Simplicial fibrations

Noname manuscript No. (will be inserted by the editor) Simplicial fibrations Desamparados Fern´andez-Ternero · Jos´e Manuel Garc´ıa-Calcines ·Enrique Mac´ıas-Virg´os ·Jos´e Antonio Vilches Received: date / Accepted: date Abstract We undertake a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of “homotopy” has been replaced by that of “contiguity”.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 that introduced by G. Minian, established in terms of a cylinder construction K×Im. 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; any fibration with a strongly collapsible base is fibrewise trivial. We also introduce the concept of “simplicial finite-fibration”, that is, a simplicial map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration PK→K×Kis a finite-fibration, where PKis the simplicial complex of Moore paths introduced by M. Grandis. This result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we prove a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber. Finally, we introduce a definition of “ˇ Svarc genus” of a simplicial map and we are able to compare the ˇ Svarc The first and the fourh authors were partially supported by MINECO Spain Research Project MTM2015–65397–P and Junta de Andaluc´ıa Research Groups FQM–326 and FQM–189. The second and third authors were partially supported by MINECO-FEDER research project MTM2016–78647–P D. Fern´andez-Ternero ·J.A. Vilches Dpto. de Geometr´ıa y Topolog´ıa, Universidad de Sevilla, Spain Tel.: +34-954-557196 Fax: +34-954-557970 E-mail: [email protected],vilc[email protected] J.M. Garc´ıa-Calcines Dpto. de Matem´aticas, Estad´ıstica e Investigaci´on Operativa. Universidad de La Laguna, Spain E-mail: [email protected] E. Mac´ıas-Virg´os Dpto. de Matem´aticas, Universidade de Santiago de Compostela, Spain E-mail: [email protected] 2 Desamparados Fern´andez-Ternero et al. genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers. Keywords Simplicial complexes ·contiguous simplicial maps ·fibrations · LS-category ·topological complexity Mathematics Subject Classification (2010) 55U05 ·55M30 ·55M99 ·55R65 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 [8] or discrete Morse theory [14], 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 [5]. 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 [4]. 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 [11,13]. Moreover, we also introduced a notion of topological complexity, defined in purely combinatorial terms [12]. Both invariants have similar properties to the classical ones and also new results arise. As a collateral result, cofibrations were studied in [13], 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 the notion of simplicial fibration in terms of a contiguity lifting property and we show that in fact it is equivalent to the notion introduced by Minian in [19]: a simplicial map p:E→Bis a simplicial fibration if for any simplicial map H:K×Im→Band any simplicial map f:K×{0} → E, there is a simplicial map ˜ H:K×Im→Esuch that p◦˜ H=Hand ˜ H◦im 0=f (see Definition 7 and Proposition 3). There is a less restrictive notion of simplicial finite-fibration if we limit the lifting property to finite complexes Kin the definition above. The finiteness condition will be necessary in order to obtain several key results that involve the path fibration studied in Section 5, as Theorems 1 and 4. Simplicial fibrations 3 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 PKof 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 PK→K×Kis a simplicial finite-fibration (Theorem 1). 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. In the next section we introduce one new idea based on a notion of “P- homotopy” modelled on Moore paths (see Definition 13), which allows us to prove the P-equivalence of the complexes Kand PKand to give a general result about the factorization of any map into a P-equivalence and a finite-fibration (Section 8). Our next result (Section 9) is a simplicial version of Varadarajan’s theorem (see Theorem 5) relating the LS-category of the total space, the base and the generic fiber of a fibration. 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 ˇ Svarc genus for simplicial maps and to discuss its relationship with the simplicial LS- category and the discrete topological complexity introduced in our previous papers [11,13,12]. 