Simplicial fibrations

TL;DR

This paper systematically studies simplicial fibrations using contiguity, establishing their properties, introducing finite-fibrations, and relating these concepts to simplicial LS-category and topological complexity.

Contribution

It defines simplicial fibrations via contiguity, proves their key properties, introduces finite-fibrations, and connects these to LS-category and topological complexity in a simplicial context.

Findings

Simplicial fibrations are equivalent to known notions via cylinder constructions.

All fibers of a simplicial fibration share the same strong homotopy type.

Path fibrations are finite-fibrations, enabling factorization of maps up to P-homotopy.

Abstract

We undertake a systematic study of the notion of fibration in the setting of abstract simplicial complexes, where the concept of `homotopy' has been replaced by that of `contiguity'. Then a fibration will be a simplicial map satisfying the `contiguity lifting property'. This definition turns out to be equivalent to a known notion introduced by G. Minian, established in terms of a cylinder construction $K \times I_m$. This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration have the same strong homotopy type, a notion that has been recently introduced by Barmak and Minian; any fibration with a strongly collapsible base is fibrewise trivial; and some other ones. We introduce the concept of `simplicial finite-fibration', that is, a map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration $PK \to K\times K$ is a finite-fibration, where PK is the space of Moore paths introduced by M. Grandis. This important result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we introduce a definition of `\v{S}varc genus' of a simplicial map and, and using the properties stated before, we are able to compare the \v{S}varc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers. Finally, another key result is a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber.

Equations

$$\phi ={\phi }_0{\sim }_c\cdots {\sim }_c{\phi }_m=\psi$$

$$K\times \{0\}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$f={\phi }_0{\sim }_c{\phi }_1{\sim }_c\cdots {\sim }_c{\phi }_m=g.$$

$${\phi }_i(\tau )\cup {\phi }_{i+1}(\tau )=H(\tau \times \{i,i+1\})\in B$$

$$K\times \{0\}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K{\times }_{M}L\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K{\times }_{B}E\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$L\times \{0\}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$L\times I_m\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K\times L\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$f_1\times f_2:K_1\times K_2\to L_1\times L_2$$

$$(f_1\times f_2)(v_1,v_2):=(f_1(v_1),v_2(v_2)),$$

$$\phi =({\phi }_1,{\phi }_2):K\times \{0\}\to E_1\times E_2$$

$$H=(H_1,H_2):K\times I_m\to B_1\times B_2$$

$$\tilde{H}:=({\tilde{H}}_1,{\tilde{H}}_2):K\times I_m\to E_1\times E_2$$

$${\gamma }^{-}:=$$

$${\gamma }^{+}:=$$

$$\overline{\gamma }(i)=\gamma (-i),$$

$$|\gamma |(i)=\gamma (i+{\gamma }^{-}).$$

$$(\gamma \ast \delta )(i)=\{\begin{array}{ll}\gamma (i-{\delta }^{-}),& i\le {\gamma }^{+}+{\delta }^{-},\\ \delta (i-{\gamma }^{+}),& i\ge {\gamma }^{+}+{\delta }^{-}.\end{array}$$

$$\gamma \ast (\delta \ast \epsilon )=(\gamma \ast \delta )\ast \epsilon .$$

$$\bigcup _{i=0}^p{f}_i(\sigma )\in L,\text{for any simplex }\sigma \in K.$$

$$\mathrm{ev}:L^K\times K\to K,(f,v)\mapsto f(v),$$

$$\mathbf{SC}(M\times K,L)\equiv \mathbf{SC}(M,L^K).$$

$$\{{\gamma }_0(i),\dots ,{\gamma }_p(i),{\gamma }_0(i+1),\dots ,{\gamma }_p(i+1)\}$$

$$K^{[a,b]}\subset \mathrm{P}K.$$

$$\pi =(\alpha ,\omega ):\mathrm{P}K\to K\times K$$

$$L\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Omega :L\times I_m\to K^{[a-m,b+m]}\subset \mathrm{P}K$$

Full text

The first and the fourh authors were partially supported by MINECO Spain Research Project MTM2015–65397–P and Junta de Andalucía Research Groups FQM–326 and FQM–189. The second and third authors were partially supported by MINECO-FEDER research project MTM2016–78647–P

1. 1 Introduction
2. 2 Simplicial complexes and contiguity
3. 3 Definitions of simplicial fibration
4. 4 Examples and properties
5. 5 The path complex $\mathrm{P}K$
6. 6 Homotopy fiber theorem
7. 7 Collapsible base
8. 8 Varadarajan’s theorem
9. 9 Factorization
10. 10 ​​Švarc genus

1. Introduction

In recent years there has been a renovated interest in abstract simplicial complexes, as a setting which is well suited for discretizing topological invariants and for designing computer algorithms. Under this point of view and boosted by the increasing computer capacities, several classical theories have been developed, thus providing new powerful tools like persistent homology [7] or discrete Morse theory [8], which are being applied in robotics, neural networks or big data mining.

However, there is a lack of development of other ideas in this new field of “applied algebraic topology”, like Lusternik-Schnirelmann category or topological complexity, which classically needed the use of notions such as homotopy, fibrations or cofibrations.

In the framework of abstract simplicial complexes, the classical notion of “contiguity” between simplicial maps [21] plays the role that “homotopy” plays in the context of topological spaces. This notion has received new attention in the last years after the work of J. Barmak and G. Minian [4]. They showed that the equivalence under contiguity classes is the same as the equivalence by “strong collapses”, a highly interesting idea which is related on one hand with the classical Whitehead collapses, and on the other hand with the theory of posets and finite topological spaces [3].

Using the ideas above, several of the authors have recently introduced a notion of LS-category in the simplicial setting, which generalizes the well known notion of “arboricity” in graph theory [12, 14]. Moreover, we also introduced a notion of topological complexity, defined in purely combinatorial terms [13]. Both invariants have similar properties to the classical ones and also new results arise.

As a collateral result, cofibrations were studied in [14], but a systematic study of the notion of simplicial fibration was lacking. This study is the aim of the present paper.

The contents are as follows:

In Section 2 we recall the basic notions of simplicial complexes and contiguity classes.

In Section 3 we introduce two possible definitions of simplicial fibration in terms of a contiguity lifting property and we show that in fact they are equivalent to a third one introduced by Minian in [19]: a simplicial map $p\colon E\to B$ is a simplicial fibration if for any simplicial map $H\colon K\times I_{m}\to B$ and any simplicial map $f:K\times\{0\}\to E$, there is a simplicial map $\tilde{H}\colon K\times I_{m}\to E$ such that $p\circ\tilde{H}=H$ and $\tilde{H}\circ i_{0}^{m}=f$ (see Definition 3.3). There is a more general notion of simplicial finite-fibration if we limit the lifting property to finite complexes $K$ in the definition above. It will be necessary to obtain several key results along the paper.

In Section 4 we give several basic examples and constructions, including products and pullbacks of simplicial fibrations. Then, we introduce (Section 5) the notion of Moore path and the space $\mathrm{P}K$ of Moore paths on a simplicial complex $K$. This notion has been developed in [15] by M. Grandis. The main result of this section is that the path map $\mathrm{P}K\to K\times K$ is a simplicial finite-fibration (Theorem 5.6).

In Section 6 we prove the important result that all the fibers of a simplicial fibration have the same strong homotopy type. In the same line, we adapt another classical result by showing (Section 7) that a simplicial fibration with a collapsible base is trivial. Here, “collapsible” means that there is a finite sequence of strong collapses and expansions transforming the base onto a point. Our next result (Section 8) is a simplicial version of Varadarajan’s theorem (see Theorem 8.2) relating the LS-category of the total space, the base and the generic fiber of a fibration.

In the last sections of the paper we introduce several new ideas. The first one is based on a notion of “P-homotopy” modelled on Moore paths (see Definition 5.7), which allows us to prove the $P$-equivalence of the complexes $K$ and $\mathrm{P}K$ and to give a general result about the factorization of any map into a P-equivalence and a finite-fibration (Section 9).

On the other hand, the $P$-homotopy equals the usual contiguity property for finite complexes. This allows us to define in Section 10 a general notion of Švarc genus for simplicial maps and to discuss its relationship with the simplicial LS-category and the discrete topological complexity introduced in our previous papers [12, 14, 13].

2. Simplicial complexes and contiguity

We start by briefly recalling the notions of simplicial complex and contiguity. We are assuming that the reader is familiarized with these notions as well as others that will be appearing throughout the paper (see, for instance, [17, 21] for more details on this topic).

Definition 2.1**.**

An (abstract) simplicial complex is a set $V$ together with a collection $K$ of finite subsets of $V$ such that if $\sigma\in K$ and $\tau\subseteq\sigma$ then $\tau\in K$.

Notice that $K$ is not necessarily finite in the above definition. As usual, $K$ will denote the simplicial complex and $V(K)$ the corresponding vertex set.

Definition 2.2**.**

Given two simplicial complexes $K$ and $L$, a simplicial map from $K$ to $L$ is a set map $\varphi\colon V(K)\to V(L)$ such that if $\sigma\in K$ then $\varphi(\sigma)\in L$.

Definition 2.3**.**

Let $K,L$ be two simplicial complexes. Two simplicial maps $\varphi,\psi\colon K\to L$ are contiguous [21, p. 130] if, for any simplex $\sigma\in K$, the set $\varphi(\sigma)\cup\psi(\sigma)$ is a simplex of $L$; that is, if $v_{0},\dots,v_{k}$ are the vertices of $\sigma$ then the vertices $f(v_{0}),\dots,f(v_{k}),g(v_{0}),\dots,g(v_{k})$ span a simplex of $L$.

This relation, denoted by $\varphi\sim_{c}\psi$, is reflexive and symmetric, but in general it is not transitive. In order to overcome this fact we use the notion of contiguity class.

Definition 2.4**.**

Two simplicial maps $\varphi,\psi\colon K\to L$ are in the same contiguity class with $m$ steps, denoted by $\varphi\sim\psi$, if there is a finite sequence

[TABLE]

of contiguous simplicial maps $\varphi_{i}\colon K\to L$, $0\leq i\leq m$.

It is straightforward to prove the following:

Lemma 2.5**.**

Let $f\colon K\to L$ be a simplicial map between the simplicial complexes $K$ and $L$. If $g\colon V(K)\to V(L)$ is a map such that $f\sim_{c}g$ then $g$ is a simplicial map.

Now we recall a formal notion of combinatorial homotopy introduced by Minian in [19]. First of all we need a triangulated version of the real interval $[0,n]$.

