Contiguity Distance between Simplicial Maps
aa r X i v : . [ m a t h . A T ] D ec CONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS
AYSE BORAT, MEHMETCIK PAMUK, AND TANE VERGILIA bstract . We study properties of contiguity distance between simplicial maps. In par-ticular, we show that simplicial versions of LS -category and topological complexity areparticular cases of this more general notion. C ontents
1. Introduction 12. Contiguity Distance 53. Acknowledgement 14References 141. I ntroduction
The Lusternik-Schnirelmann category, introduced by Lusternik and Schnirelmann[14], is an important numerical invariant concerning the critical points of smooth func-tions on manifolds.
Definition 1. [4, 14]
Lusternik Schnirelmann category of a space X, denoted by cat ( X ) , is theleast non-negative integer k if there are open subsets U , U , . . . , U k which cover X such that eachinclusion map ι i : U i ֒ → X is nullhomotopic in X for i = , , . . . , k. Topological complexity of a topological space introduced by Farber [5] is anothernumerical invariant closely related to motion planning problems.
Definition 2. [5]
Let π : PX → X × X be the path fibration. Topological complexity of a spaceX, denoted by TC ( X ) , is the least non-negative integer k if there are open subsets U , U , . . . , U k which cover X × X such that on each U i there exists a continuous section of π for i = , , . . . , k. Although these invariants seem independent, they are similar in nature both beinghomotopy invariants. Macias-Virgos and Mosquera-Lois [7] introduced homotopic dis-tance, a notion generalizing both cat and TC. One of the advantages of homotopicdistance is that since it is a number related to functions rather than spaces as in cat andTC, we have the opportunity to investigate the behaviour of the homotopic distanceunder compositions which is not possible to do with cat and TC. This feature leads us toprove the known TC- and cat-related theorems in an easy way.
Date : December 22, 2020.2010
Mathematics Subject Classification.
Key words and phrases. contiguity distance, homotopic distance, topological complexity, Lusternik-Schnirelmann category.
Definition 3. [7]
Let f , g : X → Y be continuous maps. Homotopic distance between f andg, denoted by D ( f , g ) , is the least non-negative integer k if there are open subsets U , U , . . . , U k which cover X such that f | U i ≃ g | U i for all i = , , . . . , k. In this paper, we consider the combinatorial objects, the simplicial complexes, andstudy the distance between two simplicial maps adapted from homotopic distance. Infact one can consider a geometric realization of a simplicial complex and study the ordi-nary homotopic distance between continuos maps induced by the geometric realizationof simplicial maps. However, we opt to stay in the simplicial category in order notto loose the combinatorial aspects. To do this, we consider a simplicial analogue ofhomotopic distance between simplicial maps which relies on the contiguity. Then thesimplicial analogues of cat and TC of a simplicial complex can be defined in terms of thisdistance. However we want to remark that the contiguity distance between simplicialmaps and the homotopic distance between their corresponding geometric realizationsmight di ff er, see Example 1.Given a set V , an abstract simplicial complex with a vertex set V is a set K of finitesubsets of V such that the elements of V belongs to K and for any σ ∈ K any subset of σ belongs to K . The elements of K are called the faces or the simplices of K . The dimensionof an abstract simplex is just its cardinality minus 1 and the dimension of K is the largestdimension of its simplices. For further details on abstract simplicial complexes, we referto [13, 16].The combinatorial description of any geometric simplicial complex ˜ K obviously givesrise to an abstract simplicial complex K . One can always associate a geometric simplicialcomplex ˜ K to an abstract simplicial complex K in such a way that the combinatorial de-scription of ˜ K is the same as K so that the underlying space of ˜ K is homeomorphic to thegeometric realization | K | . As a consequence, abstract simplicial complexes can be seenas topological spaces and geometric complexes can be seen as geometric realizations oftheir underlying combinatorial structure. So, one can consider simplicial complexes atthe same time as combinatorial objects that are well-suited for e ff ective computationsand as topological spaces from which topological properties can be inferred.It is a classical result that an arbitrary continuous map between geometric realizationsof simplicial complexes can be deformed (after su ffi ciently many subdivisions) to a sim-plicial map, known as the simplicial approximation theorem. But in general, simplicialapproximations to a given continuous map are not unique. An analogue of homotopy,called contiguity, is defined for simplicial maps so that di ff erent simplicial approxima-tions to the same continuous map are contiguous. Definition 4.
Let ϕ, ψ : K → K ′ be two simplicial maps between simplicial complexes. We saythat ϕ and ψ are contiguous , denoted ϕ ∼ c ψ , provided for a simplex σ = { v , . . . , v n } in K, theset of vertices ϕ ( σ ) ∪ ψ ( σ ) = { ϕ ( v ) , . . . , ϕ ( v n ) , ψ ( v ) , . . . , ψ ( v n ) } constitutes a simplex in K ′ . For simplicial complexes and simplicial maps, the notion of contiguity can be consid-ered as the discrete version of homotopy. Being contiguous is a combinatorial conditionwhich defines a reflexive and symmetric relation among simplicial maps. On the otherhand, this relation is not transitive. There is however an equivalence relation in the set of
ONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS 3 simplicial maps and the corresponding equivalence classes are called contiguity classes.