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, [16,21] for more details on this topic). Definition 1 An (abstract) simplicial complex is a set Vtogether with a collection Kof finite subsets of Vsuch that if σ∈Kand τ⊆σthen τ∈K. Notice that Kis not necessarily finite in the above definition. As usual, Kwill denote the simplicial complex and V(K) the corresponding vertex set. Definition 2 Given two simplicial complexes Kand L, a simplicial map from K to Lis a set map ϕ:V(K)→V(L) such that if σ∈Kthen ϕ(σ)∈L. Definition 3 Let K, L be two simplicial complexes. Two simplicial maps ϕ, ψ :K→Lare contiguous [21, p. 130] if, for any simplex σ∈K, the set ϕ(σ)∪ψ(σ) is a simplex of L; that is, if v0,...,vkare the vertices of σthen the vertices f(v0),...,f(vk), g(v0),...,g(vk) span a simplex of L. This relation, denoted by ϕ∼cψ, is reflexive and symmetric, but in general it is not transitive. In order to overcome this fact we use the notion of contiguity class. 4 Desamparados Fern´andez-Ternero et al. Definition 4 Two simplicial maps ϕ, ψ :K→Lare in the same contiguity class with msteps, denoted by ϕ∼ψ, if there is a finite sequence ϕ=ϕ0∼c· · · ∼cϕm=ψ of contiguous simplicial maps ϕi:K→L, 0 ≤i≤m. 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 5 For n≥1, let Inbe the one-dimensional simplicial complex whose vertices are the integers {0,...,n}and the edges are the pairs {j, j + 1}, for 0≤j < n. Definition 6 [16, Definition 4.25] Let Kand Lbe two simplicial complexes. The categorical product K×Lis the simplicial complex whose set of vertices is V(K×L) = V(K)×V(L),and whose simplices are given by the rule: σ∈K×L if and only if pr1(σ)∈Kand pr2(σ)∈L, where pr1,pr2are 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 1 [19, Prop. 2.16] Two simplicial maps f, g :K→Lare in the same contiguity class, with m≥1steps, f∼g, if and only if there exists a simplicial map H:K×Im→Lsuch that H(v, 0) = f(v)and H(v, m) = g(v), for all vertices v∈K. Remark 1 The preceding proposition holds when we consider K×Imto 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 [16]). Barmak and Minian introduced [4,5] the so-called strong homotopy type for simplicial complexes. Two simplicial complexes have the same strong homotopy type, denoted by K∼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 vis dominated by another vertex v0if every maximal simplex that contains valso contains v0. A complex is called strongly collapsible if it has the same strong homotopy type that a point. Strong homotopy type is deeply related to the notion of contiguity between simplicial maps. More precisely, the following result holds: Proposition 2 [5, Cor. 2.12] Two simplicial complexes Kand Lhave the same strong homotopy type if and only if there are simplicial maps ϕ:K→Land ψ:L→Ksuch that ψ◦ϕ∼1Kand ϕ◦ψ∼1L. Simplicial fibrations 5 3 Definition 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. The definition of fibration we give below corresponds to simplicial maps with the contiguity lifting property with respect to any simplicial complex. Definition 7 A simplicial map p:E→Bis a simplicial fibration if for any simplicial complex K, for any two simplicial maps f, g :K→Bin the same contiguity class, f∼g, with msteps, and for any map ˜ f:K→Esuch that p◦˜ f=f, there exists a simplicial map ˜g:K→Esuch that ˜ fand ˜gare in the same contiguity class with msteps, ˜ f∼˜g, and for the sequences of contiguous simplicial maps f=ϕ0∼c· · · ∼cϕm=gand ˜ f=eϕ0∼c· · · ∼ceϕm= ˜g it holds that p◦eϕi=ϕi, 0 ≤i≤m. E p  Kf// g// ˜ f == ˜g == B We can obtain an equivalent notion of simplicial fibration if we consider the notion of homotopy for simplicial complexes introduced by Minian (see Prop. 1). Proposition 3 A simplicial map p:E→Bis a simplicial fibration if and only if, given simplicial maps H:K×Im→Band ϕ:K×{0} → Eas in the following commutative diagram: K× {0} _ im 0  ϕ//E p  K×Im e H :: H//B (1) there exists a simplicial map e H:K×Im→Esuch that e H◦im 0=ϕand p◦e H=H. Proof First, let us assume that the simplicial map p:E→Bis a simplicial fibration. Consider a simplicial complex Kand simplicial maps ϕ:K× {0} → Eand H:K×Im→Bsuch that Diagram (1) is commutative. Now, we define ϕi:K→Bby ϕi(v) = H(v, i), where v∈V(K) and i∈V(Im). These maps are contiguous because given τ∈Kthe following fact holds true: ϕi(τ)∪ϕi+1(τ) = H(τ× {i, i + 1})∈B(2) since His a simplicial map. Then, by hypothesis, there exists a finite chain of simplicial maps eϕi:K→Eand direct contiguities, with msteps, eϕ0∼ceϕ1∼c · · · ∼ceϕm, such that p◦eϕi=ϕi, for all i= 0,...,m. Hence, the map e H:K×Im→Egiven by e H(v, i) = eϕi(v) is simplicial, by an argument analogous to (2), and satisfies e H◦im 0=ϕand p◦e H=H. 6 Desamparados Fern´andez-Ternero et al. Reciprocally, let us assume that the simplicial map p:E→Bsatisfies the condition indicated above and let us prove that pis a simplicial fibration. Consider a simplicial complex Kand two maps f, g :K→Bin the same contiguity class with msteps. If m= 1, that is, fand gare