Definition 1.3**.**

Let $i_{X}:A\hookrightarrow X$ be a cofibration. We say that a subset $U\subseteq X$ is relatively sectional if $A\subseteq U$ and there exists a homotopy of pairs $H:(U\times I,A\times I)\rightarrow(X,A)$ satisfying that $H(x,0)=x$ and $H(x,1)\in A$ for all $x\in U.$ Then we define $\mbox{relcat}^{op}(i_{X})$ as the least nonnegative integer $n$ such that $X$ admits a cover constituted by $n+1$ relatively sectional open subsets. If such an integer does not exist then we simply set $\mbox{relcat}^{op}(i_{X})=\infty.$

Remark 1.4*.*

We can obviously define $\mbox{relcat}^{op}(i_{X})$ when $i_{X}$ is a general map by considering the cofibration that approximates $i_{X}.$ However, for the sake of simplicity, in this paper we have preferred to work directly with cofibrations.

Now we see the relationship between $\mbox{relcat}^{op}$ and relative category in the sense of Doeraene-El Haouari. For this task we will use the following lemma, whose proof can be found, for instance, in [15], and therefore is omitted:

Lemma 1.5**.**

Let $j:A\hookrightarrow X$ be a cofibration and $f:X\rightarrow X$ a map such that $f\circ j=j$ and $f\simeq 1_{X}.$ Then there exists a map $g:X\rightarrow X$ satisfying $g\circ j=j$ and $g\circ f\simeq 1_{X}$ rel. $A.$**

Now we are in conditions to state and prove the main result in this section.

Theorem 1.6**.**

If $i_{X}:A\hookrightarrow X$ is a cofibration where $X$ is a normal space, then $\mbox{relcat}(i_{X})=\mbox{relcat}^{op}(i_{X}).$ **

Proof.

Suppose that $\mbox{relcat}^{op}(i_{X})\leq n$ and take $\{U_{i}\}_{i=0}^{n}$ a cover of $X$ by relatively sectional open subsets. Also consider homotopies $H_{i}:(U_{i}\times I,A\times I)\rightarrow(X,A)$ such that $H_{i}(x,0)=x$ and $H_{i}(x,1)\in A.$ As $X$ is normal we can take closed subsets $A_{i},B_{i}$ and open subsets $\Theta_{i}$ satisfying that $A_{i}\subseteq\Theta_{i}\subseteq B_{i}\subseteq U_{i}$ where $\{A_{i}\}_{i=0}^{n}$ is a cover of $X.$ There are also Urysohn maps $h_{i}:X\rightarrow[0,1]$ such that $h_{i}(A_{i})=\{1\}$ and $h_{i}(X\setminus\Theta_{i})=\{0\}.$ We define $L_{i}:(X\times I,A\times I)\rightarrow(X,A)$ as

[TABLE]

and take $L:(X\times I,A\times I)\rightarrow(X^{n+1},T^{n}(i_{X}))$ as $L=(L_{0},L_{1},...,L_{n}).$ Defining $f(x):=L(x,1)$, we obtain a map $f:X\rightarrow T^{n}(i_{X})$ satisfying $L|_{A\times I}:\tau_{n}\simeq f\circ i_{X}$ and $L:\Delta_{n+1}\simeq t_{n}\circ f$. Therefore, by Theorem 1.1, $\mbox{relcat}(i_{X})\leq n.$

Conversely, suppose that $\mbox{relcat}(i_{X})\leq n$ and take, again by Theorem 1.1, a map $f:X\rightarrow T^{n}(i_{X})$ satisfying $f\circ i_{X}\simeq\tau_{n}$ and $t_{n}\circ f\simeq\Delta_{n+1}.$ As $i_{X}$ is a cofibration we can suppose, without losing generality, that $f\circ i_{X}=\tau_{n}$ and $t_{n}\circ f\simeq\Delta_{n+1}.$ Take a homotopy $L:t_{n}\circ f\simeq\Delta_{n+1}$ and consider $t_{n}\circ f=(f_{0},...,f_{n})$ and $L=(L_{0},...,L_{n})$ with $f_{i}:X\rightarrow X$ and $L_{i}:X\times I\rightarrow X$ for all $i\in\{0,1,...,n\}.$ Observe that $f_{i}\circ i_{X}=i_{X}$ and $L_{i}:f_{i}\simeq 1_{X}.$ Hence, by Lemma 1.5 above, we can find a map $g_{i}:X\rightarrow X$ such that $g_{i}\circ i_{X}=i_{X}$ and a homotopy $L^{\prime}_{i}:1_{X}\simeq g_{i}\circ f_{i}$ rel $A.$ We establish

[TABLE]

Taking into account that $(\varphi_{0}(x),...,\varphi_{n}(x))\in T^{n}(i_{X})$ for all $x\in X,$ we obtain $\varphi:X\rightarrow T^{n}(i_{X})$ such that $t_{n}\circ\varphi=(\varphi_{0},...,\varphi_{n}).$ Obviously, $\varphi\circ i_{X}=\tau_{n}$ and $L^{\prime}=(L^{\prime}_{0},...,L^{\prime}_{n})$ is a homotopy $L^{\prime}:\Delta_{n+1}\simeq t_{n}\circ\varphi$ rel $A.$