Definition 2.6**.**

For $n\geq 1$, let $I_{n}$ be the one-dimensional simplicial complex whose vertices are the integers $\{0,\dots,n\}$ and the edges are the pairs $\{j,j+1\}$, for $0\leq j<n$.

Definition 2.7**.**

[17, Definition 4.25] Let $K$ and $L$ be two simplicial complexes. The categorical product $K\times L$ is the simplicial complex whose set of vertices is $V(K\times L)=V(K)\times V(L),$ and whose simplices are given by the rule: $\sigma\in K\times L$ if and only if $\mathrm{pr}_{1}(\sigma)\in K$ and $\mathrm{pr}_{2}(\sigma)\in L$, where $\mathrm{pr}_{1},\mathrm{pr}_{2}$ are the canonical projections.

Given two simplicial maps, Minian proved that belonging to the same contiguity class is equivalent to the existence of a simplicial homotopy between them, modelled by a simplicial cylinder.

Proposition 2.8**.**

[19, Prop. 2.16]** Two simplicial maps $f,g\colon K\to L$ are in the same contiguity class, with $m$ steps, $f\sim g$, if and only if there exists some $m\geq 1$ and some simplicial map $H\colon K\times I_{m}\to L$ such that $H(v,0)=f(v)$ and $H(v,m)=g(v)$, for all vertices $v\in K$.

Remark 1*.*

The preceding proposition holds when we consider $K\times I_{m}$ to be the categorical product, but notice that the proof does not work for the more usual notion of simplicial product, namely the so-called simplicial cartesian product (see [17]).

Barmak and Minian introduced [3, 4] the so-called strong homotopy type for simplicial complexes. Two simplicial complexes have the same strong homotopy type, denoted by $K\sim L$, if they are related by a finite sequence of two kind of simplicial moves, namely, strong collapses and expansions. An elementary strong collapse consists of removing the open star around a dominated vertex, where a vertex $v$ is dominated by another vertex $v^{\prime}$ if every maximal simplex that contains $v$ also contains $v^{\prime}$. A complex is called strongly collapsible if it has the same strong homotopy type of a point.

Strong homotopy type is deeply related to the notion of contiguity between simplicial maps. More precisely, the following result holds:

Proposition 2.9**.**

[4, Cor. 2.12]** Two simplicial complexes $K$ and $L$ have the same strong homotopy type if and only if there are simplicial maps $\varphi\colon K\to L$ and $\psi\colon L\to K$ such that $\psi\circ\varphi\sim 1_{K}$ and $\varphi\circ\psi\sim 1_{L}$.

3. Definitions of simplicial fibration

The goal of this section is to establish a notion of fibration in the simplicial context. As we shall see, there are several options of doing this, depending on the particular kind of lifting property we deal with.

Our first definition of fibration in the simplicial context corresponds to simplicial maps with the contiguity lifting property with respect to any simplicial complex.

Definition 3.1**.**

A simplicial map $p\colon E\to B$ is a type I simplicial fibration if for any simplicial complex $K$ (finite or not), given any two contiguous simplicial maps $f,g\colon K\to B$, $f\sim_{c}g$, and any map $\tilde{f}\colon K\to E$ such that $p\circ\tilde{f}=f$, there exists a simplicial map $\tilde{g}\colon K\to E$ such that $\tilde{f}$ and $\tilde{g}$ are contiguous, $\tilde{f}\sim_{c}\tilde{g}$, and $p\circ\tilde{g}=g$ (see Figure 3.1).

A second option to give a simplicial notion of fibration consists in generalizing the first definition to contiguity classes with a given number of steps (see Definition 2.4).

Definition 3.2**.**

A simplicial map $p\colon E\to B$ is a type II simplicial fibration if for any simplicial complex $K$, for any two simplicial maps $f,g\colon K\to B$ in the same contiguity class, $f\sim g$, with $m$ steps, and for any map $\tilde{f}\colon K\to E$ such that $p\circ\tilde{f}=f$, there exists a simplicial map $\tilde{g}\colon K\to E$ such that $\tilde{f}$ and $\tilde{g}$ are in the same contiguity class with $m$ steps, $\tilde{f}\sim\tilde{g}$, and $p\circ\tilde{g}=g$.

At this point we are ready for the third definition of simplicial fibration in terms of the notion of homotopy, introduced by Minian (see Prop. 2.8).

Definition 3.3**.**

The map $p\colon E\to B$ is a type III simplicial fibration if given simplicial maps $H\colon K\times I_{m}\to B$ and $\varphi\colon K\times\{0\}\to E$ as in the following commutative diagram:

[TABLE]

there exists a simplicial map $\widetilde{H}\colon K\times I_{m}\to E$ such that $\widetilde{H}\circ i_{0}^{m}=\varphi$ and $p\circ\widetilde{H}=H$.

Theorem 3.4**.**

The three definitions of simplicial fibration are equivalent.

Proof.

Type I $\Rightarrow$ Type II: Assuming that the simplicial map $p\colon E\to B$ satisfies Definition 3.1, let us consider a simplicial complex $K$ and two simplicial maps $f,g\colon K\to B$ in the same contiguity class with $m$ steps, that is, there exists a finite sequence of direct contiguities

[TABLE]

Taking a simplicial map $\widetilde{f}$ such that $p\circ\tilde{f}=f$, by Definition 3.1, there is a simplicial map $\widetilde{\varphi_{1}}\colon K\to E$ such that $\widetilde{f}\sim_{c}\widetilde{\varphi_{1}}$ and $p\circ\widetilde{\varphi_{1}}=\varphi_{1}$. Iterating the same argument for every $\varphi_{i}$ with $i=1,\dots,m$, we obtain a finite sequence of direct contiguities, $\widetilde{f}=\widetilde{\varphi}_{0}\sim_{c}\widetilde{\varphi}_{1}\sim_{c}\cdots\sim_{c}\widetilde{\varphi}_{m}$, where $p\circ\widetilde{\varphi_{i}}=\varphi_{i}$, $i=1,\dots,m$. In particular, for $i=m$ we get a simplicial map $\widetilde{g}=\widetilde{\varphi}_{m}$ satisfying $\tilde{f}\sim\tilde{g}$ with $m$ steps and $p\circ\tilde{g}=g$ since $\varphi_{m}=g$. So, we conclude that $P$ is a type II simplicial fibration.

Type II $\Rightarrow$ Type III: Let us assume that the simplicial map $p\colon E\to B$ satisfies Definition  3.2. Consider a simplicial complex $K$ and simplicial maps $\varphi\colon K\times\{0\}\to E$ and $H\colon K\times I_{m}\to B$ such that Diagram (1) is commutative.

Now, we define $\varphi_{i}\colon K\to B$ by $\varphi_{i}(v)=H(v,i)$, where $v\in V(K)$ and $i\in V(I_{m})$. By means of Lemma 2.5, we only need to prove the contiguity condition. These maps are contiguous because given $\tau\in K$ the following fact holds true:

[TABLE]

since $H$ is a simplicial map. Then, by hypothesis, there exists a finite chain of simplicial maps $\widetilde{\varphi}_{i}\colon K\to E$ and direct contiguities, with $m$ steps, $\widetilde{\varphi}_{0}\sim_{c}\widetilde{\varphi}_{1}\sim_{c}\cdots\sim_{c}\widetilde{\varphi}_{m}$, such that $p\circ\widetilde{\varphi}_{i}=\varphi_{i}$, for all $i=0,\dots,m$.

Hence, the map $\widetilde{H}\colon K\times I_{m}\to E$ given by $\widetilde{H}(v,i)=\widetilde{\varphi}_{i}(v)$ is simplicial, by an argument analogous to (2), and satisfies $\widetilde{H}\circ i_{0}^{m}=\varphi$ and $p\circ\widetilde{H}=H$. So, $H$ satisfies Definition 3.3.

Type III $\Rightarrow$ Type I: Let us assume that the simplicial map $p\colon E\to B$ satisfies Definition 3.3. Consider a simplicial complex $K$ and two contiguous simplicial maps $f,g\colon K\to B$. Now, by Proposition 2.8, with $m=1$, there exist a homotopy $H\colon K\times I_{1}\to B$ such that $H(v,0)=f(v)$ and $H(v,1)=g(v)$ for all $v\in V(K)$.

Consider a simplicial map $\tilde{f}\colon K\to E$ such that $p\circ\tilde{f}=f$. By Definition 3.3, there is a simplicial map $\widetilde{H}\colon K\times I_{1}\to E$ such that $\widetilde{H}\circ i_{0}^{1}=\varphi$, where $p\circ\varphi(v,0)=\widetilde{f}(v)$ for all $v\in V(K)$, and $p\circ\widetilde{H}=H$. Let $\widetilde{g}\colon K\to E$ given by $\widetilde{g}(v)=\widetilde{H}(v,1)$, where $v\in V(K)$. By Proposition 2.8, we conclude that $\widetilde{f}\sim_{c}\widetilde{g}$. ∎

Remark 2*.*

Notice that the complex $K$ that we considered in the definitions above may not be finite.

Remark 3*.*

Observe that it is possible to restrict these definitions to the cases where $K$ is finite. This allows us to introduce the corresponding notions of simplicial finite-fibration of type I, II and III, which are equivalent by the finite version of Theorem 3.4.

4. Examples and properties

In this section we will introduce some important examples of simplicial fibrations. The following proposition will give us the first basic ones. Notice that, unless otherwise specified, we will use the notion of type III simplicial fibration given in Definition 3.3.

Proposition 4.1**.**

**

1. (i)

Any simplicial isomorphism is a simplicial fibration. 2. (ii)

If $*$ denotes the one-vertex simplicial complex, then the constant simplicial map $E\rightarrow*$ is a simplicial fibration, for any simplicial complex $E$. 3. (iii)

The composition of simplicial fibrations is a simplicial fibration.

Proof.

(i) is easily checked; indeed, if $p\colon E\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}B$ is a simplicial isomorphism, then, given any $m\geq 0$ and a commutative diagram $p\circ\varphi=H\circ i^{m}_{0},$ we have that the composition $\widetilde{H}\mathrel{\mathop{:}}=p^{-1}\circ H$ satisfies the required conditions. Item (ii) is also straightforward, since for any simplicial map $\varphi\colon K\times\{0\}\to E$ the vertex map $K\times I_{m}\rightarrow E$, given by $(v,i)\mapsto\varphi(v,0)$, is simplicial.