contiguous, by Proposition 1, there exists a homotopy H:K×I1→Bsuch that H(v, 0) = f(v) and H(v, 1) = g(v) for all v∈V(K). Consider a simplicial map ˜ f:K→Esuch that p◦˜ f=f. By hypothesis,there is a simplicial map e H:K×I1→Esuch that e H◦i1 0=ϕ, where p◦ϕ(v, 0) = e f(v) for all v∈V(K), and p◦e H=H. Let eg:K→Egiven by eg(v) = e H(v, 1), where v∈V(K). By Proposition 1, we conclude that e f∼ceg. If m > 1, that is, there exists a finite sequence of direct contiguities f=ϕ0∼cϕ1∼c· · · ∼cϕm=g, when iterating the same argument for every ϕiwith i= 1,...,m, we obtain a finite sequence of direct contiguities, e f=eϕ0∼ceϕ1∼c· · · ∼ceϕm=eg, where p◦fϕi=ϕi,i= 1,...,m, that conclude the proof. ut Remark 2 Observe that, for a simplicial map to be a simplicial fibration, it is enough that the map satisfies the lifting property for contiguous maps. Remark 3 Notice that the complex Kthat we considered in Definition 7 and Proposition 3 may not be finite. It is possible to restrict the notion of simplicial fibration and the proposition above to the cases where Kis finite. This allows us to introduce the corresponding notion of simplicial finite-fibration and the corresponding equivalent notions given by the finite versions of Proposition 3 and Remark 2. 4 Examples and properties In this section we will introduce some important examples of simplicial fibrations. All the proofs are standard in algebraic homotopy theory [3] and therefore they are omitted and left to the reader. The following proposition will give us the first basic ones. Notice that we will use the characterization of simplicial fibration given by Proposition 3. Proposition 4 (i) Any simplicial isomorphism is a simplicial fibration. (ii) If ∗denotes the one-vertex simplicial complex, then the constant simplicial map E→ ∗ is a simplicial fibration, for any simplicial complex E. (iii) The composition of simplicial fibrations is a simplicial fibration. For the next result we need to recall the pullback construction for simplicial complexes. Given any pair of simplicial maps f:K→Mand g:L→Mtheir pullback is given as the following diagram: K×MLf0 // g0  L g  Kf //M Simplicial fibrations 7 where K×MLis the full simplicial subcomplex of K×Lwhose underlying vertex set is given by those pairs of vertices (v, w)∈K×Lsatisfying f(v) = g(w). The induced simplicial maps f0and g0are given by f0(v, w) = wand g0(v, w) = v. It is plain to check that this construction is the pullback of fand gin the category of simplicial complexes. Proposition 5 Let p:E→Bbe a simplicial fibration and f:K→Bany simplicial map. Then the simplicial map p0:K×BE→Kinduced by pin the pullback K×BEf0 // p0  E p  Kf //B is also a simplicial fibration. As a consequence, taking into account Proposition 4 (ii), we obtain the next result. Corollary 1 Let Kand Lbe simplicial complexes and K×Lbe their categorical product. Then the canonical projections pr1:K×L→Kand pr2:K×L→Lare simplicial fibrations. Another interesting example of simplicial fibration is given by the product of simplicial fibrations. Recall that, if f1:K1→L1and f2:K2→L2are simplicial maps, then one can construct their product simplicial map: f1×f2:K1×K2→L1×L2 defined as (f1×f2)(v1, v2):= (f1(v1), v2(v2)), for any vertex (v1, v2)∈K1×K2. Proposition 6 Let p1:E1→B1and p2:E2→B2be simplicial fibrations. Then their product p1×p2is also a simplicial fibration. 5 The path complex PK Now we study the complex PKof Moore paths. The concept of Moore path and most of the content of subsections 5.1 and 5.2 are originally due to Grandis [15, Sections 2.2 and 2.5]. 5.1 Moore paths Consider the one-dimensional simplicial complex Z, whose vertices are all the integers i∈Zand whose 1-simplices are all the consecutive pairs {i, i + 1}, that is, Zis a triangulation of the real line. Definition 8 ([15]) Let Kbe a simplicial complex. A Moore path in Kis a simplicial map γ:Z→Kwhich is eventually constant on the left and eventually constant on the right, i.e., there exist integers i−, i+∈Zsatisfying the two following conditions: 8 Desamparados Fern´andez-Ternero et al. (i) γ(i) = γ(i−),for all i≤i−, (ii) γ(i) = γ(i+),for all i≥i+. Obviously, if i−=i+we have the constant map. For a non constant Moore path γ:Z→Kwe can consider the integers γ−:= max{i−:γ(i) = γ(i−),for all i≤i−}, γ+:= min{i+:γ(i) = γ(i+),for all i≥i+}. Observe that γ−< γ+. Definition 9 The images α(γ):=γ(γ−) and ω(γ):=γ(γ+) are called the initial vertex and final vertex of γ, respectively. When γis constant we set γ−= 0 = γ+. If a, b ∈Zwith a≤b, [a, b] will denote the full subcomplex of Zgenerated by all vertices iwith a≤i≤b. Considering this notation, any Moore path γin K may be identified with the restricted