Now, using the fact that $i_{X}$ is a cofibration, there exists $A\subseteq U\subseteq X,$ an open neighbourhood of $A$ such that $A$ is a strong deformation rectract of $U$. Take $r:U\rightarrow A$ the corresponding retraction and $H:U\times I\rightarrow X$ a homotopy verifying $H(x,0)=x,$ $H(x,1)=r(x)$ and $H(a,t)=a.$ Defining $U_{i}:=\varphi_{i}^{-1}(U)$ we have that $\{U_{i}\}_{i=0}^{n}$ is an open cover of $X$ in such a way that each $U_{i}$ contains $A.$ Finally, the homotopy $G_{i}:(U_{i}\times I,A\times I)\rightarrow(X,A)$ defined as

[TABLE]

gives $G_{i}(x,0)=x$ and $G_{i}(x,1)\in A.$ This means that each $U_{i}$ is relatively sectional and therefore $\mbox{relcat}^{op}(i_{X})\leq n.$ ∎

In actual fact, taking a careful reading of the proof above, we have the following characterizations of relative category:

Corollary 1.7**.**

Let $i_{X}:A\hookrightarrow X$ be a cofibration, where $X$ is a normal space. Then the following statements are equivalent:

1. (i)

$\mbox{relcat}(i_{X})\leq n$. 2. (ii)

$\mbox{relcat}^{op}(i_{X})\leq n$. 3. (iii)

$X$* admits an open cover $U_{0},U_{1},...,U_{n}$ such that for all $i$, $A\subseteq U_{i}$ and there exists a homotopy $H_{i}:U_{i}\times I\rightarrow X$ verifying $H_{i}(x,0)=x,$ $H_{i}(x,1)\in A$ and $H_{i}(a,t)=a$, for all $x\in X,a\in A$ and $t\in I.$* 4. (iv)

$X$* admits an open cover $U_{0},U_{1},...,U_{n}$ such that for all $i$, $A\subseteq U_{i}$ and there exists a homotopy $H_{i}:U_{i}\times I\rightarrow X$ verifying $H_{i}(x,0)=x,$ $H_{i}(x,1)\in A$ and $H_{i}(a,1)=a$, for all $x\in X,a\in A$.*

Proof.

By the theorem above, (i) and (ii) are equivalent. From the second part of the proof of the theorem we deduce that both parts are also equivalent to (iii). Moreover, (iii) clearly implies (iv). Finally, if $X$ admits an open cover as in part (iv), then we can consider closed subsets $A_{i},B_{i}$ and open subsets $\Theta_{i}$ satisfying $A\subseteq A_{i}\subseteq\Theta_{i}\subseteq B_{i}\subseteq U_{i}$ and $\{A_{i}\}_{i=0}^{n}$ covers $X.$ A similar argument to that done in the firts part of the proof in the theorem above shows that $\mbox{relcat}(i_{X})\leq n.$ ∎

An interesting consequence we obtain of this characterization is the relationship between Doeraene-El Haouari’s relative category and Fadell-Husseini’s relative category [3] of a cofibred pair $(X,A).$ Recall that given $A\hookrightarrow X$ a cofibration, the Fadell-Husseini relative category of $(X,A)$, denoted as $\mbox{cat}^{FH}(X,A),$ is the least nonnegative integer $n$ such that $X$ admits a cover constituted by $n+1$ open subsets $\{U_{i}\}_{i=0}^{n}$ satisfying:

1. (i)

$A\subseteq U_{0}$ and there exists a homotopy of pairs $H:(U_{0}\times I,A\times I)\rightarrow(X,A)$ such that $H(x,0)=x$ and $H(x,1)\in A$ for all $x\in U_{0}$ (that is, $U_{0}$ is a relatively sectional open subset). 2. (ii)

For all $i\geq 1,$ $U_{i}$ is contractible in $X$.

We obtain the following inequality:

Proposition 1.8**.**

Let $i_{X}:A\hookrightarrow X$ be a cofibration where $X$ is normal and path-connected. Then, $\mbox{relcat}(i_{X})\leq\mbox{cat}^{FH}(X,A).$**

Proof.

Suppose that $\mbox{cat}_{FH}(X,A)=n$ and consider $\{U_{i}\}_{i=0}^{n}$ an open cover $X$ such that:

1. (i)

$A\subseteq U_{0}$ and there exists a homotopy of pairs $H:(U_{0}\times I,A\times I)\rightarrow(X,A)$ such that $H(x,0)=x$ and $H(x,1)\in A$ for all $x\in U_{0}.$ 2. (ii)

For all $i\geq 1,$ $U_{i}$ is contractible in $X$.