Definition 5.
We say that two simplicial maps ϕ, ψ : K → K ′ are in the same contiguity class,denoted by ϕ ∼ ψ , provided there exists a finite sequence of simplicial maps ϕ i : K → K ′ ,i = , . . . , m, such that ϕ = ϕ ∼ c ϕ ∼ c · · · ∼ c ϕ m = ψ . Barmak and Minian [1, 2] introduced the notion of strong collapse, a particular typeof collapse which is specially adapted to the simplicial structure. Actually, it can bemodelled as a simplicial map, in contrast with the standard concept of collapse whichis not a simplicial map in general: For a simplicial complex K , suppose that there is apair of simplices σ < τ in K such that σ is a face of τ and σ has no other cofaces. Sucha simplex σ is called a free face of τ . Then the simplicial complex K − { σ, τ } is complexcalled an elementary collapse of K (see Figure 1). The action of collapsing is denoted K ց K − { σ, τ } . The inverse of an elementary collapse is called an elementary expansion. bc bc bcbc bc bcbc σ τ bc bc bcbc bc bcbc F igure
1. An elementary collapseA vertex v of a simplicial complex K is dominated by another vertex v ′ if every max-imal simplex that contains v also contains v ′ . An elementary strong collapse consistsof removing the open star of a dominated vertex v from a simplicial complex K . Theinverse of an elementary strong collapse is called an elementary strong expansion. Afinite sequence of elementary strong collapses (expansions) is called a strong collapse(expansion), see Figure 2. bc bc bcbc bc bcbc v ′ v bc bc bcbc bc bc v ′ F igure
2. An elementary strong collapse
AYSE BORAT, MEHMETCIK PAMUK, AND TANE VERGILI
Two simplicial complexes K , K ′ have the same strong homotopy type, denoted by K ∼ K ′ , if they are related by a sequence of strong collapses and expansions. Surpris-ingly, this turns out to be intimately related to the classical notion of contiguity. Moreprecisely, having the same strong homotopy type is equivalent to the existence of a strongequivalence.A simplicial map ϕ : K → K ′ is called a strong equivalence if there exists ψ : K ′ → K such that ϕ ◦ ψ ∼ id K ′ and ψ ◦ ϕ ∼ id K . The theory of strong homotopy types of simplicialcomplexes was introduced in [2]. Strong homotopy types can be described by elemen-tary moves called strong collapses. From this theory, Birman and Minian obtained newresults for studying simplicial collapsibility.A natural definition of Lusternik-Schnirelman (LS) category for simplicial complexes,that is invariant under strong equivalences, is given in [10] and a notion of discretetopological complexity in the setting of simplicial complexes by means of contiguoussimplicial maps is given in [8].Let K be a simplicial complex and L ⊆ K a subcomplex. We say that L is categorical ,provided there exists a vertex v ∈ K such that the inclusion map i : L ֒ → K and theconstant map c v : L → K are in the same contiguity class. The simplicial LS category,denoted scat ( K ), is defined as the least integer n ≥ K is covered by ( n + K is strongly collapsible i.e., having the strong homotopy type of apoint if and only if scat( K ) = K and K ′ be two simplicial complexes. Then thecategorical product of K and K ′ , denoted by K Q K ′ is an simplicial complex such that(1) its vertices are pairs ( v , ω ) where v is a vertex of K and ω is a vertex of K ′ , and(2) the projections pr : K Q K ′ → K and pr : K Q K ′ → K ′ are simplicial maps andare universal with the property.Let K be a simplicial complex and K = K Q K a categorical product. Then a simpli-cial subcomplex Ω ⊂ K is a Farber subcomplex, provided there exists a simplicial map σ : Ω → K such that ∆ ◦ σ ∼ ι Ω where ι Ω : Ω ֒ −→ K is the inclusion map and ∆ : K → K isthe diagonal map ∆ ( v ) = ( v , v ). Definition 6. [8]
The discrete topological complexity TC ( K ) of the simplicial complex K is theleast integer n ≥ such that K can be covered by ( n + Farber subcomplexes.
In other words, TC( K ) ≤ n if and only if K = Ω ∪ . . . ∪ Ω n , and there exist simpli-cial maps σ j : Ω j → K such that ∆ ◦ σ j ∼ ι j where ι j : Ω j ֒ −→ K are inclusions for j = , . . . , n . ONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS 5
Before the end of this section, we remark that for a given simplicial complex K , cat( | K | )and TC( | K | ) are lower bounds for scat( K ) and TC( K ), respectively.2. C ontiguity D istance Throughout the paper, a simplicial complex is meant to be an abstract simplicial com-plex, all simplicial complexes are assumed to be (edge-) path connected, and all mapsbetween simplicial complexes are assumed to be simplicial maps.
Definition 7. [7, Definition 8.1]
For simplicial maps ϕ, ψ : K → K ′ , the contiguity distancebetween ϕ and ψ , denoted by SD ( ϕ, ψ ) , is the least integer n ≥ such that there exists a coveringof K by subcomplexes K , K , . . . , K n with the property that ϕ (cid:12)(cid:12)(cid:12) K j , ψ (cid:12)(cid:12)(cid:12) K j : K j → K ′ are in the samecontiguity class for all j = , , . . . , n. Remark 1.