simplicial map γ: [γ−, γ+]→K. The interval [γ−, γ+] will be called the support of γ. If γis a Moore path in Kwith support [γ−, γ+], then one can take the reverse Moore path γas γ(i) = γ(−i), whose support is [−γ+,−γ−]. Notice that this reparametrization describes γin the opposite direction. If γis a Moore path in Kwith support [γ−, γ+] such that γ+−γ−=m, then we define one normalized Moore path |γ|:Im→Kas |γ|(i) = γ(i+γ−). The advantage of this reparametrization is that the support of |γ|is [0, m],and therefore it will be more manageable when dealing with simplicial fibrations. Definition 10 Given γ, δ Moore paths in Ksuch that ω(γ) = α(δ), the product path γ∗δit is defined as (γ∗δ)(i) = (γ(i−δ−), i ≤γ++δ−, δ(i−γ+), i ≥γ++δ−. It is not difficult to see that the support of γ∗δis [γ−+δ−, γ++δ+].The product of Moore paths is strictly associative, that is, given Moore paths γ, δ, ε such that ω(γ) = α(δ) and ω(δ) = α(ε), then γ∗(δ∗ε) = (γ∗δ)∗ε. Moreover, if cvdenotes the constant path in a vertex v∈K, then it is immediate to check that γ∗cw=γ=cv∗γwhere v=α(γ) and w=ω(γ). 5.2 The path complex Next we will consider a suitable notion of Moore path complex associated to the 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. Simplicial fibrations 9 Indeed, if Kand Lare simplicial complexes, we define the simplicial complex LK, whose vertices are all simplicial maps f:K→Land where we consider as simplices the finite sets {f0,...,fp}of simplicial maps K→Lsuch that Sp i=0 fi(σ)∈L, for any simplex σ∈K. It is not difficult to check that this definition induces a structure of simplicial complex in LK. Moreover, denoting by × the categorical product in SC, we have that the evaluation map ev: LK×K→L, (f, v)7→ f(v), is simplicial. This fact allows us to establish a natural bijection SC(M×K, L)≡SC(M, LK).(3) Observe that for a simplicial map h:L→L0, there is a well defined map hK:LK→(L0)K, which preserves the identities and the compositions, that is, we have a functor (−)K:SC →SC. More is true, the functor (−)×K:SC →SC is left adjoint to the functor (−)K:SC →SC. Definition 11 Let Kbe a simplicial complex. We define the Moore path complex of K, denoted by PK, as the full subcomplex of KZgenerated by all the Moore paths γ:Z→K. Then, {γ0,...,γp} ⊂ PKdefines a simplex in PKif and only if {γ0(i),...,γp(i), γ0(i+ 1),...,γp(i+ 1)} is a simplex in K, for any integer i∈Z. An interesting property of PKis that, for any bounded interval [a, b]⊂Z, the complex K[a,b]is, in fact, a full subcomplex of PK. Moreover, given a simplicial map f:K→L, since the composite f◦γis a Moore path in Lfor any Moore path γin K, we obtain the Moore path complex functor P: SC →SC. One can check that this functor preserves binary products and equalizers. Therefore P preserves finite limits and, in particular, pullbacks. In general P does not preserve limits; for instance, P does not preserve infinite products [15]. Hence, P has not a left adjoint functor. Definition 12 The initial and final vertices of any given Moore path γdefine simplicial maps α: PK→Kand ω: PK→K. Remark 4 Taking into account Proposition 5 and the above comments, it is immediately seen that the category of simplicial complexes satisfies the axioms P1, P2 and P5 of a P-category (or category with a natural path object) in the sense of Baues [3]. Besides, by Proposition 4, axiom P3 is partially satisfied. However, as the path functor P has not a left adjoint we cannot pass from certain diagrams to others in which some kind of cylinder could be involved in order to use our notion of simplicial fibration. Moreover, the simplicial complex PKis always infinite, even in the case that Kis finite. These two facts prevent us to prove that any simplicial fibration satisfies the homotopy lifting property with respect to P (in the sense of Baues). For identical reasons we can not prove P4 axiom (relative path axiom). We also observe that, with our notion of simplicial fibration, if a hypothetical P-category structure held on the category of simplicial complexes, then this would not give rise to another fibration category structure agreeing with the former on finite simplicial complexes, as the functor P : SC →SC does not restrict to finite complexes. 