As $X$ is path-connected we can take a homotopy $H_{i}:U_{i}\times I\rightarrow X$ with $H_{i}(x,0)=0$ and $H_{i}(x,1)=a_{0}\in A$ for all $x\in U_{i}$ and $i\geq 1.$ We can also suppose, without losing generality, that $U_{i}\cap A=\emptyset$ for all $i\geq 1$ (otherwise we take the open cover $\{U_{0},U_{1}\setminus A,...,U_{n}\setminus A\}$ of $X$). By normality, we take an open cover $\{V_{i}\}_{i=0}^{n}$ satisfying $V_{i}\subseteq\overline{V_{i}}\subseteq U_{i}$ for all $i\geq 0.$ Then we can define

[TABLE]

which is an open neighborhood of $A$ such that $\Omega\cap V_{i}=\emptyset$ for all $i\geq 1.$ If $U^{\prime}_{0}:=U_{0}$ and $U^{\prime}_{i}:=\Omega\cup V_{i}$ for $i\geq 1,$ then it is not difficult to check that $\{U^{\prime}_{i}\}_{i=0}^{n}$ is an open cover of $X.$ Moreover, for $i\geq 1$ the homotopy $H^{\prime}_{i}:(U^{\prime}_{i}\times I,A\times I)\rightarrow(X,A)$ defined as

[TABLE]

satisfies $H^{\prime}_{i}(x,0)=x$ and $H^{\prime}_{i}(x,1)\in A.$ This fact implies that each $U^{\prime}_{i}$ is a relatively sectional open subset and therefore $\mbox{relcat}(i_{X})=\mbox{relcat}^{op}(i_{X})\leq n.$ ∎

Remark 1.9*.*

Observe that, actually, the proof above also works when the pair $(X,A)$ is 0-connected, instead of requiring path-connectedness for $X.$

Another interesting consequence is given by the nilpotence of the relative cohomology.

Proposition 1.10**.**

If $i_{X}:A\hookrightarrow X$ is a cofibration with $X$ a normal space, then $\mbox{nil}\hskip 2.0ptH^{*}(X,A)\leq\mbox{relcat}(i_{X}).$**

Proof.

Suppose $\mbox{relcat}(i_{X})=n$ and take $\{U_{i}\}_{i=0}^{n}$ a cover of $X$ constituted by $n+1$ relatively sectional open subsets, and consider $x_{0},x_{1},...,x_{n}\in H^{*}(X,A).$ For all $i$, if $x_{i}\in H^{n_{i}}(X,A),$ also consider the following part of the long exact sequence in cohomology of the triple $(X,U_{i},A)$

[TABLE]

where $q_{i}:(X,A)\hookrightarrow(X,U_{i})$ and $k_{i}:(U_{i},A)\hookrightarrow(X,A)$ denote the respective inclusions. Taking into account the existence of homotopy (of pairs) commutative diagrams of the form

[TABLE]

we have that $k_{i}^{*}=0$ so each $q_{i}^{*}$ is surjective and we can pick $\bar{x}_{i}\in H^{n_{i}}(X,U_{i})$ such that $q_{i}^{*}(\bar{x}_{i})=x_{i}$ for all $i\geq 0.$ But then

[TABLE]

where $q:(X,A)\hookrightarrow(X,\bigcup_{i=0}^{n}U_{i})$ denotes the natural inclusion. But $\{U_{i}\}_{i=0}^{n}$ is a cover of $X$, so $q^{*}=0$ and $x_{0}\cup x_{1}\cup...\cup x_{n}=0.$ ∎

Remark 1.11*.*

We point out that

[TABLE]

On the other hand, there exists a similar lower bound for $\mbox{secat}(i_{X}).$ Indeed, it is well-known that, if $i_{X}^{*}:H^{*}(X)\rightarrow H^{*}(A)$ denotes the homomorphism induced in cohomology, then $\mbox{nil}\hskip 2.0pt\mbox{ker}\hskip 2.0pt(i_{X}^{*})\leq\mbox{secat}(i_{X})$. It is natural to ask whether these two notions $\mbox{nil}\hskip 2.0pt\mbox{ker}\hskip 2.0pt(i_{X}^{*})$ and $\mbox{nil}\hskip 2.0ptH^{*}(X,A),$ are related or not. In [5, Th.21 (e)] we found an answer to this question: If $i_{X}$ has a homotopy retraction, then $\mbox{nil}\hskip 2.0pt\mbox{ker}\hskip 2.0pt(i_{X}^{*})=\mbox{cuplength}\hskip 2.0pt(X/A)=\mbox{nil}\hskip 2.0ptH^{*}(X,A).$

2 General covers

As we have previously commented, T. Srinivasan defined in [11, 12] the notion of generalized Lusternik-Schnirelmann category of a space $X,$ $\mbox{cat}_{g}(X),$ as the least nonnegative integer $n$ (or infinity) such that $X$ admits a cover constituted by $n+1$ subsets (not necessarily open) which are contractible in $X.$ It is a numerical homotopy invariant which agrees with $\mbox{cat}(X)$ when $X$ is an ANR. Our aim in this section is to extend this notion to both sectional and relative categories.

2.1 Generalized sectional category

Recall that the sectional category of a fibration $p:E\rightarrow B$, $\mbox{secat}(p),$ is defined as the least nonnegative integer $n$ (or infinity) such that $B$ admits a cover constituted by $n+1$ open subsets on each of which there exists a continuous local section of $p$. This is a numerical homotopy invariant. Requiring homotopy section instead of section permits one to extend the notion of sectional category to any map. This is a variant of Lusternik-Schnirelmann category (or L.S category, for short) and also a generalization, since $\mbox{secat}(p)=\mbox{cat}(B)$ when $E$ is a contractible space. The sectional category was introduced by Schwarz [13], under the name genus and then renamed by James [8].