Now, in order to prove (iii), consider $p\colon E\rightarrow B$ and $q\colon B\rightarrow C$ simplicial fibrations, $m\geq 1$, and the following commutative diagram of simplicial maps:

[TABLE]

Since $q$ is a simplicial fibration we can consider a simplicial map $\widehat{G}\colon K\times I_{m}\rightarrow B$ satisfying $\widehat{G}\circ i_{0}^{m}=p\circ\varphi$ and $q\circ\widehat{G}=G$. Finally, as $p$ is a simplicial fibration we can also consider a simplicial map $\widetilde{G}\colon K\times I_{m}\rightarrow E$ such that $\widetilde{G}\circ i_{0}^{m}=\varphi$ and $p\circ\widetilde{G}=\widehat{G}$. From these conditions we have that $(q\circ p)\circ\widetilde{G}=q\circ\widehat{G}=G$. ∎

For the next result we need to recall the pullback construction for simplicial complexes. Given any pair of simplicial maps $f\colon K\rightarrow M$ and $g\colon L\rightarrow M$ their pullback is given as the following diagram:

[TABLE]

where $K\times_{M}L$ is the full simplicial subcomplex of $K\times L$ whose underlying vertex set is given by those pairs of vertices $(v,w)\in K\times L$ satisfying $f(v)=g(w)$. The induced simplicial maps $f^{\prime}$ and $g^{\prime}$ are given by $f^{\prime}(v,w)=w$ and $g^{\prime}(v,w)=v$. It is plain to check that this construction is the pullback of $f$ and $g$ in the category of simplicial complexes.

Proposition 4.2**.**

Let $p\colon E\rightarrow B$ be a simplicial fibration and $f\colon K\rightarrow B$ any simplicial map. Then the simplicial map $p^{\prime}\colon K\times_{B}E\rightarrow K$ induced by $p$ in the pullback

[TABLE]

is also a simplicial fibration.

Proof.

Take any commutative diagram where $L$ is a simplicial complex:

[TABLE]

Considering the composition of this diagram with the pullback square and using the fact that $p$ is a simplicial fibration, one can take a simplicial map $\widetilde{H}\colon L\times I_{m}\to E$ satisfying $p\circ\widetilde{H}=f\circ G$ and $\widetilde{H}\circ i_{0}^{m}=f^{\prime}\circ g$. By the pullback property there is an induced simplicial map $\widetilde{G}\colon L\times I_{m}\rightarrow K\times_{B}E$ making commutative the following diagram:

[TABLE]

By the universal property of the pullback we have that $\widetilde{G}\circ i_{0}^{m}=g.$ ∎

Corollary 4.3**.**

Let $K$ and $L$ be simplicial complexes and $K\times L$ be their categorical product. Then the canonical projections $\mathrm{pr}_{1}\colon K\times L\rightarrow K$ and $\mathrm{pr}_{2}\colon K\times L\rightarrow L$ are simplicial fibrations.

Proof.

One has just to take into account part (ii) of Proposition 4.1 because the following square is a pullback:

[TABLE]

∎

Another interesting example of simplicial fibration is given by the product of simplicial fibrations. Recall that, if $f_{1}\colon K_{1}\to L_{1}$ and $f_{2}\colon K_{2}\to L_{2}$ are simplicial maps, then one can construct their product simplicial map:

[TABLE]

defined as

[TABLE]

for any vertex $(v_{1},v_{2})\in K_{1}\times K_{2}.$

Proposition 4.4**.**

Let $p_{1}\colon E_{1}\rightarrow B_{1}$ and $p_{2}\colon E_{2}\rightarrow B_{2}$ be simplicial fibrations. Then their product $p_{1}\times p_{2}$ is also a simplicial fibration.

Proof.

Let $K$ be a simplicial complex, consider

[TABLE]

and

[TABLE]

simplicial maps such that $(p_{1}\times p_{2})\circ\varphi=H\circ i_{0}^{m}$. As $p_{i}\circ\varphi_{i}=H_{i}\circ i_{0}^{m}$ and $p_{i}$ is a simplicial fibration, there is a simplicial map $\widetilde{H}_{i}\colon K\times I_{m}\to E_{i}$ such that $p_{i}\circ\widetilde{H}_{i}=H_{i}$ and $\widetilde{H}_{i}\circ i_{0}^{m}=\varphi_{i}$ for $i=1,2$. Hence, the simplicial map

[TABLE]

verifies the expected conditions. ∎

5. The path complex $\mathrm{P}K$

5.1. Moore paths

Consider the one-dimensional simplicial complex $Z$, whose vertices are all the integers $i\in\mathbb{Z}$ and whose 1-simplices are all the consecutive pairs $\{i,i+1\}$, that is, $\mathbb{Z}$ is a triangulation of the real line.

Definition 5.1** ([15]).**

Let $K$ be a simplicial complex. A Moore path in $K$ is a simplicial map $\gamma\colon Z\rightarrow K$ which is eventually constant on the left and eventually constant on the right, i.e., there exist integers $i^{-},i^{+}\in\mathbb{Z}$ satisfying the two following conditions:

1. (i)

$\gamma(i)=\gamma(i^{-}),$ for all $i\leq i^{-}$, 2. (ii)

$\gamma(i)=\gamma(i^{+}),$ for all $i\geq i^{+}$.

Obviously, if $i^{-}=i^{+}$ we have the constant map. For a non constant Moore path $\gamma\colon Z\rightarrow K$ we can consider the integers

[TABLE]

Observe that $\gamma^{-}<\gamma^{+}$.

Definition 5.2**.**

The images $\alpha(\gamma)\mathrel{\mathop{:}}=\gamma(\gamma^{-})$ and $\omega(\gamma)\mathrel{\mathop{:}}=\gamma(\gamma^{+})$ are called the initial vertex and final vertex of $\gamma$, respectively. When $\gamma$ is constant we set $\gamma^{-}=0=\gamma^{+}$.

If $a,b\in\mathbb{Z}$ with $a\leq b$, $[a,b]$ will denote the full subcomplex of $\mathbb{Z}$ generated by all vertices $i$ with $a\leq i\leq b$. Considering this notation, any Moore path $\gamma$ in $K$ may be identified with the restricted simplicial map $\gamma\colon[{\gamma}^{-},\gamma^{+}]\rightarrow K$. The interval $[\gamma^{-},\gamma^{+}]$ will be called the support of $\gamma$.

If $\gamma$ is a Moore path in $K$ with support $[\gamma^{-},\gamma^{+}]$, then one can take the reverse Moore path $\overline{\gamma}$ as

[TABLE]

whose support is $[-\gamma^{+},-\gamma^{-}]$. Notice that this reparametrization describes $\gamma$ in the opposite direction.

If $\gamma$ is a Moore path in $K$ with support $[\gamma^{-},\gamma^{+}]$ such that $\gamma^{+}-\gamma^{-}=m$, then we define one normalized Moore path $|\gamma|\colon I_{m}\to K$ as

[TABLE]

The advantage of this reparametrization is that the support of $|\gamma|$ is $[0,m],$ and therefore it will be more manageable when dealing with simplicial fibrations.

Definition 5.3**.**

Given $\gamma,\delta$ Moore paths in $K$ such that $\omega(\gamma)=\alpha(\delta)$, the product path $\gamma*\delta$ it is defined as

[TABLE]

It is not difficult to see that the support of $\gamma*\delta$ is $[\gamma^{-}+\delta^{-},\gamma^{+}+\delta^{+}].$ The product of Moore paths is strictly associative, that is, given Moore paths $\gamma,\delta,\varepsilon$ such that $\omega(\gamma)=\alpha(\delta)$ and $\omega(\delta)=\alpha(\varepsilon)$, then

[TABLE]

Moreover, if $c_{v}$ denotes the constant path in a vertex $v\in K$, then it is immediate to check that $\gamma*c_{w}=\gamma=c_{v}*\gamma$ where $v=\alpha(\gamma)$ and $w=\omega(\gamma)$.

5.2. The path complex

Next we will consider a suitable notion of Moore path complex associated to a simplicial complex $K$. In order to do so we need to recall some categorical properties in the category SC of simplicial complexes and simplicial maps.

Indeed, if $K$ and $L$ are simplicial complexes, we define the simplicial complex $L^{K}$, whose vertices are all simplicial maps $f\colon K\rightarrow L$ and where we consider as simplices the finite sets $\{f_{0},\dots,f_{p}\}$ of simplicial maps $K\rightarrow L$ such that

[TABLE]

It is not difficult to check that this definition induces a structure of simplicial complex in $L^{K}$. Moreover, denoting by $\times$ the categorical product in SC, we have that the evaluation map

[TABLE]

is simplicial. This fact allows us to establish a natural bijection

[TABLE]

Observe that for a simplicial map $h\colon L\rightarrow L^{\prime}$, there is a well defined map $h^{K}\colon L^{K}\rightarrow(L^{\prime})^{K}$, which preserves the identities and the compositions, that is, we have a functor $(-)^{K}\colon\mathbf{SC}\rightarrow\mathbf{SC}$. More is true, the functor $(-)\times K\colon\mathbf{SC}\rightarrow\mathbf{SC}$ is left adjoint to the functor $(-)^{K}\colon\mathbf{SC}\rightarrow\mathbf{SC}$.

Definition 5.4**.**

Let $K$ be a simplicial complex. We define the Moore path complex of $K$, denoted by $\mathrm{P}K$, as the full subcomplex of $K^{Z}$ generated by all the Moore paths $\gamma\colon Z\rightarrow K$.

Then, $\{\gamma_{0},\dots,\gamma_{p}\}\subset\mathrm{P}K$ defines a simplex in $\mathrm{P}K$ if and only if

[TABLE]

is a simplex in $K$, for any integer $i\in\mathbb{Z}$.

An interesting property of $\mathrm{P}K$ is that, for any bounded interval $[a,b]\subset\mathbb{Z}$, the complex $K^{[a,b]}$ is, in fact, a full subcomplex of $\mathrm{P}K$:

[TABLE]

Moreover, given a simplicial map $f\colon K\to L$, since the composite $f\circ\gamma$ is a Moore path in $L$ for any Moore path $\gamma$ in $K$, we obtain the Moore path complex functor $\mathrm{P}\colon\mathbf{SC}\rightarrow\mathbf{SC}$. One can check that this functor preserves binary products and equalizers. Therefore $\mathrm{P}$ preserves finite limits and, in particular, pullbacks. In general $\mathrm{P}$ does not preserve limits; for instance, $\mathrm{P}$ does not preserve infinite products.