16 Desamparados Fern´andez-Ternero et al. gives a lift: Fb× {0}ib// _  E p  Fb×In+m(γ∗δ)◦pr2 // g γ∗δ 66 B This proves that (γ∗δ)]and δ]◦γ]are in the same contiguity class. Using the above reasonings and the fact that γ∗γ∼cb0rel. {0, n}and γ∗γ∼cb rel. {0, n}, we conclude the proof of the result. ut 7 Collapsible base Minian’s Lemma (Lemma 1) and the Simplicial Pasting Lemma (Lemma 2) will be crucial for the next results. These results should be compared with those included in [21]. We start with this fairly general proposition. Proposition 9 Let p:E→Bbe a simplicial fibration and let F0, F1:K×In→E be simplicial maps such that p◦F0and p◦F1are in the same contiguity class with msteps. Let H:p◦F0∼p◦F1be the map given by Proposition 1 and analogously, let G:F0|K×{0}∼F1|K×{0}be the map given by Proposition 1. Consider the following commutative diagram: K× {0} × Im G// _  E p  K×In×ImH //B Then, for suitable q≥nand p≥m, there exist subdivision maps t1:Iq→Inand t2:Ip→Imand a simplicial map H0:K×Iq×Ip→Esuch that H0defines a homotopy F0◦(1K×t1)∼F1◦(1K×t1), which is an extension of G◦(1K×{0}×t2). Proof Consider the commutative diagram K×Jnm f// _  E p  K×In×ImH //B where Jnm is the full subcomplex of In×Imgiven in the proof of Theorem 2 and fis the simplicial map (see Lemma 2) defined as f(v, i, 0) = F0(v, i), f(v, 0, j) = G(v, 0, j) and f(v, i, m) = F1(v, i). Now we take the simplicial maps φ:In0×Im0→ In×Imand φ0:In00 ×Im00 →In0×Im0given by Lemma 1. As φ◦(In0×{0})⊂Jnm Simplicial fibrations 17 we have a lift e H:K×In0×Im0→E, represented by the dotted arrow in the composite diagram: K×In0× {0}1K×φ// _  K×Jnm f// _  E p  K×In0×Im0 1K×φ // e H 22 K×In×ImH //B Renaming q=n00,p=m00, and taking into account that φ0(Jn00 m00 )⊂In0× {0} and that φ◦φ0=t1×t2,we have that the simplicial map H0:=e H◦(1K×φ0) fulfills all the expected requirements. ut Now, for our next result, we need to establish the following natural definition: Definition 15 Let p:E→Bbe a simplicial fibration and consider f, g :K→E simplicial maps. Then fand gare in the same class of fibrewise contiguity, denoted by f∼pg, if there exists a finite sequence of simplicial maps f=f0∼c· · · ∼cfn=g such that p◦f=p◦fi, for all i= 0,...,n. In other words, fand gare in the same class of fibrewise contiguity if there exists a simplicial map F:K×In→Esatisfying (i) F(v, 0) = f(v) and F(v, n) = g(v), for all v∈K; (ii) p(F(v, i)) = p(f(v)), for all v∈Kand i∈In(therefore p◦f=p◦g). Remark 5 By carefully using the simplicial isomorphism In ∼ = →Ingiven by i7→ n−i, one can check that the existence of the homotopy lift does not depend on the base point of In. As a corollary of Proposition 9, we have: Corollary 3 Let p:E→Bbe a simplicial fibration and let F0, F1:K×In→E be two lifts of the same map K× {0}f// _  E p  K×In// F0 F1 77 B Then, for a suitable q≥n, there exists a subdivision map t:Iq→Insuch that F0◦(1K×t)∼pF1◦(1K×t). Proof Just apply Proposition 9 to the simplicial maps G, H defined as G(v, 0, j) = f(v, 0) and H(v, i, j) = (p◦f)(v, 0),for all v∈Kand j∈In.ut Observe that the previous corollary also holds true when we consider K× {n}in the diagram instead of K× {0}. For our next result we will consider the notion of having the same type of fibrewise contiguity. 18 Desamparados Fern´andez-Ternero et al. Definition 16 We say that two simplicial fibrations p1:E1→B,p2:E2→ Bhave the same type of fibrewise contiguity when there exist simplicial maps f:E1→E2and g:E2→E1such that p2◦f=g,p1◦g=fand g◦f∼p11E1 and f◦g∼p21E2. 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 3 Let p:E→Bbe a simplicial fibration and let f0, f1:K→Bsimplicial maps. Consider, for each i= 1,2, the pullback of fialong p: Ei pi  f0 i//E p  Kfi //B If f0and f1are in the same class of contiguity, then the simplicial fibrations p0 and p1have the same type of fibrewise contiguity. Proof Consider a simplicial map F:K×In→Bsuch that F(v, 0) = f0(v) and F(v, n) = f1(v),for all v∈K. Take F0 0,F0 1lifts of the following diagrams (for the second diagram see Remark 5): E0× {0}f0 0// _  E p  E1× {n}f0 1// _  E p  E0×In F0 0 66 F◦(p0×1) //B E1×In F0 1 66 F◦(p1×1) //B By the pullback property there are well defined simplicial maps g0:E0→E1and g1:E1→E0satisfying the commutativities given in the following diagrams: E0 g0 !! F0 0(−,n)  p0 ## E1 g1 !! F0 1(−,0)  p1 ## E1 f0 1// p1  E p  E0 f0 0// p0  E p  Kf1 //B K f0 //B Let us first check that g0◦g1∼p11E1.Indeed, observe that F0 0◦(g1×1) and F0 1 are lifts of the same map E1× {0}f0 0◦g1// _  E p  E1×InF◦(p1×1) // 66 B Simplicial fibrations 19 Corollary 3 assures the existence of a suitable subdivision map t:In0→Inand a simplicial map G:E1×In0×Im→Esatisfying that G:F0 0◦(g1×1) ◦(1 ×t)∼pF0 1◦(1 ×t) Again, using the pullback property, there is an induced simplicial map e G:E1× Im→E1satisfying f0 1◦e G=G(−, n0,−) and p1◦e G=p1◦pr. A simple inspection proves that e G:g0◦g1∼p11E1.Analogously, by applying Corollary 3 to the diagram p◦f0 1◦g0=F◦(p0×1)|E0×{n}for the common lifts F0 1◦(g0×1) and F0 0, and taking into account Remark 5, one can find a simplicial map Hsatisfying H:F0 1◦(g0×1)◦(1×s)∼pF0 0◦(1×s) for a suitable subdivision map s:In00 →In. Taking e Hthe simplicial map characterized by the equalities f0 0◦e H=G(−,0,−) and p0◦e H=p0◦pr, we obtain e H:g1◦g0∼p01E0.ut Corollary 4 Let p:E→Bbe a simplicial fibration where Bis strongly collapsible. Then phas the same class of fibrewise contiguity of the trivial fibration B×p−1(b0)→B, for any vertex b0∈B. Proof Just take into account that 1B:B→Band the constant path cb0:B→B are in the same class of contiguity and use Theorem 3. ut We finish this section with an interesting property whose proof relies on the proof of Theorem 3. Indeed, remember that, in the statement of such theorem, we have pullbacks p◦f0 i=fi◦pi(i= 0,1) where f0∼f1(i.e., f0and f1are in the same class of contiguity). From the proof one obtains simplicial maps g0:E0→E1 and g1:E1→E0(with p1◦g0=p0and p0◦g1=p1) and Minian simplicial homotopies F0 0:E0×In→Eand F0 1:E1×In→Esatisfying F0 0:f0 0∼f0 1◦g0 and F0 1:f0 0◦g1∼f0 1. Moreover, g0and g1satisfy g1◦g0∼p01E0and g0◦g1∼p11E1. In particular g1◦g0∼1E0and g0◦g1∼1E1. We easily have the following lemma: Lemma 3 Consider the pullback of a simplicial fibration p:E→Balong a simplicial map f:B→Bsuch that f∼1B. If f0:E0→Edenotes the base change of f, then f0∼g0and f0◦g1∼1E.In particular, f0is a strong equivalence with g1:E→E0as a homotopy inverse. Proof Just use the argument above to the pullback of palong the identity 1B:B→B.ut And finally our result. Observe that such result completes Proposition 5. Proposition 10 Consider the pullback of a simplicial fibration p:E→Balong a simplicial map f:K→B: E0 p0  f0 //E p  Kf //B If fis a strong equivalence, then f0is also a strong equivalence. 20 Desamparados Fern´andez-Ternero et al. Proof Suppose g:B→Kis a simplicial map such that g◦f∼1Kand f◦g∼1B. Consider first the following diagram, where E00 is the pullback of p0along g: E00 g0 // p00  E0 p0  f0 //E p  Bg//Kf //B As f◦g∼1B, by the lemma above (note that the composite of pullbacks is a pullback) one can find simplicial maps g0:E00 →Eand g1:E→E00 such that g1◦g0∼1E00 ,g0◦g1∼1E,f0◦g0∼g0and f0◦g0◦g1∼1E. Therefore, g0has a left homotopy inverse and f0has a right homotopy inverse: (g1◦f0)◦g0∼1E00 and f0◦(g0◦g1)∼1E. Similarly, from the diagram of pullbacks E000 f000 // p000  E00 p00  g0 //E0 p0  Kf //Bg//K we have that g0has a right homotopy inverse. It straightforwardly follows that g0is a strong equivalence. Since f0◦g0∼g0we conclude that f0is a strong equivalence. ut 8 Factorization In this section we will see that any simplicial map may be considered, in a homotopical sense, as a simplicial finite-fibration. For our main result in this section we will use the fact that (α, ω): PK→K×Kis a simplicial finite-fibration, for any simplicial complex K(see Theorem 1). Theorem 4 Let f:K→Lbe a simplicial map. Then there is a factorization f=p◦j, where jis a P-homotopy equivalence and pis a simplicial finite-fibration. Proof We consider Pf=K×LPLthe pullback of f:K→Lalong α: PL→L, where αis the initial vertex map. In this way we have a commutative square: Pf α0  f0 //PL α  Kf //L As we have previously observed, the simplicial complex Pfis defined as the full subcomplex of K×PLwhose set of vertices is given as Pf={(v, γ)∈K×PL:f(v) = α(γ)} Simplicial fibrations 21 and α0, f0are the restrictions of the obvious projections. We define the simplicial map j:K→Pfas j(v):= (v, cf(v)),where cf(v):Z→Ldenotes the constant Moore path at f(v). On the other hand, the simplicial map p:Pf→Lis defined as p(v, γ):=ω(γ), where ωis the final vertex map. Obviously, we have that p◦j=f. Let us first check that pis a simplicial finite-fibration. Indeed, consider the following commutative diagram: Pf f0 // (α0,p)  PL (α,ω)  K×Lf×1L// pr1  L×L pr1  Kf//L 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 (α, ω) a simplicial finite-fibration, the simplicial map (α0, p): Pf→K×Lmust be also a simplicial finite-fibration. Therefore, the composition of the finite-fibrations p= pr2◦(α0, p): Pf→Lis a finite-fibration. Next we check that j:K→Pfis a P-homotopy equivalence. As α0◦j= 1Kit only remains to see that j◦α0≃1Pf. We directly define the homotopy H:Pf→ PPfas H(v, γ)(i):= (v, γi) where γi∈PLis defined as γi(j) = (γ(j), j ≤i, γ(i), j ≥i, for any γ∈PLand i∈Z. Using the exponential law for simplicial complexes one can see that His indeed a well-defined simplicial map defining a homotopy between j◦α0and 1Pf. We leave the details to the reader. ut It could be interesting to compare the above theorem and its proof with Theorem 29 of [19]. As a consequence of this theorem new finite-fibrations appear. Example 2 For instance, for any simplicial complex Kand a base vertex v0∈K one can consider the full subcomplex of PK, P0K={γ∈PK:α(γ) = v0}. Observe that