There is another simplicial version of homotopic distance, called simplicial distance,introduced in [3] and given in the sense of Gonzalez [6] . The underlying idea in the simplicialdistance
SimpD lies in [6, Lemma 1.1] even though
SimpD has nothing to do with sections ofa path fibration. It is not easy to compare simplicial distance and contiguity distance as it is noteasy to compare simplicial complexity [6] and discrete topological complexity [8] . It is easy to see that the contiguity distance defines a symmetric relation on the setof simplicial maps and the contiguity distance between two maps is zero if and only ifthey are in the same contiguity class. The next proposition tells us that this notion iswell-defined on the set of equivalence classes of simplicial maps.
Proposition 1. If ϕ ∼ ¯ ϕ, ψ ∼ ¯ ψ : K → K ′ , then SD ( ϕ, ψ ) = SD ( ¯ ϕ, ¯ ψ ) .Proof. Suppose first that SD( ϕ, ψ ) = n . By definition this means that there exists acovering of K by subcomplexes K , K , . . . , K n with the property that ϕ (cid:12)(cid:12)(cid:12) K j , ψ (cid:12)(cid:12)(cid:12) K j : K → K ′ are in the same contiguity class for all j . Since ϕ ∼ ¯ ϕ and ψ ∼ ¯ ψ , so their restrictions to K j are also in the same contiguity classes for all j . Also recall that, contiguity classes areequivalence classes, so we have ¯ ϕ (cid:12)(cid:12)(cid:12) K j ∼ ϕ (cid:12)(cid:12)(cid:12) K j ∼ ψ (cid:12)(cid:12)(cid:12) K j ∼ ¯ ψ (cid:12)(cid:12)(cid:12) K j for all j . Therefore SD( ¯ ϕ, ¯ ψ ) ≤ n .Starting with SD( ¯ ϕ, ¯ ψ ) gives us SD( ϕ, ψ ) ≤ SD( ¯ ϕ, ¯ ψ ), which finishes the proof. (cid:3) We can use a finite covering of a complex K to produce an upper bound for the sim-plicial distance between maps. Proposition 2.
Given two simplicial maps ϕ, ψ : K → K ′ and a finite covering of K by subcom-plexes K , K , . . . , K n , we haveSD ( ϕ, ψ ) ≤ n X i = SD ( ϕ (cid:12)(cid:12)(cid:12) K j , ψ (cid:12)(cid:12)(cid:12) K j ) + n . Proof.
Suppose SD( ϕ | K j , ψ | K j ) = m j for each j = , , . . . , n . So there exists a covering of K j by subcomplexes K j , K j , . . . , K m j j such that ϕ | K ij ∼ ψ | K ij . AYSE BORAT, MEHMETCIK PAMUK, AND TANE VERGILI
The collection K = n K , . . . , K m , K , . . . , K m , . . . , K n , . . . , K m n n o is a covering for K satisfying ϕ | L ∼ ψ | L for all L ∈ K . So since the cardinality of K is ( m + m + . . . + m n ) + n +
1, therequired inequality holds. (cid:3)
Next, we mention the relation between the simplicial LS -category and the contiguitydistance between simplicial maps. First note that for a subcomplex L of a simplicialcomplex K , if id K (cid:12)(cid:12)(cid:12) L and c v (cid:12)(cid:12)(cid:12) L are in the same contiguity class then L is categorical in K .From this observation it is easy to see that for a simplicial complex K and a vertex v of K , we have scat( K ) = SD(id K , c v ) . Let K be a simplicial complex and | K | denote its geometric realization. We know thatboth scat( K ) and TC( K ) might di ff er from cat( | K | ) and TC( | K | ) (see, Theorem 4 and [8,Theorem 5.2]). The simplicial category and discrete topological complexity depend onthe simplicial structure more than on the geometric realization of the complex [8, 10].This implies that for simplicial complexes K , K ′ and simplicial maps ϕ, ψ : K → K ′ , weexpect SD( ϕ, ψ ) is not necessarily the same as D( | ϕ | , | ψ | ), where | ϕ | , | ψ | : | K | → | K ′ | arecontinuous maps between their corresponding geometric realizations [7]. Proposition 3.
For simplicial maps ϕ, ψ : K → L, we have D( | ϕ | , | ψ | ) ≤ SD ( ϕ, ψ ) .Proof. Let SD( φ, ψ ) = n so that there exist subcomplexes K , K , . . . , K n in such a way thatthe inclusion map ι i : K i → K and the constant map c v : K i → L are in the same contiguityclass, ι i ∼ c v . Note that the union of the closed subsets | K | , | K | , . . . , | K n | of | K | covers | K | and the geometric realizations of ι i and c v , | ι i | , | c v | : | K i | → | K | are homotopic continuous maps. (cid:3) The following is an example for the strict form of the inequality given in Proposition 3.
Example 1.
