aa r X i v : . [ m a t h . A T ] S e p SIMPLICIAL DISTANCE
AYSE BORAT
Abstract.
In this paper we will introduce and give topological propertiesof a new concept named simplicial distance which is the simplicial analog ofthe homotopic distance (in the sense of Marcias-Virgos and Mosquera-Lois intheir paper [6]). According to our definition of simplicial distance, simplicialcomplexity is a particular case of this new concept. Introduction
Simplicial complexity as defined by Gonzalez in [5] and its higher analogs as definedby Borat in [1] have benefits such as the motion planners in this setting can becomputed with a help of a computer. In this paper we will introduce a new conceptnamed simplicial distance of which simplicial complexity is a particular case. Soone open question in this paper is to carry the work to the digital world.To understand the simplicial complexity more clearly we need the definition of thetopological complexity. Therefore we will start this section by recalling some basicdefinitions such as TC, cat and homotopic distance.
Definition 1.1. [6] Let f, g : X → Y be continuous maps. Homotopic distanceD ( f, g ) between f and g is the least non-negative integer k if there exist opensubsets U , U , . . . , U k covering X such that f | U i ≃ g | U i for each i = 0 , , . . . , k . Definition 1.2. [4] Lusternik Schnirelmann category cat ( X ) of a space X is theleast non-negative integer k if there exist open subsets U , U , . . . , U k covering X such that each inclusion ι : U i ֒ → X is nullhomotopic. Definition 1.3. [3] Let π : P X → X × X , π ( γ ) = ( γ (0) , γ (1)) be the path fibrationwhere P X stands for the path space of X . Topological complexity TC ( X ) is theleast non-negative integer k if there exist open subsets U , U , . . . , U k covering X × X such that π admits a continuous section on each U i . Definition 1.4. [5] Let K be an ordered complex. For b, c ∈ Z ≥ , the ( b, c ) -simplicial distance SC bc ( K ) is the least non-negative integer k if there are subcom-plexes J , J , · · · , J k covering Sd b ( K ) such that π : J i ֒ → Sd b ( K × K ) ι −→ K × K pr −−→ K and π : J i ֒ → Sd b ( K × K ) ι −→ K × K pr −−→ K Date : September 4, 2020.
Key words and phrases.
Topological complexity, simplicial complexity, simplicial complex,barycentric subdivision, contiguous maps. are c -contiguous for each i ∈ { , , · · · , k } .If there is no such covering, then we set SC bc ( K ) = ∞ . Definition 1.5. [5] There is a monotonic sequenceSC b ( K ) ≥ SC b ( K ) ≥ SC b ( K ) ≥ . . . SC b ( K ) is defined to be the stabilized value of this sequence. Another monotonicsequence can be given bySC ( K ) ≥ SC ( K ) ≥ SC ( K ) ≥ . . . The simplicial complexity SC ( K ) is defined to be the stabilized value of the lattersequence.Notice that topological complexity in [3] and Lusternik Schnirelmann category in [4]are defined in "nonreduced" terms. Throughout this paper we will use the reducedversion of homotopic distance, TC, cat and their simplicial analogs.If we replace f = pr : X × X → X and g = pr : X × X → X with theprojection maps to the factors, then D ( pr , pr ) = TC ( X ) , [6]. Moreover, one cansee homotopic distance as a generalization of Lusternik Schnirelmann category aswell, as D ( id X , c ) = cat ( X ) where c is a constant map on X , [6]. One anothercharacterization of LS category via homotopic distance can be given as follows.For a base point x ∈ X , consider the inclusions i , i : X → X × X given by i ( x ) = ( x, x ) and by i ( x ) = ( x , x ) . Then D ( i , i ) = cat ( X ) , [6].One of the main results of this paper is that we can have a similar relation betweenSC and SimpD as we have a relation between TC and D in the topological realm, seeProposition 3.1. We also have a result which gives the relation between SimpD andthe TC of the geometric realization of some particular complex, see Proposition 3.1.One another main result is that if α : L → M is a simplicial map with a left stronghomotopy type inverse, then SimpD ( α ◦ ϕ, α ◦ ψ ) = SimpD ( ϕ, ψ ) , see Corollary 3.8.We have more results in the last section such as how being in the same contiguityclass effects the simplicial distance.We would like to mention that for more work on homotopic distance one can see[2] where homotopic distance is introduced in the higher dimensional case and [7]in which homotopic distance between functors is carried out.2. Simplicial Distance
Definition 2.1.