this simplicial complex is nothing else than the construction Pf given in the theorem above, where f:{v0},→Kis the inclusion map. Then, the simplicial finite-fibration associated to fis precisely the map P0K→K, where γ7→ ω(γ). Example 3 There is also a 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 22 Desamparados Fern´andez-Ternero et al. ∆:K→K×Kthrough a P-homotopy equivalence followed by a simplicial finitefibration K c!! ∆//K×K PK (α,ω) :: We already know that (α, ω) is a simplicial finite-fibration. On the other hand, c:K→PKis defined as the simplicial map sending each v∈Kto the constant Moore path at v,c(v):=cv. This map is, indeed, a P-homotopy equivalence: the simplicial map α: PK→Ksatisfies α◦c= 1K. Moreover, the homotopy H: PK→P2Kdefined as H(γ)(i) = γi, where γiis the i-truncated Moore path of γ, satisfies α◦H=c◦αand ω◦H= 1PK; that is, H:c◦α≃1PK. 9 Varadarajan’s theorem As an application of Theorem 2 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 Mac´ıas in [6]. Definition 17 The simplicial LS-category scat(K) of the simplicial complex Kis the least integer n≥0 such that Kcan be covered by n+1 subcomplexes Kjsuch that each inclusion ιj:Kj⊂Kbelongs to the contiguity class of some constant map cvj:Kj→K. This notion is the simplicial version of the well known homotopic invariant cat(X), the so-called Lusternik-Schnirelmann category of the topological space X [7]. It has been introduced by the authors in [11,13], its most important property being the invariance by strong homotopy equivalences. Accordingly to Theorem 2, all the fibers p−1(v), v∈B, of a fibration with connected base p:E→Bhave the same strong homotopy type, so we call generic fiber Fof the fibration any simplicial complex into that equivalence class, and its simplicial category scat(F) is well defined. Theorem 5 Let p:E→Bbe a simplicial fibration with connected base Band generic fiber F. Then scat(E) + 1 ≤(scat(B) + 1)(scat(F) + 1). Proof Let scat B=m, and take a categorical covering U0,...,Umof B. From Theorem 2 we know that all the fibers have the same strong homotopy type. We identify Fto the fiber p−1(v) over some base point v∈B. Let scat F=n, with V0,...,Vna categorical covering of F. For each i∈ {0,...,m}let ιi:Ui⊂Bbe the inclusion. By definition of simplicial category, the map ιiis in the contiguity class of a constant map, say ci:Ui→B. Since Bis path connected, we can assume that ciis the constant map corresponding to the base point v. Consider the map pi=ιi◦p:p−1(Ui)→B. If i:p−1(Ui)⊂Eis the inclusion, then p◦i=pi. Now, from ιi∼cvit follows that pi∼cv, the latter being the constant map with domain p−1(Ui). By the contiguity Simplicial fibrations 23 lifting property, there exists a simplicial map Gi:p−1(Ui)→Esuch that i∼Gi and p◦Gi=cv. The latter means that the map Gitakes its values in F. We denote gi:p−1(Ui)→Fthe map given by gi(v) = Gi(v). In this way Gi=ιF◦gi, where ιF:F⊂Eis the inclusion. For each i∈ {0,...,m},j∈ {0,...,n}, we take the subcomplex Wij =g−1 i(Vj)⊂p−1(Ui)⊂E. Since B=U0∪ · · · ∪ Umimplies E=p−1(U0)∪ · · · ∪ p−1(Um), and since F=V0∪ · · · ∪ Vnimplies p−1(Ui) = Wi0∪ · · · ∪ Win, it follows that {Wij }is a covering of E. It only remains to prove that each Wij is categorical in E. Let gij :Wij →Vjbe the restriction of gito Wij ⊂p−1(Ui). We know that each Vj⊂Fis categorical, so the inclusion Jj:Vj⊂Fis in the contiguity class of some constant map cj:Vj→ F, whose image is some vertex fjof F. Then the composition Wij gij →Vi Jj ⊂F belongs to the contiguity class of the constant map fj:Wij →Fsince we have that Jj◦gij ∼cj◦gij =fj.Let ij :Wij ⊂Ebe the inclusion, that is, the restriction of i:p−1(Vj)⊂Eto Wij. Since i∼Giit follows that ij ∼Gij , the latter being the restriction of Gito Wij. Finally we have ij ∼Gij =ιF◦Jj◦gij =ιF◦fj, so the inclusion Wij ⊂Eis in the contiguity class of a constant map. ut 10 ˇ Svarc genus In the classical topological setting, the Lusternik-Schnirelmann category can be seen as a particular case of the so-called “ˇ Svarc genus” or sectional category of a continuous map. In this section we recall the simplicial version of this invariant introduced by the authors in [12] and establish some key links with simplicial fibrations. Definition 18 The simplicial ˇ Svarc genus of a simplicial map ϕ:K→Lis the minimun integer n≥0 such that Lis the union L0∪· · ·∪Lnof n+1 subcomplexes, and for each jthere exists a section σjof ϕ, that is, a simplicial map σj:Lj→K such that ϕ◦σjis the inclusion ιj:Lj⊂L. We denote this genus by Sg(ϕ). A slight modification in the above definition is to change the equality by “being in the same contiguity class”. Definition 19 The homotopy simplicial ˇ Svarc genus of a simplicial map ϕ:K→ L, denoted by hSg(ϕ), is the minimum n≥0 such that L=L0∪ · · · ∪ Ln, with Lj⊂Lsubcomplex, and for each j∈ {0,...,n}there exists an “up to contiguity class” simplicial section σjof ϕ, that is, a simplicial map σj:Lj→Ksuch that ϕ◦σj∼ιj. Remark 6 Note that when the complex Lis finite, the condition ϕ◦σj∼ιj(same contiguity class) can be changed to ϕ◦σj≃ιj(P-homotopy), by Proposition 8. Obviously, hSg(ϕ)≤Sg(ϕ). The equality holds for some particular classes of maps. Theorem 6 Let p:E→Bbe a simplicial fibration. Then hSg(p) = Sg(p). 