We begin this subsection by establishing the notion of generalized sectional category.

Definition 2.1**.**

Let $p:E\rightarrow B$ be a fibration. The generalized sectional category of $p$, denoted as $\mbox{secat}_{g}(p),$ is the least nonnegative integer $n$ (or $\infty$) such that $B$ admits a cover by $n+1$ subsets $A_{0},...,A_{n}$ on each of which there exists a continuous local section of $p$, that is, a commutative diagram of the form

[TABLE]

We clearly have the inequality $\mbox{secat}_{g}(p)\leq\mbox{secat}(p).$ In addition, it is possible to extend this definition to any map $f:E\rightarrow B$, not necessarily a fibration, by just requiring continuous local sections, up to homotopy. When $f$ is a fibration both notions agree.

Generalized sectional category is a homotopy invariant in the next sense. The proof of the result we are going to state is omitted as it is just a simple repetition of the classical case.

Proposition 2.2**.**

Consider the following homotopy commutative diagram, where $\alpha$ and $\beta$ are homotopy equivalences:

[TABLE]

Then $\mbox{secat}_{g}(f)=\mbox{secat}_{g}(f^{\prime}).$

The first example of generalized sectional category is the generalized L.-S. category. As in the classical case, in order to establish $\mbox{cat}_{g}(X)$ when $X$ is a path-connected space with base point $x_{0}\in X$, we can suppose that every subset $A_{i}$ is contractible in $X$ to the point $x_{0}.$ This means that the inclusion map $A_{i}\hookrightarrow X$ is homotopic to the constant map at $x_{0}.$ Therefore, whenever we have a path-connected pointed space $X$, $\mbox{cat}_{g}(X)$ can be defined as

[TABLE]

where $x_{0}:*\rightarrow X$ denotes the map carrying $*$ to $x_{0}.$ In particular, taking the path fibration $p:P(X)\rightarrow X$, where $P(X)=\{\alpha\in X^{I}:\alpha(0)=x_{0}\}$ and $p(\alpha)=\alpha(1)$, we have by Proposition 2.2 that

[TABLE]

A basic property, relating the generalized versions of sectional category with L.-S. category, is given in the next proposition. Observe that in any homotopy commutative diagram of the form

[TABLE]

we have the inequality $\mbox{secat}_{g}(f^{\prime})\leq\mbox{secat}_{g}(f).$ Indeed, if $A$ is a subset of $B$ together with $s:A\rightarrow E$ a local homotopy section of $f$, then, we clearly have that $s^{\prime}:=\alpha\circ s:A\rightarrow E^{\prime}$ is a local homotopy section of $f^{\prime}.$

Proposition 2.3**.**

Let $f:X\rightarrow Y$ be a map where $Y$ is a path-connected based space. Then $\mbox{secat}_{g}(f)\leq\mbox{cat}_{g}(Y).$ Moreover, if $X$ is contractible, then $\mbox{secat}_{g}(f)=\mbox{cat}_{g}(Y).$

Proof.

By the path-connectedness of $Y$, the following diagram is commutative, up to homotopy:

[TABLE]

so we can apply the argument above. Also observe that whenever $X$ is contractible, $*\rightarrow X$ is a homotopy equivalence so we can apply Proposition 2.2. ∎

Another interesting example of generalized sectional category is the generalized topological complexity of a space $X$. It is simply defined as

[TABLE]

where $\pi_{X}$ denotes the path fibration, given as $\pi_{X}(\alpha)=(\alpha(0),\alpha(1)).$ Again, as $\pi_{X}$ is the fibration that approximates the diagonal map $\Delta_{X}:X\rightarrow X\times X,$ by Proposition 2.2, we have $\mbox{TC}_{g}(X)=\mbox{secat}_{g}(\Delta_{X}:X\rightarrow X\times X).$ Also note by the same proposition that $\mbox{TC}_{g}$ is a homotopy invariant. Moreover:

Proposition 2.4**.**

Let $X$ be a path-connected space. Then

[TABLE]

Proof.

By Proposition 2.3, $\mbox{TC}_{g}(X)\leq\mbox{cat}_{g}(X\times X)$ so it only remains to prove the inequality $\mbox{cat}_{g}(X)\leq\mbox{TC}_{g}(X).$ In order to prove this one just has to follow the same proof as in the classical case (see [4]). ∎

Now we prove that generalized sectional category agrees with sectional category of a fibration provided we are dealing with absolute neighborhood retracts (for metric spaces). Recall that an absolute neighborhood retract, or ANR, is a metric space $Y$ such that for any metric space $X$ and closed subset $A$, any continuous map $f:A\rightarrow Y$ has an extension $\tilde{f}:U\rightarrow Y$ for some neighborhood $U$ of $A\subseteq X.$

In order to fulfill our purposes we first recall two crucial results about ANR spaces. Such results are also used in [11, 12]. Given $\mathcal{U}=\{U_{\lambda}\}_{\lambda\in\Lambda}$ a cover of a space $Y$, we say that two maps $f,g:X\rightarrow Y$ are $\mathcal{U}$-close, if for any $x\in X$ there exists $\lambda\in\Lambda$ such that $f(x)$ and $g(x)$ belong to $U_{\lambda}.$

Lemma 2.5**.**

[6]** Let $Y$ be an ANR space. Then there exists a function (not necessarily continuous) $\varepsilon:Y\rightarrow(0,+\infty)$ such that the cover

[TABLE]

satisfies that any pair of $\mathcal{V}$-close maps $f,g:X\rightarrow Y$ are homotopic.