Definition 5.5**.**

The initial and final vertices of any given Moore path $\gamma$ define simplicial maps $\alpha\colon\mathrm{P}K\to K$ and $\omega\colon\mathrm{P}K\to K$.

5.3. The path fibration

The aim of this subsection is to establish the following important example of simplicial finite-fibration (see Remark 3).

Theorem 5.6**.**

If $K$ is any simplicial complex, then the following simplicial map

[TABLE]

is a simplicial finite-fibration where $\alpha$ and $\omega$ are the maps given in Definition 5.5.

Proof.

Let $L$ be a finite simplicial complex, $m\geq 1$, and a commutative diagram of simplicial maps

[TABLE]

We recall that, as $L$ is finite, there exists a factorization of $\varphi$ of the form

[TABLE]

Indeed, if $[a(v),b(v)]$ denotes the support of $\varphi(v)$ for any vertex $v\in L$, then we may take $a=\min\{a(v)\colon v\in L\}$ and $b=\max\{b(v)\colon v\in L\}$ due to the fact that $L$ is finite. We therefore obtain a commutative diagram

[TABLE]

where $\mathrm{ev}_{a}$ and $\mathrm{ev}_{b}\colon K^{[a,b]}\rightarrow K$ denote the evaluation simplicial maps at $a$ and $b$, respectively. Our aim is to construct a simplicial map

[TABLE]

satisfying $\pi\circ\Omega=(G,H)$ and $\Omega\circ i_{0}^{m}=\varphi$. Notice that there is natural inclusion $K^{[a,b]}\hookrightarrow K^{[a-m,b+m]}$.

[TABLE]

For this task, consider the simplicial maps $\widehat{G},\widehat{H}\colon L\rightarrow K^{I_{m}}$ respectively associated to $G,H\colon L\times I_{m}\to K$ from the natural adjunction (3). Then, for any fixed vertex $v\in L$, we have three Moore paths

[TABLE]

satisfying $\varphi(v)(a)=\widehat{G}(v)(0)$ and $\varphi(v)(b)=\widehat{H}(v)(0)$.

For any fixed $i\in I_{m}$ we may also consider the $i$-truncated Moore paths, $\widehat{G}(v)_{i}$ and $\widehat{H}(v)_{i}$, whose supports are contained in $[0,i]$ (and therefore in $[0,m]$), given by:

[TABLE]

Observe that $\widehat{G}(v)_{i}$ starts at $\widehat{G}(v)(0)=\varphi(v)(a)$ and ends at $\widehat{G}(v)(i)=G(v,i)$; moreover, when $i=0$ we obtain the constant path at $G(v,0)$. Similarly, $\widehat{H}(v)_{i}$ starts at $\widehat{H}(v)(0)=\varphi(v)(b)$ and ends at $\widehat{H}(v)(i)=H(v,i);$ we also have that, when $i=0$, it is the constant path at $H(v,0)$.

Let us denote by $-\widehat{G}(v)_{i}$ the reverse of the normalized Moore path of $\widehat{G}(v)_{i}$, that is,

[TABLE]

In this way, $-\widehat{G}(v)_{i}$, $\varphi(v)$ and $\widehat{H}(v)_{i}$ can be multiplied and its multiplication $(-\widehat{G}(v)_{i})*\varphi(v)*\widehat{H}(v)_{i}$ is a Moore path starting at $G(v,i)$ and ending at $H(v,i)$, so that for $i=0$ it equals $\varphi(v)$. Such multiplication gives rise to the desired simplicial map $\Omega$ by establishing the identity

[TABLE]

Its explicit expression is as follows:

[TABLE]

At this point we will prove that $\Omega$ is a simplicial map. Taking into account the exponential law, this is equivalent to prove that the following map is simplicial:

[TABLE]

Now, given $\sigma\in L$, $i\in I_{m}$ and $j\in[a-m,b+m]$, we will prove that

[TABLE]

is a simplex in $K$. Taking into account that $\widetilde{\Omega}$ is piecewise defined, we have to consider the following cases:

• •

If $i=0$ and $j=a-i$, then

[TABLE]

• •

If $i=0$ and $j=b+i$, then

[TABLE]

• •

If $i>0$ and $j=a-i$, then $\mu=G(\sigma\times\{i-1,i\})$.

• •

If $i>0$ and $j=a-1$, then $\mu=G(\sigma\times\{0,1\})$.

• •

If $i>0$ and $j=b$, then $\mu=H(\sigma\times\{0,1\})$.

• •

If $i>0$ and $j=b+i$, then $\mu=H(\sigma\times\{i,i+1\})$.

Since $\varphi$, $G$ and $H$ are simplicial maps, in all cases we obtain a simplex in $K$ and hence $\widetilde{\Omega}$ is a simplicial map.

Finally, it is not difficult to check that $\Omega$ satisfies the commutativity in Diagram (4). ∎

5.4. $P$-homotopy

The maps $\alpha$ and $\omega$ of Definition 5.5 allow us to introduce the following notion of homotopy:

Definition 5.7**.**

Given $f,g\colon K\rightarrow L$ simplicial maps, we will say that $f$ is P-homotopic to $g$, denoted by $f\simeq g$, when there exists a simplicial map

[TABLE]

such that $\alpha\circ H=f$ and $\omega\circ H=g$.

This relation is certainly reflexive and symmetric but presumably non transitive (see [15, p. 123]). More is true, it is compatible with left and right compositions.

Definition 5.8**.**

Let $f\colon K\rightarrow L$ be a simplicial map. Then $f$ is said to be a $P$-homotopy equivalence if there exists a simplicial map $g\colon L\rightarrow K$ such that $g\circ f\simeq 1_{K}$ and $f\circ g\simeq 1_{L}$.

Taking into account the links between strong homotopy type and contiguity classes established by Barmak and Minian in [3] and [4], notice that if $P$-homotopies of the above definition are switched by finite sequences of contiguous maps, then we conclude that $f$ is a strong equivalence.

An alternative equivalent form for the notion of $P$-homotopy is given the following result:

Proposition 5.9**.**

Let $f,g\colon K\rightarrow L$ be simplicial maps. Then $f\simeq g$ if and only if there exists a sequence of simplicial maps $\{f_{i}\colon K\rightarrow L\}_{i\in\mathbb{Z}}$ indexed by the integer numbers, such that:

1. (i)

$f_{i}\sim_{c}f_{i+1}$* are contiguous maps;* 2. (ii)

For all vertex $v\in K$ there exist integers $n_{v}$ and $m_{v}$ such that $f_{i}(v)=f(v),$ for all $i\leq n_{v}$ and $f_{i}(v)=g(v)$ for all $i\geq m_{v}$.

Proof.

If $H\colon K\to\mathrm{P}L$ is a homotopy between $f$ and $g$ then we define the simplicial map $f_{i}\mathrel{\mathop{:}}=H(-)(i)\colon K\to L$. Obviously, as $H$ is simplicial and $\{i,i+1\}$ is a simplex in $\mathbb{Z}$, we have that $f_{i}$ and $f_{i+1}$ are contiguous, so condition (i) holds. Moreover, since for any $v\in K$ we have that $H(v)\in\mathrm{P}L$ is a Moore path with support $[n_{v},m_{v}]$, condition (ii) is also fulfilled as $\alpha\circ H=f$ and $\omega\circ H=g$.

Conversely, given a sequence of simplicial maps $\{f_{i}\colon K\to L\}_{i\in\mathbb{Z}}$ satisfying (i) and (ii) we define $H\colon K\to\mathrm{P}L$ as

[TABLE]

It is straightforward to check from (i) that $H$ is simplicial and from (ii) that $\alpha\circ H=f$ and $\omega\circ H=g$. ∎

Now we will see the relationship between the homotopy $\simeq$ and the class of contiguity $\sim$. It is clear that Proposition 2.8 can be rewritten as follows: $f$ and $g$ are in the same class of contiguity if and only if there exist integers $a\leq b$ and a simplicial map $H\colon K\times[a,b]\rightarrow L$ such that $H(v,a)=f(v)$ and $H(v,b)=g(v),$ for all vertex $v\in K$.

Proposition 5.10**.**

Let $f,g\colon K\rightarrow L$ be simplicial maps. Then

1. (i)

If $f\sim g$, then $f\simeq g$. 2. (ii)

If $K$ is finite and $f\simeq g$, then $f\sim g$.

In particular, if $K$ is finite, then $f$ and $g$ are in the same contiguity class, $f\sim g$, if and only if they are $\mathrm{P}$-homotopic, $f\simeq g$.

Proof.

First consider $H\colon K\times[a,b]\rightarrow L$ such that $H(v,a)=f(v)$ and $H(v,b)=g(v)$. Taking the composition of the adjoint simplicial map $K\to L^{[a,b]}$, given by the bijection (3), with the inclusion $L^{[a,b]}\subset\mathrm{P}L$, we obtain a simplicial map $H^{\prime}\colon K\to\mathrm{P}L$ satisfying $\alpha\circ H^{\prime}=f$ and $\omega\circ H^{\prime}=g$.

Conversely, suppose that $K$ is finite and consider a simplicial map $G\colon K\to\mathrm{P}L$ satisfying $\alpha\circ G=f$ and $\omega\circ G=g$. For any $v\in K$ we have that $G(v)$ is a Moore path in $L$ with support $[a_{v},b_{v}].$ Taking $a=\min\{a_{v}\colon v\in K\}$ and $b=\max\{b_{v}\colon v\in K\}$ we obtain a factorization

[TABLE]

Considering the adjoint of $K\to L^{[a,b]}$ we obtain a simplicial map $G^{\prime}\colon K\times[a,b]\to L$ such that $G^{\prime}(v,a)=f(v)$ and $G^{\prime}(v,b)=g(v),$ for all $v\in K$. Hence, this means that $f\sim g$. ∎

We have the following immediate result:

Corollary 5.11**.**

Let $f\colon K\rightarrow L$ be a simplicial map between finite simplicial complexes. Then $f$ is a $P$-homotopy equivalence if and only if $f$ is a strong equivalence.

6. Homotopy fiber theorem

Recall that for any natural number $n$, $I_{n}$ is the full subcomplex of $\mathbb{Z}$ generated by all vertices $0\leq i\leq n$. If $n^{\prime}\geq n$, then a subdivision map $t\colon I_{n^{\prime}}\rightarrow I_{n}$ is any simplicial map satisfying $t(0)=0$ and $t(n^{\prime})=n$. The proof of the following Lemma can be found in [19].