Consider the simplicial complex K given in Figure 3 [2] . Let id K and c be theidentity simplicial map and constant simplicial map on K, respectively. We know that scatK = ( [10, Example 3.2] ) so that SD ( id K , c ) = . Notice that the homotopic distance D( | id K | , | c | ) iszero which follows from the fact that the geometric realization | K | , of K is contractible. Therefore D( | id K | , | c | ) < SD ( id K , c ) . bc bcbcbc bcbc F igure | K | is contractible whereas K is not strongly collapsible ONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS 7
Before we study the behaviour of contiguity distance under barycentric subdivision,we recall some ingredients and talk about how scat behaves under barycentric subdivi-sion.
Definition 8.
The barycentric subdivison of a given simplicial complex K is the simplicial com-plex sdK whose the set of vertices is K and each n-simplex in sdK is of form { σ , σ , . . . , σ n } where σ ( σ ( . . . ( σ n . Definition 9.
For a simplicial map ϕ : K → L, the induced map sd ϕ : sdK → sdL is given by ( sd ϕ )( { σ , . . . , σ q } ) = { ϕ ( σ ) , . . . , ϕ ( σ q ) } . Notice that sd ϕ is a simplicial map and sd(id) = id. Proposition 4. [9]
If the simplicial maps ϕ, ψ : K → L are in the same contiguity class, so aresd ϕ and sd ψ . The relation between the contiguity distance of two maps and the contiguity distanceof their induced maps on barycentric subdivisions can be given as follows.
Theorem 1.
For simplicial maps ϕ, ψ : K → K ′ , SD ( sd ϕ, sd ψ ) ≤ SD ( ϕ, ψ ) .Proof. Let SD( ϕ, ψ ) = n . Then there are subcomplexes K , K , . . . , K n covering K such that ϕ | K i ∼ ψ | K i for all i = , , . . . , n .Take the cover { sd( K ) , sd( K ) , . . . , sd( K n ) } of sd K . By Proposition 4, if ϕ | K i ∼ ψ | K i thensd( ϕ | K i ) ∼ sd( ψ | K i ).On the other hand, sd( ϕ | K i ) = sd ϕ | sd ( K i ) . More precisely, if { σ , . . . , σ q } ∈ sd( K i ),sd( ϕ | K i )( { σ , . . . , σ q } ) = n sd ϕ | K i ( σ ) , . . . , sd ϕ | K i ( σ q ) } = { sd ϕ ( σ ) , . . . , sd ϕ ( σ q ) o = sd ϕ (cid:12)(cid:12)(cid:12) sd ( K i ) . Hence, since we have sd( ϕ | K i ) ∼ sd( ψ | K i ), it follows that sd ϕ (cid:12)(cid:12)(cid:12) sd ( K i ) ∼ sd ψ (cid:12)(cid:12)(cid:12) sd ( K i ) for all i . (cid:3) Although the below corollary is given as a consequence of some theorems related tofinite spaces in [10, Corollary 6.7] and a direct proof is given in [9, Theorem 3.1.1], wegive the following alternative proof using the contiguity distance for the consistency ofthe paper
Corollary 1.
For a simplicial complex K, scat ( sdK ) ≤ scat ( K ) .Proof. In Theorem 1, take ϕ = id and ψ = c as the identity map and a constant map,respectively. So the induced maps sd(id) and sd( c ) are also the identity and constant mapon sd( K ). Thus the corollary follows. (cid:3) Observe that, for a simplicial complex K being strongly collapsible is equivalent tosaying that scat( K ) = SD(id K , c v ) =
0. Hence, for a strongly collapsible complex K , we AYSE BORAT, MEHMETCIK PAMUK, AND TANE VERGILI have id K ∼ c v . The following theorem tells that the same is true for arbitrary maps. Theorem 2.
For any maps ϕ, ψ : K → K ′ , SD ( ϕ, ψ ) = , provided K or K ′ is strongly collapsible.Proof. Suppose K is strongly collapsible, then we have id K ∼ c v where v is a vertex in K .We have the following diagram K id K ( ( c v K ϕ / / K ′ which implies that ϕ ◦ id K ∼ ϕ ◦ c v (constant) . Similarly we have K id K ( ( c v K ψ / / K ′ so that ψ ◦ id K ∼ ψ ◦ c v (constant) . Since K ′ is edge-path connected, all the constant maps are in the same contiguity class.Hence we have ϕ = ϕ ◦ id K ∼ ψ ◦ id K = ψ .On the other hand, if K ′ is strongly collapsiblescat( K ′ ) = = SD ( id K ′ , c ω )where ω is a vertex in K ′ . That is, id K ′ ∼ c ω . This time we have the following diagram K ϕ / / K ′ id K ′ ) ) c ω K ′ so that id K ′ ◦ ϕ ∼ c ω ◦ ϕ (constant) . Similarly we have K ψ / / K ′ id K ′ ) ) c ω K ′ so that id K ′ ◦ ψ ∼ c ω ◦ ψ (constant) . Note that K ′ is edge-path connected since it is strongly collapsible. Hence we have ϕ = id K ′ ◦ ϕ ∼ id K ′ ◦ ψ = ψ . (cid:3) For the converse we have the following result.
Corollary 2.
Let K be a simplicial complex. If SD ( ϕ, ψ ) = for any pair of simplicial maps ϕ, ψ : K → K, then K is strongly collapsible.
ONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS 9
Proof.
If we take ϕ = id K and ψ = c v on a fixed vertex v ∈ K , our assumption SD ( id K , c v ) = K ) = K is stronglycollapsible. (cid:3) Theorem 3.
Let v be a vertex of the simplicial complex K. For the simplicial mapsi , i : K → K defined as i ( σ ) = ( σ, v ) and i ( σ ) = ( v , σ ) , we have scat ( K ) = SD ( i , i ) .Proof. First we prove that SD( i , i ) ≤ scat( K ). Let L ⊆ K be categorical. That is, thereexists a vertex v of K such that the inclusion map ι : L ֒ −→ K and the constant map c v : L → K are in the same contiguity class. We want to show that i (cid:12)(cid:12)(cid:12) L and i (cid:12)(cid:12)(cid:12) L are also inthe same contiguity class. Consider the following composition of simplicial maps L ∆ L / / L ι Q c v ) ) c v Q ι K where ∆ L is the diagonal map of L , defined on the set of vertices by v → ( v , v ), and ι Q c v and c v Q ι is the categorical product of ι and c v . Then i (cid:12)(cid:12)(cid:12) L = ( ι Y c v ) ◦ ∆ L , and i (cid:12)(cid:12)(cid:12) L = ( c v Y ι ) ◦ ∆ L . Since L is categorical, then ι ∼ c v . We have ι Y c v ∼ c v Y c v , c v Y ι ∼ c v Y c v . This implies ι Y c v ∼ c v Y ι so that ( ι Q c v ) ◦ ∆ L ∼ ( c v Q ι ) ◦ ∆ L , which proves our claim.Next, we show that scat( K ) ≤ SD( i , i ). Assume that L is a subcomplex of K with i (cid:12)(cid:12)(cid:12) L ∼ i (cid:12)(cid:12)(cid:12) L . Let p i : K → K be the projection maps for i = ,
2. Then p ◦ i (cid:12)(cid:12)(cid:12) L ∼ p ◦ i (cid:12)(cid:12)(cid:12) L sothat ι ∼ c v . (cid:3) The Proposition 3 leads to the following theorem.
Theorem 4.
Let K be a simplicial complex and | K | its geometric realization. cat ( | K | ) ≤ scat ( K ) .Proof. Consider the simplicial maps i : K → K and i : K → K defined in Theorem 3 sothat scat( K ) = SD( i , i ). In that case, their geometric realizations | i | , | i | : | K | → | K | are continuous maps. By Lemma 5.1 in [8], we know that | K | and | K | × | K | are homotopyequivalent spaces. Let u : | K | → | K | × | K | be the homotopy equivalence. Therefore the inclusion maps ι : | K | → | K | × | K | and ι : | K | → | K | × | K | are homotopic to u ◦ | i | and u ◦ | i | respectively. By Proposition 3 and [7, Proposition 3.1], we havecat( | K | ) = D( ι , ι ) = D( u ◦ | i | , u ◦ | i | ) ≤ D( | i | , | i | ) ≤ SD( i , i ) = scat( K ) . (cid:3) Our next aim is to prove Theorem 5. So we need Corollary 3 and Corollary 4 whichfollow from Proposition 5 and Proposition 6, respectively.
Proposition 5.
Let ϕ, ψ : K → K ′ and µ : M → K be simplicial maps. Then we haveSD ( ϕ ◦ µ, ψ ◦ µ ) ≤ SD ( ϕ, ψ ) . Proof.
Let SD( ϕ, ψ ) = n . Then there exist subcomplexes K , . . . , K n of K such that ϕ (cid:12)(cid:12)(cid:12) K j ∼ ψ (cid:12)(cid:12)(cid:12) K j for all j . Define M ⊃ M j : = µ − ( K j ) and the restriction map µ j : M j → K . Then( ϕ ◦ µ ) j = ϕ ◦ µ j = ϕ ◦ ι j ◦ ¯ µ j = ϕ j ◦ ¯ µ j ∼ ψ j ◦ ¯ µ j = ψ ◦ ι j ◦ ¯ µ j = ψ ◦ µ j = ( ψ ◦ µ ) j where ι j : K j ֒ −→ K is the inclusion and ¯ µ j : M j → K j , ¯ µ j ( x ) = µ j ( x ) is a map satisfying µ j = ι j ◦ ¯ µ j . Therefore SD( ϕ ◦ µ, ψ ◦ µ ) ≤ n . (cid:3) Corollary 3.
Let ϕ, ψ : K → K ′ be simplicial maps and β : M → K be a simplicial map whichhas a right strong equivalence (that is, β satisfies β ◦ α ∼ id K where α : K → M). ThenSD ( ϕ ◦ β, ψ ◦ β ) = SD ( ϕ, ψ ) .Proof. Since β ◦ α ∼ id K it follows that ϕ ◦ β ◦ α ∼ ϕ and ψ ◦ β ◦ α ∼ ψ . SoSD( ϕ, ψ ) = SD( ϕ ◦ β ◦ α, ψ ◦ β ◦ α ) ≤ SD( ϕ ◦ β, ψ ◦ β ) ≤ SD( ϕ, ψ )where the equality follows from Proposition 1 and the inequalities follow from Proposi-tion 5. Hence we have SD( ϕ ◦ β, ψ ◦ β ) = SD( ϕ, ψ ). (cid:3) Proposition 6.