The other result we need is a version of a lemma that appears in [14]. Observe that our statement is slightly different to the one given in T. Srinivasan’s work. However, its proof only needs very small straightforward modifications of the one given in [12]. This way it actually consists of an almost identical argument so we have decided to omit it.

Lemma 2.6**.**

Let $X$ be a metric space, $A\subseteq X$, $K$ an ANR space and $f:A\rightarrow K$ a map. Then, for any function $\varepsilon:K\rightarrow(0,+\infty)$ (not necessarily continuous) there exist an open neighborhood $U\supseteq A$ in $X$ and a map $g:U\rightarrow K$ such that:

1. (i)

For all $u\in U$ there is $a_{u}\in A$ satisfying

[TABLE] 2. (ii)

$g_{|A}\simeq f.$**

Now we are in conditions to state and prove one of the main results of this section.

Theorem 2.7**.**

Let $p:E\rightarrow B$ be a fibration where $E$ and $B$ are ANR spaces. Then $\mbox{secat}_{g}(p)=\mbox{secat}(p).$

Proof.

It only remains to check the inequality $\mbox{secat}(p)\leq\mbox{secat}_{g}(p)$. As $B$ is an ANR, by Lemma 2.5 there exists a function $\varepsilon:B\rightarrow(0,+\infty)$ such that the cover $\mathcal{W}=\{B(b,\varepsilon(b)):b\in B\}$ of $B$ satisfies that any pair of $\mathcal{W}$-close maps $f,g:Z\rightarrow B$ are homotopic. By continuity of $p:E\rightarrow B$, for each $e\in E$ pick $\delta(e)>0$ such that $p(B(e,\delta(e)))\subseteq B(p(e),\varepsilon(p(e))).$ Then we define the funcion $\bar{\varepsilon}:E\rightarrow(0,+\infty)$ as $\bar{\varepsilon}(e)=\min\{\delta(e),\varepsilon(p(e))\}.$ Now suppose we have a subset $A\subseteq B$ together with a strict local section $s:A\rightarrow E$ of $p:$

[TABLE]

By Lemma 2.6 there exist an open neighborhood $U\supseteq A$ in $B$ and a map $s^{\prime}:U\rightarrow E$ satisfying that $s^{\prime}_{|A}\simeq s$ and for all $u\in U$ there exists $a_{u}\in A$ such that $\max\{d(s^{\prime}(u),s(a_{u})),d(u,a_{u}))\}<\bar{\varepsilon}(s(a_{u}))$. If $inc_{U}:U\hookrightarrow B$ denotes the canonical inclusion, then we want to prove that $p\circ s^{\prime}$ and $inc_{U}:U\rightarrow B$ are $\mathcal{W}$-close. Indeed, given $u\in U$ we have that $d(s^{\prime}(u),s(a_{u}))<\bar{\varepsilon}(s(a_{u}))\leq\delta(s(a_{u}))$; in other words, $s^{\prime}(u)\in B(s(a_{u}),\delta(s(a_{u})))$, so that

[TABLE]

that is, $p(s^{\prime}(u))\in B(a_{u},\varepsilon(a_{u}))$. Moreover, $d(u,a_{u})<\bar{\varepsilon}(s(a_{u}))\leq\varepsilon(p(s(a_{u})))=\varepsilon(a_{u}),$ so $u\in B(a_{u},\varepsilon(a_{u})).$ This proves that $p\circ s^{\prime}$ and $inc_{U}:U\rightarrow B$ are $\mathcal{W}$-close and therefore $p\circ s^{\prime}\simeq inc_{U}.$ Finally, as $p$ is a fibration we can obtain a map $s^{\prime\prime}:U\rightarrow E$ such that $p\circ s^{\prime\prime}=inc_{U}.$ Using this argument for covers we conclude the proof. ∎

As a consequence of this theorem we obtain the next results:

Corollary 2.8**.**

Let $X$ be a space having the homotopy type of a CW-complex (equivalently, of an ANR). Then $\mbox{TC}(X)=\mbox{TC}_{g}(X).$

Proof.

As $\mbox{TC}_{g}(-)$ y $\mbox{TC}(-)$ are homotopy invariants, we can suppose that $X$ is an ANR space. In this case it is well-known that $X\times X$ y $X^{I}$ are ANR spaces (see [6]). Then, applying Theorem 2.7 above to the fibration $\pi:X^{I}\rightarrow X\times X$ we conclude the proof. ∎

Remark 2.9*.*

Another interesting example of TC-like numerical homotopy invariant agreeing with its generalized version (i.e., using general covers) is given by the higher analogs of topological complexity $\mbox{TC}_{m}(-)$ [9].