Lemma 6.1**.**

Given natural numbers $n,m$, there exist $n^{\prime},m^{\prime}$ with $n^{\prime}\geq n$ and $m^{\prime}\geq m$ and a simplicial map $\phi\colon I_{n^{\prime}}\times I_{m^{\prime}}\rightarrow I_{n}\times I_{m}$ with the following sketches on the boundaries:

where $a^{\prime}\rightarrow a,$ $b^{\prime}\rightarrow b$, etc. are subdivision maps. Moreover, there exist $n^{\prime\prime}\geq n^{\prime}$, $m^{\prime\prime}\geq m^{\prime}$ and a simplicial map $\phi^{\prime}\colon I_{n^{\prime\prime}}\times I_{m^{\prime\prime}}\rightarrow I_{n^{\prime}}\times I_{m^{\prime}}$ with the opposite sketches on the boundaries such that the composition

[TABLE]

has the form $\phi\circ\phi^{\prime}=t_{1}\times t_{2},$ where $t_{1}\colon I_{n^{\prime\prime}}\rightarrow I_{n}$ and $t_{2}\colon I_{m^{\prime\prime}}\rightarrow I_{m}$ are subdivision maps.

Another technical lemma is the so-called Simplicial Pasting Lemma, whose proof can be found in [20].

Lemma 6.2**.**

Let $U,V$ be subcomplexes of a simplicial complex $K$ and let $f\colon U\to L$, $g\colon V\to L$ be simplicial maps such that $f(v)=g(v),$ for all vertex $v\in U\cap V$. Then, the vertex function $h\colon U\cup V\to L,$ defined as

[TABLE]

is a simplicial map.

Given $v,v^{\prime}$ vertices in a simplicial complex $K$ and a Moore path $\gamma$ joining $v$ and $v^{\prime}$, that is, $\alpha(\gamma)=v$ and $\omega(\gamma)=v^{\prime}$, let us consider the Moore path $\widehat{\gamma}\mathrel{\mathop{:}}=|\overline{\gamma}|,$ that is the normalized of the reverse Moore path of $\gamma$. It is plain to check that the simplicial map $H\colon I_{2n}\times I_{n}\to B$ defined as

[TABLE]

gives a homotopy $H$ between $\widehat{\gamma}*\gamma$ and the constant Moore path $c_{b^{\prime}}$ relative to $\{0,n\}$. Similarly, one can also check that $\gamma*\widehat{\gamma}\sim c_{b}$ rel. $\{0,n\}$.

Using this language, a simplicial complex $B$ is connected if and only for any pair of vertices $b,b^{\prime}\in B$ there exists a normalized Moore path $\gamma\colon[0,n]\to B$ such that $\gamma(0)=b$ and $\gamma(n)=b^{\prime}$.

Given a simplicial fibration $f\colon E\to B$ and a vertex $b_{0}\in B$, the simplicial fiber of $p$ over $b_{0}$, denoted by $F_{b_{0}}$, is the full subcomplex of $E$ generated by all the vertices $e\in E$ such that $p(e)=b_{0}$. In other words, $F_{b_{0}}=p^{-1}(b_{0})$.

The following theorem is one of the main results of this paper, since it shows that our notion of simplicial fibration has nice properties like the homotopy invariance of the fiber.

Theorem 6.3**.**

Let $p\colon E\to B$ be a simplicial fibration where $B$ is a connected simplicial complex. Then any two simplicial fibers of $p$ have the same strong homotopy type.

Proof.

Let $\gamma\colon I_{n}\to B$ be a Moore path such that $\gamma(0)=b$ and $\gamma(n)=b^{\prime}$. Let us first check that there is a simplicial map $F_{b}\rightarrow F_{b^{\prime}}$. Indeed, if $i_{b}$ denotes the inclusion $F_{b}\subset E$ and $\mathrm{pr}_{2}\colon F_{b}\times I_{n}\rightarrow I_{n}$ the natural projection, then take a lift in the diagram:

[TABLE]

Clearly, we have that $\widetilde{\gamma}(v,n)\in F_{b^{\prime}},$ for all $v\in F_{b}$. Therefore there is an induced simplicial map $\gamma^{\sharp}:F_{b}\rightarrow F_{b^{\prime}}$ given by $\gamma^{\sharp}(v)\mathrel{\mathop{:}}=\widetilde{\gamma}(v,n)$.

Suppose $\delta\colon I_{n}\rightarrow B$ is another Moore path satisfying that $\delta(0)=b$, $\delta(n)=b^{\prime}$ and there is $H\colon I_{n}\times I_{m}\rightarrow B$ such that $H\colon\gamma\sim\delta$ rel. $\{0,n\}$. We shall prove that $\gamma^{\sharp}$ and $\delta^{\sharp}$ are in the same contiguity class. In particular we will prove that $\gamma^{\sharp}$ is independent, up to contiguity class, of the chosen lift $\widetilde{\gamma}$.

First, let us consider the commutative diagram:

[TABLE]

where:

• •

$J_{nm}=(I_{n}\times\{0\})\cup(\{0\}\times I_{m})\cup(I_{n}\times\{m\})$,

• •

$\mathrm{pr}\colon F_{b}\times I_{n}\times I_{m}\to F_{b}\times I_{n}$ is the projection on the two first complexes,

• •

$f$ is the simplicial map (see Lemma 6.2 above) defined as

[TABLE]

Now, take the simplicial maps $\phi\colon I_{n^{\prime}}\times I_{m^{\prime}}\rightarrow I_{n}\times I_{m}$ and $\phi^{\prime}\colon I_{n^{\prime\prime}}\times I_{m^{\prime\prime}}\rightarrow I_{n^{\prime}}\times I_{m^{\prime}}$ given by Lemma 6.1. As $\phi(I_{n^{\prime}}\times\{0\})\subset J_{nm}$, we have a lift $\widetilde{H}\colon F_{b}\times I_{n^{\prime}}\times I_{m^{\prime}}\rightarrow E$, represented by the dotted arrow in the composite diagram:

[TABLE]

Taking into account that $\phi^{\prime}(J_{n^{\prime\prime}m^{\prime\prime}})\subset I_{n^{\prime}}\times\{0\}$ and that $\phi\circ\phi^{\prime}=t_{1}\times t_{2},$ we have that the simplicial map

[TABLE]

satisfies that $H^{\prime}(v,n^{\prime\prime},j)\in F_{b^{\prime}},$ for all $v\in F_{b}$ and $j\in I_{m^{\prime\prime}}$. Therefore, there is an induced simplicial map

[TABLE]

It is not difficult to see that $\overline{H}(v,0)=\gamma^{\sharp}(v)$ and $\overline{H}(v,m^{\prime\prime})=\delta^{\sharp}(v);$ that is, $\gamma^{\sharp}\sim\delta^{\sharp}$ are in the same contiguity class.

It is immediate to check that, if $c_{b}\colon I_{n}\rightarrow B$ denotes the constant Moore path in $b\in B$, then $c_{b}^{\sharp}$ is (up to contiguity class) the identity $1_{F_{b}}\colon F_{b}\rightarrow F_{b}$. Moreover, if $\gamma^{\prime}\colon I_{n}\rightarrow B$ and $\delta^{\prime}\colon I_{m}\rightarrow B$ are Moore path spaces such that $\gamma(0)=b,$ $\gamma(n)=b^{\prime}=\delta(0)$ and $\delta(m)=b^{\prime\prime},$ then we can prove that

[TABLE]

belongs to the same contiguity class of the composition $\delta^{\sharp}\circ\gamma^{\sharp}$. Indeed, the simplicial map $\widetilde{\gamma*\delta}\colon F_{b}\times I_{n+m}\ \to E$ defined as

[TABLE]

gives a lift:

[TABLE]

This proves that $(\gamma*\delta)^{\sharp}$ and $\delta^{\sharp}\circ\gamma^{\sharp}$ are in the same contiguity class.

Using the above reasonings and the fact that $\overline{\gamma}*\gamma\sim c_{b^{\prime}}$ rel. $\{0,n\}$ and $\gamma*\overline{\gamma}\sim c_{b}$ rel. $\{0,n\}$, we conclude the proof of the result. ∎

7. Collapsible base

Minian’s Lemma (Lemma 6.1) and the Simplicial Pasting Lemma (Lemma 6.2) will be crucial for the next results. We start with this fairly general proposition.

Proposition 7.1**.**

Let $p\colon E\rightarrow B$ be a simplicial fibration and let $F_{0},F_{1}\colon K\times I_{n}\rightarrow E$ be simplicial maps such that $p\circ F_{0}$ and $p\circ F_{1}$ are in the same contiguity class with $m$ steps. Let $H\colon p\circ F_{0}\sim p\circ F_{1}$ be the map given by Proposition 2.8 and analogously, let $G\colon F_{0}|_{K\times\{0\}}\sim F_{1}|_{K\times\{0\}}$ be the map given by Proposition 2.8. Consider the following commutative diagram:

[TABLE]

Then, for suitable $q\geq n$ and $p\geq m,$ there exist subdivision maps $t_{1}\colon I_{q}\rightarrow I_{n}$ and $t_{2}\colon I_{p}\rightarrow I_{m}$ and a simplicial map

[TABLE]

such that $H^{\prime}$ defines a homotopy $F_{0}\circ(1_{K}\times t_{1})\sim F_{1}\circ(1_{K}\times t_{1})$, which is an extension of $G\circ(1_{K\times\{0\}}\times t_{2}).$

Proof.

Consider the commutative diagram

[TABLE]

where $J_{nm}$ is the full subcomplex of $I_{n}\times I_{m}$ given in the proof of Theorem 6.3 and $f$ is the simplicial map (see Lemma 6.2) defined as

[TABLE]

Now we take the simplicial maps $\phi\colon I_{n\prime}\times I_{m\prime}\rightarrow I_{n}\times I_{m}$ and $\phi^{\prime}\colon I_{n^{\prime\prime}}\times I_{m^{\prime\prime}}\rightarrow I_{n^{\prime}}\times I_{m^{\prime}}$ given by Lemma 6.1. As $\phi\circ(I_{n^{\prime}}\times\{0\})\subset J_{nm}$ we have a lift $\widetilde{H}\colon K\times I_{n^{\prime}}\times I_{m^{\prime}}\rightarrow E$, represented by the dotted arrow in the composite diagram:

[TABLE]

Renaming $q=n^{\prime\prime}$, $p=m^{\prime\prime}$, and taking into account that

[TABLE]

and that $\phi\circ\phi^{\prime}=t_{1}\times t_{2},$ we have that the simplicial map $H^{\prime}\mathrel{\mathop{:}}=\widetilde{H}\circ(1_{K}\times\phi^{\prime})$ fulfills all the expected requirements. ∎

Now, for our next result, we need to establish the following natural definition:

Definition 7.2**.**

Let $p\colon E\rightarrow B$ be a simplicial fibration and consider $f,g\colon K\rightarrow E$ simplicial maps. Then it is said that $f$ and $g$ are in the same class of fibrewise contiguity, denoted by $f\sim_{p}g,$ if there exists a finite sequence of simplicial maps

[TABLE]

such that $p\circ f=p\circ f_{i}$, for all $i=0,\dots,n$.