Let ϕ, ψ : K → K ′ and φ, φ ′ : K ′ → M be simplicial maps. If φ ∼ φ ′ , thenSD ( φ ◦ ϕ, φ ′ ◦ ψ ) ≤ SD ( ϕ, ψ ) .Proof. Suppose SD( ϕ, ψ ) = n . Then there exist subcomplexes K ′ , K ′ , . . . , K ′ n of K ′ suchthat ϕ (cid:12)(cid:12)(cid:12) K ′ i and ψ (cid:12)(cid:12)(cid:12) K ′ j are in the same contiguity class for all i , j . So( φ ◦ ϕ ) (cid:12)(cid:12)(cid:12) K ′ i = φ ◦ ϕ (cid:12)(cid:12)(cid:12) K ′ i ∼ φ ′ ◦ ϕ (cid:12)(cid:12)(cid:12) K ′ i ∼ φ ′ ◦ ϕ (cid:12)(cid:12)(cid:12) K ′ j = ( φ ′ ◦ ϕ ) (cid:12)(cid:12)(cid:12) K ′ j . Hence SD( φ ◦ ϕ, φ ′ ◦ ψ ) ≤ n . (cid:3) Corollary 4.
Let ϕ, ψ : K → K ′ be simplicial maps and α : K ′ → M be a simplicial mapwhich has a left strong equivalence (that is, α satisfies β ◦ α ∼ id K ′ where β : M → K ′ ). ThenSD ( α ◦ ϕ, α ◦ ψ ) = SD ( ϕ, ψ ) . ONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS 11
Proof.
Since β ◦ α ∼ id K ′ it follows that β ◦ α ◦ ϕ ∼ ϕ and β ◦ α ◦ ψ ∼ ψ . SoSD( ϕ, ψ ) = SD( β ◦ α ◦ ϕ, β ◦ α ◦ ψ ) ≤ SD( α ◦ ϕ, α ◦ ψ ) ≤ SD( ϕ, ψ )where the equality follows from Proposition 1 and the inequalities follow from Proposi-tion 6. Hence we have SD( α ◦ ϕ, α ◦ ψ ) = SD( ϕ, ψ ). (cid:3) Theorem 5. If β : K ′ ∼ K and α : L ∼ L ′ have the same strong homotopy type and if simplicialmaps ϕ, ψ : K → L and ϕ ′ , ψ ′ : K ′ → L ′ make the following diagrams commutative with respectto f and g respectively, in the sense of contiguity (that is, α ◦ ϕ ◦ β ∼ ϕ ′ and α ◦ ψ ◦ β ∼ ψ ′ ),then we have SD ( ϕ, ψ ) = SD ( ϕ ′ , ψ ′ ) . K LK ′ L ′ ϕψ αβ ϕ ′ ψ ′ Proof.
SD( ϕ ′ , ψ ′ ) = SD( α ◦ ϕ ◦ β, α ◦ ψ ◦ β ) = SD( ϕ ◦ β, ψ ◦ β ) = SD( ϕ, ψ )where the second equality follows from Corollary 4 and the last equality follows fromCorollary 3. (cid:3)
Remark 2.
Notice that the result of Theorem 5 is still valid even if we consider β and α as rightand left strong equivalences, respectively. The simplicial LS category of a simplicial map is defined as in the following definition.
Definition 10. [15]
Let ϕ : K → K ′ be a simplicial map and ω be a vertex of K ′ . Simplicial LScategory scat ( ϕ ) of ϕ is defined to be the least integer n such that there exists a covering of K bysubcomplexes K , K , . . . , K n such that ϕ (cid:12)(cid:12)(cid:12) K j : K j → K ′ and constant map c ω : K j → K are in thesame contiguity class for all j.
Corollary 5.
Let ϕ : K → K ′ be a simplicial map. Then scat ( ϕ ) ≤ min { scat ( K ) , scat ( K ′ ) } .Proof. Let id K : K → K be the identity map and c v : K → K be the constant map at thevertex v in K .scat( ϕ ) = SD( ϕ, ϕ ◦ c v ) = SD( ϕ ◦ id K , ϕ ◦ c v ) ≤ SD(id K , c ϕ ( v ) ) = scat( K )where the inequality follows from Proposition 6. Hence scat( ϕ ) ≤ scat( K ).On the other hand, we havescat( K ′ ) = SD(id K ′ , c ω ) ≥ SD(id K ′ ◦ ϕ, c ω ◦ ϕ ) = scat( ϕ )where id K ′ : K ′ → K ′ is the identity map and c ω : K ′ → K ′ is the constant map at thevertex ω in K ′ . Thus scat( ϕ ) ≤ scat( K ′ ). (cid:3) Let K be a simplicial complex and p , p : K → K projection maps onto the first andsecond factors, respectively. The following theorem is first proved in [8, Theorem 3.4] (seealso [7, Example 8.2]). Here, we provide an alternative proof using contiguity distance. Theorem 6.