Finally, we also recover an already known result, prove by T. Srinivasan [11, 12]:

Corollary 2.10**.**

Let $X$ be a space having the homotopy type of a connected CW-complex (equivalently, of a connected ANR space). Then $\mbox{cat}(X)=\mbox{cat}_{g}(X).$

Proof.

As $P(X)$ is an ANR space, one has just to follow for the path fibration $p:P(X)\rightarrow X$ a similar argument to that done in the previous corollary. ∎

2.2 Generalized relative category

We finish our study with the generalized relative category in the sense of Doeraene-El Haouari. Recall from Corollary 1.7, that for a given cofibration $i_{X}:A\hookrightarrow X$ where $X$ is a normal space, $\mbox{relcat}(i_{X})$ can be defined as the least nonnegative integer $n$ (or infinity) such that $X$ admits a cover by $n+1$ relatively sectional open subsets. This way, we can straightforwardly define the generalized relative category, as follows:

Definition 2.11**.**

Let $i_{X}:A\hookrightarrow X$ be a cofibration. The generalized relative category of $i_{X},$ $\mbox{relcat}_{g}(i_{X}),$ is defined as the least nonnegative integer $n$ (or infinity) such that $X$ admits a cover by $n+1$ relatively sectional subsets.

Generalized relative category is a homotopy invariant in the following sense:

Proposition 2.12**.**

Consider the following homotopy commutative diagram, where $\alpha$ and $\beta$ are homotopy equivalences and $i_{X}$, $i_{X^{\prime}}$ are cofibrations:

[TABLE]

Then $\mbox{relcat}_{g}(i_{X})=\mbox{relcat}_{g}(i_{X^{\prime}}).$

Proof.

As $i_{X}$ is a cofibration we can suppose that the diagram is strictly commutative and therefore we have $\beta:(X,A)\rightarrow(X^{\prime},A^{\prime})$ a homotopy equivalence of pairs where $\beta_{|A}=\alpha:A\rightarrow A^{\prime}.$ This way there exists $\beta^{\prime}:(X^{\prime},A^{\prime})\rightarrow(X,A)$ a homotopy inverse of $\beta$ and therefore we can take $G:(X\times I,A\times I)\rightarrow(X,A)$ a homotopy of pairs such that $G(x,0)=x$ and $G(x,1)=(\beta^{\prime}\circ\beta)(x)$, for all $x\in X$.

Let $U^{\prime}$ be a relatively sectional subset with respect to $i_{X^{\prime}}$ and take $H:(U^{\prime}\times I,A^{\prime}\times I)\rightarrow(X^{\prime},A^{\prime})$ such that $H(x,0)=x$ and $H(x,1)\in A^{\prime}.$ If we define $U:=\beta^{-1}(U^{\prime})$, then the homotopy of pairs $F:(U\times I,A\times I)\rightarrow(X,A)$ given by

[TABLE]

proves that $U$ is a relatively sectional subset with respect to $i_{X}.$ Considering a cover argument we obtain the inequality $\mbox{relcat}_{g}(i_{X})\leq\mbox{relcat}_{g}(i_{X^{\prime}}).$ The other inequality is similarly proved. ∎

Now we want to prove that, when dealing among ANR spaces, we have the equality $\mbox{relcat}(i_{X})=\mbox{relcat}_{g}(i_{X}).$ In order to prove this we begin by giving the next technical lemma:

Lemma 2.13**.**

Let $i_{X}:A\hookrightarrow X$ be a cofibration between ANR spaces. Then there exists a factorization:

[TABLE]

where $\widehat{A}$ is an ANR space, $j$ a homotopy equivalence and $p$ a fibration.

Proof.

The space $\widehat{A}$ is given from the classical mapping track (or mapping cocylinder) construction $\widehat{A}=L_{i_{X}}=\{(a,\alpha)\in A\times X^{I}:i_{X}(a)=\alpha(0)\}$, in other words, from the pullback

[TABLE]

Such a construction gives us the classical factorization of the map $i_{X}$ through a homotopy equivalence followed by a fibration. Therefore it only remains to check that $\widehat{A}$ is, actually, an ANR space. Indeed, we first observe that $\widehat{A}$ is homeomorphic to $p^{-1}(A)$, since $i_{X}$ is an inclusion. But the inclusion $p^{-1}(A)\hookrightarrow X^{I}$ is a cofibration (see Theorem 12 in [10]). As $p^{-1}(A)$ is a closed subspace of $X^{I}$, whose corresponding inclusion is a cofibration, we have by Theorem 2.4 of chapt. VI and Theorem 3.2 of chapt. IV in [6], that $p^{-1}(A)$ is an ANR space. ∎

Another lemma we need is related to certain kind of open covers.