In other words, $f$ and $g$ are in the same class of fibrewise contiguity if there exists a simplicial map $F\colon K\times I_{n}\rightarrow X$ satisfying

1. (i)

$F(v,0)=f(v)$ and $F(v,n)=g(v)$, for all $v\in K;$ 2. (ii)

$p(F(v,i))=p(f(v))$, for all $v\in K$ and $i\in I_{n}$ (therefore $p\circ f=p\circ g$).

As a corollary of Proposition 7.1, we have:

Corollary 7.3**.**

Let $p\colon E\rightarrow B$ be a simplicial fibration and let $F_{0},F_{1}\colon K\times I_{n}\rightarrow E$ be two lifts of the same map

[TABLE]

Then, for a suitable $q\geq n,$ there exists a subdivision map $t\colon I_{q}\rightarrow I_{n}$ such that $F_{0}\circ(1_{K}\times t)\sim_{p}F_{1}\circ(1_{K}\times t).$

Proof.

Just apply Proposition 7.1 to the simplicial maps $G,H$ defined as $G(v,0,j)=f(v,0)$ and $H(v,i,j)=(p\circ f)(v,0),$ for all $v\in K$ and $j\in I_{n}$. ∎

Remark 4*.*

Observe that the previous corollary also holds true when we consider $K\times\{n\}$ in the diagram instead of $K\times\{0\}.$ In this case one just have apply the corollary by carefully using the simplicial isomorphism $I_{n}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}I_{n}$ given by $i\mapsto n-i$. Using this trick one can also check that a fibration $p\colon E\rightarrow B$ may be also characterized by the existence of a lift in any diagram of the form

[TABLE]

For our next result we will consider the notion of having the same type of fibrewise contiguity.

Definition 7.4**.**

We say that two simplicial fibrations $p_{1}\colon E_{1}\rightarrow B$, $p_{2}\colon E_{2}\rightarrow B$ have the same type of fibrewise contiguity when there exist simplicial maps $f\colon E_{1}\rightarrow E_{2}$ and $g\colon E_{2}\rightarrow E_{1}$ such that $p_{2}\circ f=g$, $p_{1}\circ g=f$ and $g\circ f\sim_{p_{1}}1_{E_{1}}$ and $f\circ g\sim_{p_{2}}1_{E_{2}}$.

The following important result asserts that simplicial maps in the same class of contiguity induce simplicial fibrations having the same type of fibrewise contiguity.

Theorem 7.5**.**

Let $p\colon E\rightarrow B$ be a simplicial fibration and let $f_{0},f_{1}\colon K\rightarrow B$ simplicial maps. Consider, for each $i=1,2$, the pullback of $f_{i}$ along $p$:

[TABLE]

If $f_{0}$ and $f_{1}$ are in the same class of contiguity, then the simplicial fibrations $p_{0}$ and $p_{1}$ have the same type of fibrewise contiguity.

Proof.

Consider a simplicial map $F\colon K\times I_{n}\rightarrow B$ such that $F(v,0)=f_{0}(v)$ and $F(v,n)=f_{1}(v),$ for all $v\in K.$ Take $F^{\prime}_{0}$, $F^{\prime}_{1}$ lifts of the following diagrams (for the second diagram see Remark 4):

[TABLE]

By the pullback property there are well defined simplicial maps $g_{0}\colon E_{0}\rightarrow E_{1}$ and $g_{1}\colon E_{1}\rightarrow E_{0}$ satisfying the commutativities given in the following diagrams:

[TABLE]

Let us first check that $g_{0}\circ g_{1}\sim_{p_{1}}1_{E_{1}}.$ Indeed, observe that $F^{\prime}_{0}\circ(g_{1}\times 1)$ and $F^{\prime}_{1}$ are lifts of the same map

[TABLE]

Corollary 7.3 assures the existence of a suitable subdivision map $t\colon I_{n^{\prime}}\rightarrow I_{n}$ and a simplicial map $G\colon E_{1}\times I_{n^{\prime}}\times I_{m}\rightarrow E$ satisfying that

[TABLE]

Again, using the pullback property, there is an induced simplicial map $\widetilde{G}\colon E_{1}\times I_{m}\rightarrow E_{1}$ satisfying

[TABLE]

A simple inspection proves that $\widetilde{G}\colon g_{0}\circ g_{1}\sim_{p_{1}}1_{E_{1}}.$

Analogously, by applying Corollary 7.3 to the diagram

[TABLE]

for the common lifts $F^{\prime}_{1}\circ(g_{0}\times 1)$ and $F^{\prime}_{0}$, and taking into account Remark 4, one can find a simplicial map $H$ satisfying

[TABLE]

for a suitable subdivision map $s\colon I_{n^{\prime\prime}}\rightarrow I_{n}.$ Taking $\widetilde{H}$ the simplicial map characterized by the equalities $f^{\prime}_{0}\circ\widetilde{H}=G(-,0,-)$ and $p_{0}\circ\widetilde{H}=p_{0}\circ\mathrm{pr}$ we obtain $\widetilde{H}\colon g_{1}\circ g_{0}\sim_{p_{0}}1_{E_{0}}.$ ∎

Corollary 7.6**.**

Let $p\colon E\rightarrow B$ be a simplicial fibration where $B$ is strongly collapsible. Then $p$ has the same class of fibrewise contiguity of the trivial fibration $B\times p^{-1}(b_{0})\rightarrow B$, for any vertex $b_{0}\in B.$

Proof.

Just take into account that $1_{B}\colon B\rightarrow B$ and the constant path $c_{b_{0}}\colon B\rightarrow B$ are in the same class of contiguity and use Theorem 7.5. ∎

We finish this section with an interesting property whose proof relies on the proof of Theorem 7.5. Indeed, remember that, in the statement of such theorem, we have pullbacks ($i=0,1$):

[TABLE]

where $f_{0}\sim f_{1}$ (i.e., $f_{0}$ and $f_{1}$ are in the same class of contiguity). From the proof one obtains simplicial maps $g_{0}:E_{0}\rightarrow E_{1}$ and $g_{1}:E_{1}\rightarrow E_{0}$ (with $p_{1}\circ g_{0}=p_{0}$ and $p_{0}\circ g_{1}=p_{1}$) and Minian simplicial homotopies

[TABLE]

satisfying $F^{\prime}_{0}:f^{\prime}_{0}\sim f^{\prime}_{1}\circ g_{0}$ and $F^{\prime}_{1}:f^{\prime}_{0}\circ g_{1}\sim f^{\prime}_{1}$. Moreover, $g_{0}$ and $g_{1}$ satisfy $g_{1}\circ g_{0}\sim_{p_{0}}1_{E_{0}}$ and $g_{0}\circ g_{1}\sim_{p_{1}}1_{E_{1}}.$ In particular $g_{1}\circ g_{0}\sim 1_{E_{0}}$ and $g_{0}\circ g_{1}\sim 1_{E_{1}}.$

We easily have the following lemma:

Lemma 7.7**.**

Consider the pullback of a simplicial fibration $p:E\rightarrow B$ along a simplicial map $f:B\rightarrow B$ such that $f\sim 1_{B}:$

[TABLE]

Then $f^{\prime}\sim g_{0}$ and $f^{\prime}\circ g_{1}\sim 1_{E}.$ In particular, $f^{\prime}$ is a strong equivalence with $g_{1}:E\rightarrow E^{\prime}$ as a homotopy inverse.

Proof.

Just observe that the following square is a pullback

[TABLE]

and apply the argument above. ∎

And finally our result. Observe that such result completes Proposition 4.2.

Proposition 7.8**.**

Consider the pullback of a simplicial fibration $p:E\rightarrow B$ along a simplicial map $f:K\rightarrow B$:

[TABLE]

If $f$ is a strong equivalence, then $f^{\prime}$ is also a strong equivalence.

Proof.

Suppose $g:B\rightarrow K$ a simplicial map such that $g\circ f\sim 1_{K}$ and $f\circ g\sim 1_{B}.$ Consider first the following diagram, where $E^{\prime\prime}$ is the pullback of $p^{\prime}$ along $g$:

[TABLE]

As $f\circ g\sim 1_{B}$, by the lemma above (note that the composite of pullbacks is a pullback) one can find simplicial maps $g_{0}:E^{\prime\prime}\rightarrow E$ and $g_{1}:E\rightarrow E^{\prime\prime}$ such that $g_{1}\circ g_{0}\sim 1_{E^{\prime\prime}}$, $g_{0}\circ g_{1}\sim 1_{E}$, $f^{\prime}\circ g^{\prime}\sim g_{0}$ and $f^{\prime}\circ g^{\prime}\circ g_{1}\sim 1_{E}$. Therefore, $g^{\prime}$ has a left homotopy inverse and $f^{\prime}$ has a right homotopy inverse:

[TABLE]

Similarly, from the diagram of pullbacks

[TABLE]

we have that $g^{\prime}$ has a right homotopy inverse. It straightforwardly follows that $g^{\prime}$ is a strong equivalence. Since $f^{\prime}\circ g^{\prime}\sim g_{0}$ we conclude that $f^{\prime}$ is a strong equivalence. ∎

8. Varadarajan’s theorem

As an application of Theorem 6.3 about the strong homotopy type of the fibers of a fibration, we prove in this section a simplicial version of a well-known result from Varadarajan for topological fibrations [22], establishing a relationship between the LS-categories of the total space, the base and the homotopic fiber. A more general result for smooth foliations was proved by Colman and Macías in [6].

Definition 8.1**.**

The simplicial LS-category $\mathop{\mathrm{scat}}(K)$ of the simplicial complex $K$ is the least integer $n\geq 0$ such that $K$ can be covered by $n+1$ subcomplexes $K_{j}$ such that each inclusion $\iota_{j}\colon K_{j}\subset K$ belongs to the contiguity class of some constant map $c_{v_{j}}\colon K_{j}\to K$.