24 Desamparados Fern´andez-Ternero et al. Proof We only have to prove that Sg(ϕ)≤hSg(ϕ), so let hSg(ϕ) = nand L0,...,Lnbe a covering of Bby subcomplexes such that there exist simplicial maps σj:Lj→Ewith p◦σj∼ιj, as in Definition 19. By Proposition 3, for each jwe have simplicial maps Hj:Lj×[0, mj]→Bwith Hj(−,0) = p◦σjand Hj(−, mj) = ιjthe inclusion Lj⊂B. Take a lift ˜ Hj:Lj×[0, mj]→Ein the following diagram, Lj× {0} _  σj//E p  Lj×[0, mj]Hj// ˜ Hj :: B in such a way that p◦˜ H=Hand ˜ H(v, 0) = σj(v). Then the map ξj:Lj→E given by ξj(v) = ˜ H(v, mj) is simplicial and verifies p◦ξj(v) = p◦˜ H(v, mj) = H(v, mj) = ιj(v) = v, so it is a true section of p. Then Sg(ϕ)≤n.ut Remark 7 Note that if p:E→Bis a simplicial finite-fibration then Theorem 6 above is still true when the base Bis finite. 10.1 Simplicial L-S category Taking into account the notion of simplicial LS-category (Definition 17), what we want to see is that it equals (as in the classical case) the ˇ Svarc genus of a certain fibration. Assume that the complex Kis connected. Then every two constant maps cvi, cvj, belong to the same contiguity class, as can be easily seen by considering a path γ: [0, m]→Kconnecting viand vjand the sequence of contiguous maps given by the truncated paths γk: [0, k]→K. So we can choose a base point v0 and assume that all the constant maps in Definition 17 equal cv0. Consider P0Kthe full subcomplex of the path complex PKconsisting on all Moore paths γ∈PKwhose initial point is the base vertex v0(see Example 2). The map ω: P0K→K, sending each path to its final point, is then the pullback of the simplicial finite-fibration (α, ω): PK→K×Kby the map f, where f(v) = (v0, v): P0K ω  //PK (α,ω)  Kf//K×K Theorem 7 Let Kbe a connected finite complex. Then the simplicial LS-category scat(K)equals the ˇ Svarc genus of the simplicial finite-fibration ω: P0K→K. Proof We have to prove that scat(K) = Sg(ω), but we know that the latter equals hSg(ω). So, first assume that scat(K) = nand let K=K0∪ · · · ∪ Knsuch that each inclusion ιj:Kj⊂Kbelongs to the contiguity class of the constant map Simplicial fibrations 25 v0, that is, there is a Minian simplicial homotopy Hj:Kj×[0, mj]→Kwith Hj(v, 0) = v0and Hj(v, mj) = v. Take a lift ˜ Hjof Hjin the following diagram: Kj× {0} _  cj//P0K ω  Kj×[0, mj]Hj// ˜ Hj 99 K where cj(v) is the constant Moore path at v0, for all v. Define σj:Kj→P0Kas σj(v) = ˜ Hj(v, mj). Then ω◦σj(v) = Hj(v, mj) = v, so σj is a section of the fibration. The map σjis simplicial since v∈Kj7→ (v, mj)∈Kj×[0, mj] is a simplicial map. We have then proved that Sg(ω)≤n. Conversely, if s:L→P0Kis a section of ω, each path s(v) has initial vertex v0and final vertex v. Denote by [v−, v+] the support of s(v). Then, since Kis finite, there is an interval [m, n] containing all the supports, so we can define H:L×[m, n]→Kas H(v, i) = s(v)(i). Then H(v, m) = s(v)(m) = s(v)(v−) = α(s(v)) = v0, while H(v, m) = s(v)(n) = s(v)(v+) = ω(s(v)) = v, showing that the inclusion L⊂Kbelongs to the contiguity class of the constant map v0. So each covering of Kby subcomplexes Lverifying the Definition 18 gives the same covering verifying Definition 17. Then scat(K)≤Sg(ω). ut 10.2 Discrete topological complexity In [12], a subset of the authors introduced a notion of discrete topological complexity which is a version of Farber’s topological complexity [10], 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 20 [12] The discrete topological complexity TC(K) of the simplicial complex Kis the least integer n≥0 such that K×Kcan be covered by n+ 1 “Farber subcomplexes” Ωj, where the latter means that there exist simplicial maps σj:Ωj→Ksuch that ∆◦σjis in the contiguity class of the inclusion ιj:Ωj⊂K×K. Here ∆:K→K×Kdenotes the diagonal map v7→ (v, v). In other words: Proposition 11 The discrete topological complexity of the abstract simplicial complex Kis the homotopic ˇ Svarc genus of the diagonal map ∆:K→K×K, i.e., TC(K) = hSg(∆). Our main result in this section is the following one. Theorem 8 Let Kbe a finite complex. The discrete topological complexity of K equals the ˇ Svarc genus of the finite-fibration (α, ω): PK→K×K.