Lemma 2.14**.**

Let $i_{X}:A\hookrightarrow X$ be a cofibration with $X$ a normal space. Also suppose $\{U_{i}\}_{i=0}^{n}$ an open cover of $X$ in such a way that for any $i$ we have $A\subseteq U_{i}$ and there exists a homotopy $H_{i}:U_{i}\times I\rightarrow X$ satisfying that $H_{i}(x,0)=x,$ $H_{i}(x,1)\in A$ and $H_{i}(-,1)_{|A}\simeq 1_{A}.$ Then $\mbox{relcat}(i_{X})\leq n.$

Proof.

Following a similar reasoning to that done in the proof of Theorem 2.7 we can consider a chain of inclusions $A\subseteq A_{i}\subseteq\theta_{i}\subseteq B_{i}\subseteq U_{i}$ with $A_{i},B_{i}$ closed subsets and $\theta_{i}$ open subsets, such that $\{A_{i}\}_{i=0}^{n}$ covers $X.$ If $h_{i}:X\rightarrow[0,1]$ is the Urysohn map given as $h_{i}(A_{i})=\{1\}$ y $h_{i}(X\setminus\theta_{i})=\{0\}$ then the map $L_{i}:X\times I\rightarrow X$ defined by

[TABLE]

is clearly continuous. Taking $L=(L_{0},...,L_{n}):X\times I\rightarrow X^{n+1}$ we have a homotopy $L:\Delta_{n+1}\simeq t_{n}\circ f,$ where $f:=L(-,1):X\rightarrow T^{n}(i_{X}).$ What is more, taking a homotopy $G_{i}:A\times I\rightarrow A$ satisfying $G_{i}:1_{A}\simeq H_{i}(-,1)_{|A}$ it is not difficult to check that $G:=(G_{0},...,G_{n}):A\times I\rightarrow A^{n+1}\subseteq T^{n}(i_{X})$ gives a homotopy $G:\tau_{n}\simeq f\circ i_{X}.$ These facts prove that the following diagram commutes, up to homotopy:

[TABLE]

Therefore, by Theorem 1.1 we have the inequality $\mbox{relcat}(i_{X})\leq n.$ ∎

Remark 2.15*.*

Observe, that the converse in the lemma above also holds, that is, if $\mbox{relcat}(i_{X})\leq n$, then there exists an open cover as in the statement of the lemma. Therefore we actually have another equivalent characterization of $\mbox{relcat}(i_{X})$ (compare with Corollary 1.7 (iv)).

Now we can state and prove the main result.

Theorem 2.16**.**

Let $i_{X}:A\hookrightarrow X$ be a cofibration between ANR spaces. Then $\mbox{relcat}(i_{X})=\mbox{relcat}_{g}(i_{X}).$

Proof.

We only have to check the inequality $\mbox{relcat}(i_{X})\leq\mbox{relcat}_{g}(i_{X}).$ Suppose we have an arbitrary subset $U\subseteq X$ such that $A\subseteq U$ and a homotopy of pairs $H:(U\times I,A\times I)\rightarrow(X,A)$ satisfying $H(x,0)=x$ and $H(x,1)\in A$, for all $x\in X$. Considering a factorization $i_{X}=p\circ j$ through the space $\widehat{A}$ as in Lemma 2.13 above, we can define a map $s:U\rightarrow\widehat{A}$ as the composite

[TABLE]

Clearly $p\circ s\simeq inc_{U}$ and $s_{|A}\simeq j,$ where $inc_{U}:U\hookrightarrow X$ stands for the natural inclusion. Observe the, since $p$ is a fibration, we can suppose without losing any generality that $p\circ s=inc_{U}$ and $s_{|A}\simeq j.$ Following the proof in Theorem 2.7 one has an open neighborhood $V\supseteq U$ in $X$ and a map $\sigma:V\rightarrow\widehat{A}$ such that $p\circ\sigma=inc_{V}$ and $\sigma_{|U}\simeq s.$ In particular, one also has

[TABLE]

Besides that, if $j^{\prime}:\widehat{A}\rightarrow A$ denotes an homotopy inverse of $j$, then

[TABLE]

In other words, there exists a homotopy $G:V\times I\rightarrow X$ such that $G:inc_{V}\simeq i_{X}\circ j^{\prime}\circ\sigma$. Obviously, $G(x,1)\in A$ and $G(-,1)_{|A}=j^{\prime}\circ(\sigma_{|A})\simeq j^{\prime}\circ j\simeq 1_{A}.$ Taking into account an argument by covers and using Lemma 2.14, we conclude the proof. ∎

As a consequence of the above theorem we can compare the monoidal topological complexity $\mbox{TC}^{M}(X)$ and its generalized version. Recall that the monoidal topological complexity of a space $X$ (see [7]) is defined as the least nonnegative integer $n$ such that $X\times X$ admits an open cover $\{U_{i}\}_{i=0}^{n}$ such that each $U_{i}$ contains $\Delta_{X}(X)=\{(x,x):x\in X\}$ and admits a strict local section $s_{i}:U_{i}\rightarrow X^{I}$ of the path fibration $\pi_{X}:X^{I}\rightarrow X\times X$ satisfying $s_{i}(x,x)=c_{x}$ (here, $c_{x}$ denotes the constant path in $X$ at the point $x$). Unlike the topological complexity, $\mbox{TC}^{M}(-)$ is not a homotopy invariant. For instance, consider $X$ the subspace of $\mathbb{R}^{2}$