This notion is the simplicial version of the well known homotopic invariant $\mathrm{cat}(X)$, the so-called Lusternik-Schnirelmann category of the topological space $X$ [5]. It has been introduced by the authors in [12, 14], its most important property being the invariance by strong homotopy equivalences.

Accordingly to Theorem 6.3, all the fibers $p^{-1}(v)$, $v\in B$, of a fibration with connected base $p\colon E\to B$ have the same strong homotopy type, so we call generic fiber $F$ of the fibration any simplicial complex into that equivalence class, and its simplicial category $\mathop{\mathrm{scat}}(F)$ is well defined.

Theorem 8.2**.**

Let $p\colon E\to B$ be a simplicial fibration with connected base $B$ and generic fiber $F$.Then

[TABLE]

Proof.

Let $\mathop{\mathrm{scat}}B=m$, and take a categorical covering $U_{0},\dots,U_{m}$ of $B$. From Theorem 6.3 we know that all the fibers have the same strong homotopy type. We identify $F$ to the fiber $p^{-1}(v)$ over some base point $v\in B$. Let $\mathop{\mathrm{scat}}F=n$, with $V_{0},\dots,V_{n}$ a categorical covering of $F$.

For each $i\in\{0,\dots,m\}$ let $I_{i}\colon U_{i}\subset B$ be the inclusion. By definition of simplicial category, the map $I_{i}$ is in the contiguity class of a constant map, say $c_{i}\colon U_{i}\to B$. Since $B$ is path connected we can assume that $c_{i}$ is the constant map corresponding to the base point $v$. Consider the map

[TABLE]

If $\epsilon_{i}\colon p^{-1}(U_{i})\subset E$ is the inclusion, then $p\circ\epsilon_{i}=p_{i}$. Now, from $I_{i}\sim c_{v}$ it follows that $p_{i}\sim c_{v}$, the latter being the constant map with domain $p^{-1}(U_{i})$. By the contiguity lifting property, there exists

[TABLE]

such that $\epsilon_{i}\sim G_{i}$ and $p\circ G_{i}=c_{v}$. The latter means that the map $G_{i}$ takes its values in $F$. We denote

[TABLE]

the map given by $g_{i}(v)=G_{i}(v)$. In this way $G_{i}=\iota_{F}\circ g_{i}$, where $\iota_{F}\colon F\subset E$ is the inclusion.

For each $i\in\{0,\dots,m\}$, $j\in\{0,\dots,n\}$, we take the subcomplex

[TABLE]

Since $B=U_{0}\cup\cdots\cup U_{m}$ implies $E=p^{-1}(U_{0})\cup\cdots\cup p^{-1}(U_{m})$, and since $F=V_{0}\cup\cdots\cup V_{n}$ implies $p^{-1}(U_{i})=W_{i0}\cup\cdots\cup W_{in}$, it follows that $\{W_{ij}\}$ is a covering of $E$.

It only remains to prove that each $W_{ij}$ is categorical in $E$. Let $g_{ij}\colon W_{ij}\to V_{j}$ be the restriction of $g_{i}$ to $W_{ij}\subset p^{-1}(U_{i})$. We know that each $V_{j}\subset F$ is categorical, so the inclusion $J_{j}\colon V_{j}\subset F$ is in the contiguity class of some constant map $c_{j}\colon V_{j}\to F$, whose image is some vertex $f_{j}$ of $F$. Then the composition

[TABLE]

belongs to the contiguity class of the constant map $f_{j}\colon W_{ij}\to F$ since

[TABLE]

Let $\epsilon_{ij}\colon W_{ij}\subset E$ be the inclusion, that is, the restriction of $\epsilon_{i}\colon p^{-1}(V_{j})\subset E$ to $W_{ij}$. Since $\epsilon_{i}\sim G_{i}$ it follows that $\epsilon_{ij}\sim G_{ij}$, the latter being the restriction of $G_{i}$ to $W_{ij}$. Finally we have

[TABLE]

so the inclusion $W_{ij}\subset E$ is in the contiguity class of a constant map. ∎

9. Factorization

In this section we will see that any simplicial map may be considered, in a homotopical sense, as a simplicial fibration.

For our main result in this section we will use the fact that $(\alpha,\omega)\colon\mathrm{P}K\to K\times K$ is a simplicial finite-fibration, for any simplicial complex $K$ (see Theorem 5.6).

Theorem 9.1**.**

Let $f\colon K\to L$ be a simplicial map. Then there is a factorization

[TABLE]

where $j$ is a $P$-homotopy equivalence and $p$ is a simplicial finite-fibration.

Proof.

We shall denote by $P_{f}=K\times_{L}\mathrm{P}L$ the pullback of $f\colon K\to L$ along $\alpha\colon\mathrm{P}L\to L$, where $\alpha$ is the initial vertex map. In this way we have a commutative square:

[TABLE]

As we have previously observed, the simplicial complex $P_{f}$ is defined as the full subcomplex of $K\times\mathrm{P}L$ whose set of vertices is given as

[TABLE]

and $\alpha^{\prime},f^{\prime}$ are the restrictions of the obvious projections. We define the simplicial map $j\colon K\to P_{f}$ as $j(v)\mathrel{\mathop{:}}=(v,c_{f(v)}),$ where $c_{f(v)}\colon\mathbb{Z}\to L$ denotes the constant Moore path at $f(v)$. On the other hand, the simplicial map $p\colon P_{f}\to L$ is defined as $p(v,\gamma)\mathrel{\mathop{:}}=\omega(\gamma)$, where $\omega$ is the final vertex map. Obviously, we have that $p\circ j=f$.

Let us first check that $p$ is a simplicial fibration. Indeed, consider the following commutative diagram:

[TABLE]

It is not difficult to check that the bottom diagram is a pullback. As, by construction, the composition diagram is also a pullback, we obtain that the top diagram must be a pullback. But then, being $(\alpha,\omega)$ a simplicial finite-fibration, the simplicial map $(\alpha^{\prime},p)\colon P_{f}\to K\times L$ must be also a simplicial finite-fibration. Therefore, the composition of the fibrations $p=\mathrm{pr}_{2}\circ(\alpha^{\prime},p)\colon P_{f}\to L$ is a finite-fibration.

Next we check that $j\colon K\to P_{f}$ is a $P$-homotopy equivalence. As $\alpha^{\prime}\circ j=1_{K}$ it only remains to see that $j\circ\alpha^{\prime}\simeq 1_{P_{f}}$. We directly define the homotopy $H\colon P_{f}\to\mathrm{P}P_{f}$ as $H(v,\gamma)(i)\mathrel{\mathop{:}}=(v,\gamma_{i})$ where $\gamma_{i}\in\mathrm{P}L$ is defined as

[TABLE]

for any $\gamma\in\mathrm{P}L$ and $i\in\mathbb{Z}$. Observe that $\gamma_{i}$ is the $i$-truncated Moore path of $\gamma$ and hence, $\alpha(\gamma_{i})=\alpha(\gamma)$ and $\omega(\gamma_{i})=\gamma(i)$, so $(v,\gamma_{i})$ is a well defined element in $P_{f}$. Moreover, if $(v,\gamma)\in P_{f}$ is fixed, we are going to check that $H(v,\gamma)\in\mathrm{P}P_{f}$; equivalently, we have to prove that $H(v,\gamma)\colon\mathbb{Z}\to P_{f}$ is a simplicial map and a eventually constant Moore path, that is, for any integer $i\in\mathbb{Z}$, the image

[TABLE]

is a simplex in $P_{f}$, or equivalently, that $\{\gamma_{i},\gamma_{i+1}\}$ is a simplex in $\mathrm{P}L$. But this is true, as, for any integer $j\in\mathbb{Z}$, the set of vertices

[TABLE]

is precisely $\gamma(\{j,j+1\})$ or $\gamma(\{i,i+1\})$, depending on whether $j\leq i$ or $j\geq i$. Thus $H$ is a simplicial map. Now, taking into account that

[TABLE]

we have that $H(v,\gamma)\in\mathrm{P}P_{f}$. Moreover, $\alpha\circ H=j\circ\alpha^{\prime}$ and $\omega\circ H=1_{P_{f}}$. Therefore, to conclude that $H$ is a P-homotopy equivalence between $j\circ\alpha^{\prime}$ and $1_{P_{f}}$, we just need to justify that it is a simplicial map. For this final task, considering the exponential law and the fact that $P_{f}$ is a full subcomplex of $K\times L^{\mathbb{Z}}$, it suffices to check that the following map is simplicial:

[TABLE]

Now if consider $\sigma$ a simplex in $K$, $\{\gamma_{0},\dots,\gamma_{p}\}$ a simplex in $L^{\mathbb{Z}}$ and $\{i,i+1\}$ a simplex in $\mathbb{Z}$, then we have to prove that the image

[TABLE]

that is,

[TABLE]

is a simplex in $K\times L^{\mathbb{Z}},$ or equivalently

[TABLE]

is a simplex in $L^{\mathbb{Z}}$. So take any integer $j\in\mathbb{Z}$ and consider

[TABLE]

Then, it is not difficult to check that this set of vertices is precisely

[TABLE]

or

[TABLE]

depending on whether $j\leq i$ or $j\geq i$. But whatever the case is, we have a simplex in $L$ because $\{\gamma_{0},\dots,\gamma_{p}\}$ is a simplex in $L^{\mathbb{Z}}$. We conclude the proof of the theorem. ∎

As a consequence of this theorem new finite-fibrations appear.

Example 9.2**.**

For instance, for any simplicial complex $K$ and a based vertex $v_{0}\in K$ one can consider the full subcomplex of $\mathrm{P}K$

[TABLE]

Observe that this simplicial complex is nothing else than the construction $P_{f}$ given in the theorem above, where $f\colon\{v_{0}\}\hookrightarrow K$ is the inclusion map. Then, the simplicial finite-fibration associated to $f$ is precisely the map

[TABLE]

Example 9.3**.**

There is also a special factorization that we are specially interested in. Although it does not come from the general construction of the above theorem, one can check that the following diagram gives a factorization of the diagonal map $\Delta\colon K\to K\times K$ through a $P$-homotopy equivalence followed by a simplicial finite-fibration

[TABLE]

We already know that $(\alpha,\omega)$ is a simplicial finite-fibration. On the other hand, $c\colon K\to\mathrm{P}K$ is defined as the simplicial map sending each $v\in K$ to the constant Moore path at $v$, $c(v)\mathrel{\mathop{:}}=c_{v}$. This map is, indeed, a $P$-homotopy equivalence: the simplicial map $\alpha\colon\mathrm{P}K\to K$ satisfies $\alpha\circ c=1_{K}$. Moreover, the homotopy $H\colon\mathrm{P}K\to\mathrm{P}^{2}K$ defined as $H(\gamma)(i)=\gamma_{i}$, where $\gamma_{i}$ is the $i$-truncated Moore path of $\gamma$, satisfies $\alpha\circ H=c\circ\alpha$ and $\omega\circ H=1_{\mathrm{P}K};$ that is, $H\colon c\circ\alpha\simeq 1_{\mathrm{P}K}$.