For a simplicial complex K, we have SD ( p , p ) = TC ( K ) .Proof. We first show that TC( K ) ≤ SD( p , p ). Suppose TC( K ) = n . Then there is acovering for K which consists of Farber subcomplexes L , L , . . . , L n . Since each L i is aFarber subcomplex, there exits a simplicial map σ i : L i → K such that ∆ ◦ σ i ∼ ι L i . ∆ ◦ σ i ∼ ι L i p ◦ ( ∆ ◦ σ i ) ∼ p ◦ ι L i = p (cid:12)(cid:12)(cid:12) L i p ◦ ( ∆ ◦ σ i ) ∼ p ◦ ι L i = p (cid:12)(cid:12)(cid:12) L i Since p ◦ ( ∆ ◦ σ i ) = p ◦ ( ∆ ◦ σ i ), we have p (cid:12)(cid:12)(cid:12) L i ∼ p (cid:12)(cid:12)(cid:12) L i .Next we will show that TC( K ) ≤ SD( p , p ). Suppose SD( p , p ) = n . Then there existsubcomplexes L , L , . . . , L n which cover K and p (cid:12)(cid:12)(cid:12) L i ∼ p (cid:12)(cid:12)(cid:12) L i for i = , , . . . , n . WLOG,we assume p (cid:12)(cid:12)(cid:12) L i ∼ c p (cid:12)(cid:12)(cid:12) L i . This means that for an element (cid:16) [ x ] , [ y ] (cid:17) in L i where [ x ] = { x , x , . . . , x k } and [ y ] = { y , y , . . . , y m } , p (cid:18)(cid:16) [ x ] , [ y ] (cid:17)(cid:19) ∪ p (cid:18)(cid:16) [ x ] , [ y ] (cid:17)(cid:19) = { x , . . . , x k , y , . . . , y m } is a simplex in K .We define a simplicial map σ i : L i → K so that L i σ / / K ∆ / / K ∆ ◦ σ i ∼ c ι L i .Define σ i (cid:18)(cid:16) [ x ] , [ y ] (cid:17)(cid:19) = p (cid:12)(cid:12)(cid:12) L i (cid:18)(cid:16) [ x ] , [ y ] (cid:17)(cid:19) ∪ p (cid:12)(cid:12)(cid:12) L i (cid:18)(cid:16) [ x ] , [ y ] (cid:17)(cid:19) = { x , . . . , x k , y , . . . , y m } = { x , . . . , x k , y , . . . , y m } . ∆ ◦ σ i (cid:18)(cid:16) [ x ] , [ y ] (cid:17)(cid:19) = (cid:16) { x , . . . , x k , y , . . . , y m } , { x , . . . , x k , y , . . . , y m } (cid:17) . ι L i (cid:18)(cid:16) [ x ] , [ y ] (cid:17)(cid:19) = (cid:16) [ x ] , [ y ] (cid:17) = (cid:16) { x , . . . , x k } , { y , . . . , y m } (cid:17) . So, L i is also a Farber subcomplex. (cid:3) There is a well-known inequality between topological complexity and LS -categoryof a topological space X . The same inequality holds for simplicial complexes (see [8,Theorem 4.3]). In the following, we provide a proof in terms of contiguity distance. Theorem 7.
For a simplicial complex K, we have scat( K ) ≤ TC ( K ) .Proof. Consider the following composition of maps K i / / K p / / K , v i / / ( v , v ) p / / v , ONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS 13 and note that p ◦ i = id K . Similarly consider the composition of maps K i / / K p / / K , v i / / ( v , v ) p / / v , and we have p ◦ i = c v . By Proposition 6, SD ( p ◦ i , p ◦ i ) ≤ SD ( p , p ) = TC ( K ) ⇒ SD ( id K , c v ) ≤ TC ( K ) ⇒ scat( K ) ≤ TC ( K ) . (cid:3) Corollary 6.
Let ϕ, ψ : K → K ′ be two simplicial maps (and K ′ be edge path connected). ThenSD ( ϕ, ψ ) ≤ scat ( K ) .Proof. If we take K ′′ = K , η = id K and η ′ = c v a constant map in Proposition 7, then theconstant maps ϕ ◦ c v and ψ ◦ c v : K → K ′ are in the same contiguity class since K ′ is edgepath connected. By Proposition 5 and Theorem 1 we haveSD( ϕ, ψ ) = SD( ϕ ◦ id K , ψ ◦ id K ) ≤ SD(id K , c v ) = scat( K ) . (cid:3) Corollary 7. TC ( K ) ≤ scat ( K ) .Proof. If we consider the projection maps p , p : K → K respectively in Corollary 6, wehave SD( p , p ) = TC( K ) ≤ scat( K ) . (cid:3) Corollary 8.
Let ϕ, ψ : K → K ′ be two simplicial maps. Then SD ( ϕ, ψ ) ≤ TC ( K ′ ) .Proof. Consider K Q K ϕ Q ψ / / K ′ Q K ′ p * * p K ′ where each p i is a projection map for i = ,
2. Then, using Proposition 3, we haveSD( ϕ, ψ ) = SD (cid:16) p ◦ ( ϕ Y ψ ) , p ◦ ( ϕ Y ψ ) (cid:17) ≤ SD( p , p ) = TC( K ′ ) . (cid:3) Remark 3.