[TABLE]

where $L_{n}=\{(t,\frac{t}{n}):t\in[0,1]\}$ is the segment joining $(0,0)$ with $(1,\frac{1}{n})$ for $n\geq 1$ and $L_{0}=[0,1]\times\{0\}$. Clearly $X$ is a contractible space. Nevertheless, on the other hand we have that $\mbox{TC}^{M}(X)\neq 0.$ Indeed, if $\mbox{TC}^{M}(X)=0$ and $s:X\times X\rightarrow X^{I}$ is a global continuous section of the path fibration $\pi:X^{I}\rightarrow X\times X$ satisfying $s(x,x)=c_{x}$, then for $n\geq 1$ we can take the sequence of points $x_{n}=(1,\frac{1}{n})$ and $x_{0}=(1,0).$ Obviously, $(x_{n},x_{0})\rightarrow(x_{0},x_{0})$ in $X\times X,$ when $n\rightarrow\infty$ and by continuity we have that $s(x_{n},x_{0})\rightarrow s(x_{0},x_{0})=c_{x_{0}}.$ This is not possible, as for all $n\geq 1$, $s(x_{n},x_{0})$ is a path from $x_{n}$ to $x_{0}$ that must pass through the origin $(0,0)$. This example also shows that, in general, $\mbox{TC}(X)\neq\mbox{TC}^{M}(X)$. We point out that this does not refute Iwase-Sakai’s conjecture [7] which asserts that $\mbox{TC}(X)=\mbox{TC}^{M}(X)$ for any locally finite simplicial complex $X$.

As we are going to see, restricting to a suitable large class of spaces, monoidal topological complexity becomes a homotopy invariant. Recall that a locally equiconnected space is a topological space $X$ in which the diagonal map $\Delta_{X}:X\rightarrow X\times X$ is a cofibration. This class of spaces is sufficiently large. For instance, it is known that CW-complexes and ANR spaces belong to such a class.

Proposition 2.17**.**

[1, Th. 12]** If $X$ is a locally equiconnected space, then $\mbox{TC}^{M}(X)=\mbox{relcat}(\Delta_{X}).$ In particular, $\mbox{TC}^{M}(-)$ is a homotopy invariant when we restrict to locally equiconnected spaces.

If we define the generalized monoidal topological complexity of a locally equiconnected space $X$ as $\mbox{TC}^{M}_{g}(X):=\mbox{relcat}_{g}(\Delta_{X})$, then we have the following corollary of Theorem 2.16:

Corollary 2.18**.**

If $X$ is a locally equiconnected space having the homotopy type of a CW-complex (equivalently, of an ANR-space), then $\mbox{TC}^{M}(X)=\mbox{TC}^{M}_{g}(X).$

Proof.

Just take into account Proposition 2.12, Proposition 2.17 and Theorem 2.16 above. ∎

Remark 2.19*.*

There is an alternative definition of $\mbox{TC}^{M}_{g}(X)$ for a non necessary locally equiconnected space. We can take subsets $U\subseteq X\times X$ such that $\Delta_{X}(X)\subseteq U$ together with a strict local section $s:U\rightarrow X^{I}$ of $\pi_{X}:X^{I}\rightarrow X\times X$ satisfying $s(x,x)=c_{x},$ for all $x\in X.$ Then, define $\mbox{TC}^{M}_{g}(X)$ as the least nonnegative integer $n$ (or infinity) such that $X\times X$ admits a cover of this kind of subsets. With this alternative definition we do not know if $\mbox{TC}^{M}_{g}(X)=\mbox{relcat}_{g}(\Delta_{X})$ when $X$ is a locally equiconnected space. What we have is the inequality $\mbox{relcat}_{g}(\Delta_{X})\leq\mbox{TC}^{M}_{g}(X)$. Indeed, suppose a subset $U\subseteq X\times X$ verifying the properties above mentioned. Then $H:U\times I\rightarrow X\times X$, defined as $H(x,y,t):=(s(x,y)(t),y)$ verifies $H(x,y,0)=(x,y),$ $H(x,y,1)=(y,y)\in\Delta_{X}(X)$ and $H(x,x,t)=(x,x).$ In other words, $U$ is a relatively sectional subset with respect to the cofibration $\Delta_{X}.$ Considering covers one can straightforwardly deduce that $\mbox{relcat}_{g}(\Delta_{X})\leq\mbox{TC}^{M}_{g}(X)$. In this case, concerning Corollary 2.18, the more we can say is: “If $X$ is an ANR space, then $\mbox{TC}^{M}(X)=\mbox{TC}^{M}_{g}(X)$”. Indeed, combining the inequality $\mbox{relcat}_{g}(\Delta_{X})\leq\mbox{TC}^{M}_{g}(X)$ with Proposition 2.17 and Theorem 2.16 we obtain

[TABLE]

concluding that $\mbox{TC}^{M}(X)=\mbox{TC}^{M}_{g}(X).$