10. ​​Švarc genus

In the classical topological setting, the Lusternik-Schnirelmann category can be seen as a particular case of the so-called “Švarc genus” or sectional category of a continuous map.

In this Section we adapt this definition to the simplicial setting.

Definition 10.1**.**

The simplicial Švarc genus of a simplicial map $\varphi\colon K\to L$ is the minimun integer $n\geq 0$ such that $L$ is the union $L_{0}\cup\dots\cup L_{n}$ of $n+1$ subcomplexes, and for each $j$ there exists a section $\sigma_{j}$ of $\varphi$, that is, a simplicial map $\sigma_{j}\colon L_{j}\to K$ such that $\varphi\circ\sigma_{j}$ is the inclusion $\iota_{j}\colon L_{j}\subset L$.

We denote this genus by $\mathrm{Sg}(\varphi)$. A slight modification in the above definition is to change the equality by “being in the same contiguity class”.

Definition 10.2**.**

The homotopy simplicial Švarc genus of a simplicial map $\varphi\colon K\to L$, denoted by $\mathrm{hSg}(\varphi)$, is the minimum $n\geq 0$ such that $L=L_{0}\cup\dots\cup L_{n}$ , and for each $j\in\{0,\dots,n\}$ there exists an “up to contiguity class” simplicial section $\sigma_{j}$ of $\varphi$, that is, a simplicial map $\sigma_{j}\colon L_{j}\to K$ such that $\varphi\circ\sigma_{j}\sim\iota_{j}$.

Remark 5*.*

Note that when the complex $L$ is finite, the condition $\varphi\circ\sigma_{j}\sim\iota_{j}$ (same contiguity class) can be changed to $\varphi\circ\sigma_{j}\simeq\iota_{j}$ (P-homotopy), by Proposition 5.10.

Obviously, $\mathrm{hSg}(\varphi)\leq\mathrm{Sg}(\varphi)$. The equality holds for some particular classes of maps.

Theorem 10.3**.**

Let $p\colon E\to B$ be a simplicial fibration. Then $\mathrm{hSg}(p)=\mathrm{Sg}(p)$.

Proof.

We only have to prove that $\mathrm{Sg}(\varphi)\leq\mathrm{hSg}(\varphi)$, so let $\mathrm{hSg}(\varphi)=n$ and $L_{0},\dots,L_{n}$ be a covering of $B$ by subcomplexes such that there exist simplicial maps $\sigma_{j}\colon L_{j}\to E$ with $p\circ\sigma_{j}\sim\iota_{j}$, as in Definition 10.2. By Theorem 3.4, for each $j$ we have simplicial maps

[TABLE]

with $H_{j}(-,0)=p\circ\sigma_{j}$ and $H_{j}(-,m_{j})=\iota_{j}$ the inclusion $L_{j}\subset B$. Take a lift $\tilde{H}_{j}\colon L_{j}\times[0,m_{j}]\to E$ in the following diagram,

\textstyle{\ \ L_{j}\times\{0\}\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ \ \ \ \sigma_{j}\ }$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{L_{j}\times[0,m_{j}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ \ \ H_{j}}$$\scriptstyle{\tilde{H}_{j}}$$\textstyle{B}

in such a way that $p\circ\tilde{H}=H$ and $\tilde{H}(v,0)=\sigma_{j}(v)$. Then the map $\xi_{j}\colon L_{j}\to E$ given by $\xi_{j}(v)=\tilde{H}(v,m_{j})$ is simplicial and verifies

[TABLE]

so it is a true section of $p$. Then $\mathrm{Sg}(\varphi)\leq n$. ∎

Remark 6*.*

Note that if $p\colon E\to B$ is a simplicial finite-fibration then Theorem 10.3 above is still true when the base $B$ is finite.

10.1. Simplicial L-S category

Taking into account the notion of simplicial LS-category (Definition 8.1), what we want to see is that it equals (as in the classical case) the Švarc genus of a certain fibration.

Assume that the complex $K$ is connected. Then every two constant maps $c_{v_{i}}$, $c_{v_{j}}$, belong to the same contiguity class, as can be easily seen by considering a path $\gamma\colon[0,m]\to K$ connecting $v_{i}$ and $v_{j}$ and the sequence of contiguous maps given by the truncated paths $\gamma_{k}\colon[0,k]\to K$. So we can choose a base point $v_{0}$ and assume that all the constant maps in Definition 8.1 equal $c_{v_{0}}$.

Let $P_{0}K$ be the full subcomplex of the path complex $PK$ consisting on all Moore paths whose initial point is the base point,

[TABLE]

The map $\omega\colon P_{0}K\to K$, sending each path to its final point, is then the pullback of the simplicial finite-fibration $(\alpha,\omega)\colon PK\to K\times K$ by the map $f$, where $f(v)=(v_{0},v)$:

\textstyle{\ P_{0}K\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{PK\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\alpha,\omega)}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{K\times K}

Theorem 10.4**.**

Let $K$ be a connected finite complex. Then the simplicial LS-category $\mathop{\mathrm{scat}}(K)$ equals the Švarc genus of the simplicial finite-fibration $\omega\colon P_{0}K\to K$.

Proof.

We have to prove that $\mathop{\mathrm{scat}}(K)=\mathrm{Sg}(\omega)$, but we know that the latter equals $\mathrm{hSg}(\omega)$.

So, first, assume that $\mathop{\mathrm{scat}}(K)=n$ and let $K=K_{0}\cup\dots\cup K_{n}$ such that each inclusion $\iota_{j}\colon K_{j}\subset K$ belongs to the contiguity class of the constant map $v_{0}$, that is, there is a Minian simplicial homotopy $H_{j}\colon K_{j}\times[0,m_{j}]\to K$ with $H_{j}(v,0)=v_{0}$ and $H_{j}(v,m_{j})=v$.

Take a lift $\tilde{H}_{j}$ of $H_{j}$ in the following diagram:

\textstyle{\ \ K_{j}\times\{0\}\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ \ \ \ c_{j}\ }$$\textstyle{P_{0}K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{K_{j}\times[0,m_{j}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ \ \ H_{j}}$$\scriptstyle{\tilde{H}_{j}}$$\textstyle{K}

where $c_{j}(v)$ is the constant Moore path at $v_{0}$, for all $v$. Define $\sigma_{j}\colon K_{j}\to P_{0}K$ as $\sigma_{j}(v)=\tilde{H}_{j}(v,m_{j})$. Then $\omega\circ\sigma_{j}(v)=H_{j}(v,m_{j})=v$, so $\sigma_{j}$ is a section of the fibration. The map $\sigma_{j}$ is simplicial since

[TABLE]

is a simplicial map. We have then proved that $\mathrm{Sg}(\omega)\leq n$.

Conversely, if $s\colon L\to P_{0}K$ is a section of $\omega$, each path $s(v)$ has initial vertex $v_{0}$ and final vertex $v$. Denote by $[v^{-},v^{+}]$ the support of $s(v)$. Then, since $K$ is finite, there is an interval $[m,n]$ containing all the supports, so we can define $H\colon L\times[m,n]\to K$ as $H(v,i)=s(v)(i)$. Then $H(v,m)=s(v)(m)=s(v)(v^{-})=\alpha(s(v))=v_{0}$, while $H(v,m)=s(v)(n)=s(v)(v^{+})=\omega(s(v))=v$, showing that the inclusion $L\subset K$ belongs to the contiguity class of the constant map $v_{0}$.

We must check that $H$ is simplicial: if $\{v_{0},\dots,v_{p}\}$ is a simplex of $L$ and $\{i,i+1\}$ is a simplex in $[m,n]$, the image by $H$ of $\sigma\times\{i,i+1\}$ is the set

[TABLE]

which is a simplex of $K$ because $s(\sigma)$ is a simplex of $P_{0}K$.

So each covering of $K$ by subcomplexes $L$ verifying the Definition 10.1 gives the same covering verifying Definition 8.1. Then $\mathop{\mathrm{scat}}(K)\leq\mathrm{Sg}(\omega)$. ∎

10.2. Discrete topological complexity

In [13], a subset of the authors introduced a notion of discrete topological complexity which is a version of Farber’s topological complexity [11], adapted to the simplicial setting.

By using well known equivalences in the topological setting, the simplicial definition avoids the use of any path complex.

Definition 10.5**.**

[13] The discrete topological complexity $\mathop{\mathrm{TC}}(K)$ of the simplicial complex $K$ is the least integer $n\geq 0$ such that $K\times K$ can be covered by $n+1$ “Farber subcomplexes” $\Omega_{j}$, where the latter means that there exist simplicial maps $\sigma_{j}\colon\Omega_{j}\to K$ such that $\Delta\circ\sigma_{j}$ is in the contiguity class of the inclusion $\iota_{j}\colon\Omega_{j}\subset K\times K$. Here $\Delta:K\to K\times K$ denotes the diagonal map $v\mapsto(v,v)$.

In other words:

Proposition 10.6**.**

The discrete topological complexity of the abstract simplicial complex $K$ is the homotopic Švarc genus of the diagonal map $\Delta\colon K\to K\times K$, i.e., $\mathop{\mathrm{TC}}(K)=\mathrm{hSg}(\Delta)$.

Our main result in this ection is the following one.

Theorem 10.7**.**

Let $K$ be a finite complex. The discrete topological complexity of $K$ equals the Švarc genus of the finite-fibration $(\alpha,\omega)\colon PK\to K\times K$.

Proof.

As was stated in Example 9.3, there is a $P$-equivalence $c$ between the complexes $K$ and $PK$, in such a way that the diagonal factors through the finite-fibration $(\alpha,\omega)\colon PK\to K\times K$. Since $K\times K$ is finite, Remark 5 applies and the homotopic Švarc genus of $\Delta$ and $(\alpha,\omega)$ can be computed by means of $P$-homotopies. Then it is clear that

[TABLE]

where the latter equality follows from Proposition 10.6. ∎