Observe that Theorem 2 also follows from Corollaries 6 and 8.
Proposition 7.
Let K , K ′ , and K ′′ be simplicial complexes, η, η ′ : K ′′ → K and ϕ, ψ : K → K ′ besimplicial maps. If ϕ ◦ η ′ ∼ ψ ◦ η ′ , then SD ( ϕ ◦ η, ψ ◦ η ) ≤ SD ( η, η ′ ) .Proof. Let SD( η, η ′ ) = n . Then there exists a covering { L , L , . . . , L n } for K ′′ such that η (cid:12)(cid:12)(cid:12) L i ∼ η ′ (cid:12)(cid:12)(cid:12) L i for i = , , . . . , n . η (cid:12)(cid:12)(cid:12) L i ∼ η ′ (cid:12)(cid:12)(cid:12) L i ϕ ◦ η (cid:12)(cid:12)(cid:12) L i ∼ ϕ ◦ η ′ (cid:12)(cid:12)(cid:12) L i η (cid:12)(cid:12)(cid:12) L i ∼ η ′ (cid:12)(cid:12)(cid:12) L i ψ ◦ η (cid:12)(cid:12)(cid:12) L i ∼ ψ ◦ η ′ (cid:12)(cid:12)(cid:12) L i Since ϕ ◦ η ′ ∼ ψ ◦ η ′ , by the transitivity of ∼ we have ϕ ◦ η (cid:12)(cid:12)(cid:12) L i ∼ ψ ◦ η (cid:12)(cid:12)(cid:12) L i and this completesour proof. (cid:3)
3. A cknowledgement
The authors thank Enrique Maci´as-Virg ´os and David Mosquera-Lois for their helpfulcomments and suggestions. The second author was partially supported by the Scientificand Technological Research Council of Turkey (T ¨UB˙ITAK) [grant number 11F015].R eferences [1] J. A. Barmak,
Algebraic topology of finite topological spaces and applications , Lecture notes in mathematics,Vol. 2032, Springer Heidelberg, 2011.[2] J. A. Barmak, E. G. Minian,
Strong homotopy types, nerves and collapses , Discrete Comput. Geom. 47 (2)(2012) 301–328.[3] A. Borat, Simplicial distance, submitted. arXiv: 2009.01640.[4] O. Cornea, G. Lupton, J. Oprea, D. Tanre, Lusternik-Schnirelmann category. Mathematical Surveys andMonographs, Vol. 103, American Mathematical Society 2003.[5] M. Farber, Topological complexity of motion planning, Discrete and Computational Geometry 29 (2003),211-221.[6] J. Gonzalez, Simplicial complexity: piecewise linear motion planning in robotics, New York Journal ofMathematics, Vol 24 (2018), 279-292.[7] E. Maci´as-Virg´os, D. Mosquera-Lois, Homotopic distance between maps, to appear in Math. Proc.Cambridge Philos. Soc. ArXiv: 1810.12591.[8] D. Fern´andez-Ternero, E. Macias-Virg´os, E. Minuz, J. A. Vilches, Discrete topological complexity, Proc.Amer. Math. Soc. 146, 4535-4548 (2018).[9] D. Fern´andez-Ternero, E. Macias-Virg´os, E. Minuz, J. A. Vilches, Simplicial Lusternik-Schnirelmanncategory, Publicacions Matematiques, Volume 63, Number 1, 265-293 (2019).[10] D. Fern´andez-Ternero, E. Macias-Virg´os, J. A. Vilches, Lusternik-Schnirelmann category of simplicialcomplexes and finite spaces, Topology and its Applications, 194, 37-50 (2015).[11] M. Farber, M. Grant, Symmetric motion planning. In M Farber, R Ghrist, M Burger, and D Koditschekeditors, Topology and Robotics, Contemporary Mathematics Series, pages 85 - 104, United States, 2007.American Mathematical Society.[12] M. Farber, M. Grant, Robot motion planning, weights of cohomology classes, and cohomology operations.Proc. Amer. Math. Soc., 136 (9), 3339 - 3349 (2008).[13] D. Kozlov, Combinatorial Algebraic Topology, Springer Science and Business Media, 2008.[14] L. Lusternik, L. Schnirelmann, Methodes Topoligiques dans les Problemes Variationnets. Herman, Paris(1934).
ONTIGUITY DISTANCE BETWEEN SIMPLICIAL MAPS 15 [15] N. A. Scoville, W. Swei, On the Lusternik–Schnirelmann category of a simplicial map, Topology and ItsApplications, 216, 116-128 (2017).[16] E.H. Spanier, Algebraic Topology. McGraw-Hill Book Co., New York-Toronto, Ony.-London, 1966.D epartment of M athematics , B ursa T echnical U niversity , B ursa , T urkey Email address : [email protected] D epartment of M athematics , M iddle E ast T echnical U niversity , A nkara , T urkey Email address : [email protected] D epartment of M athematics , K aradeniz T echnical U niversity , T rabzon , T urkey Email address ::