Let ϕ, ψ : K → L be two simplicial maps between simplicial com-plexes. ϕ and ψ are said to be contiguous, if σ = { v , . . . , v n } is a simplex in K ,then ϕ ( σ ) ∪ ψ ( σ ) = { ϕ ( v ) , . . . , ϕ ( v n ) , ψ ( v ) , . . . , ψ ( v n ) } constitute a simplex in L . Definition 2.2. [5] Let
K, L be complexes. ϕ, ψ : K → L are called c -contiguous,denoted by ϕ ∼ c ψ , if there exists a sequence of maps H i : K → L , i = 0 , , . . . , c satisfying H = f , H c = g and that for each i = 1 , , · · · , c , the pair ( H i − , H i ) iscontiguous. IMPLICIAL DISTANCE 3
Topological realization of the product of abstract complexes is homeomorphic tothe product of the topological realizations of these complexes. So throughout thispaper, all complexes and their products will be taken in the category of orderedcomplexes unless stated otherwise.For b ∈ Z ≥ , we will fix an approximation ι : Sd b +1 ( K ) → Sd b ( K ) of the identitymap id || K || . Since approximations behave well with respect to compositions (see, forexample, Remark 2.4 in [5]), by abuse of notation, we will denote the iterated com-positions of the maps between barycentric subdivisions by ι : Sd b ( K ) → Sd b ′ ( K ) . Definition 2.3.
Let
K, L be an ordered complex and ϕ, ψ : K → L be simplicialmaps. For b, c ∈ Z ≥ , the ( b, c ) -simplicial distance SimpD bc ( ϕ, ψ ) is the least non-negative integer k if there are k + 1 subcomplexes J , J , · · · , J k covering Sd b ( K ) such that γ ϕ : J i ֒ → Sd b ( K ) π ϕ −−→ L and γ ψ : J i ֒ → Sd b ( K ) π ψ −−→ L are c -contiguous for each i ∈ { , , · · · , k } where π ϕ : Sd b ( K ) ι ֒ −→ K ϕ −→ L and π ψ : Sd b ( K ) ι ֒ −→ K ψ −→ L .If there is no such covering, then we set SimpD bc ( ϕ, ψ ) = ∞ . Proposition 2.4.
If pr i : K × K → K is the projection to the i -th factor, thenSimpD bc ( pr , pr ) = SC bc ( K ) . (cid:3) The following proposition follows by Proposition 2.4 and (3.4) in [5].
Proposition 2.5.
If pr i : K × K → K is the projection to the i -th factor, thenTC ( || K || ) ≤ SimpD bc ( pr , pr ) . (cid:3) Remark 2.6.
For simplicial maps ϕ, ψ : K → L , we have the sequenceSimpD b ( ϕ, ψ ) ≥ SimpD b ( ϕ, ψ ) ≥ SimpD b ( ϕ, ψ ) ≥ . . . So we can define SimpD b ( ϕ, ψ ) := lim c →∞ SimpD bc ( ϕ, ψ ) . Remark 2.7. [8]
Composites of 1-contiguous maps are 1-contiguous, that is, if ϕ, e ϕ : K → L and ψ, e ψ : L → M are 1-contiguous, so is ψ ◦ ϕ ∼ e ψ ◦ e ϕ . It followsthat we have a well-defined composition of contiguity classes [ ψ ] ◦ [ ϕ ] = [ ψ ◦ ϕ ] . Remark 2.8. [8]
Two approximations to the same continuous maps are 1-contiguous.
AYSE BORAT
Remark 2.9. [5, Remark 2.6] If ϕ and ψ are c -contiguous, then the compositions βϕα, βψα : J α −→ K ϕ,ψ −−→ L β −→ M are c -contiguous. Proposition 2.10.
Let ϕ, ψ : K → L be simplicial maps. SimpD b ( ϕ, ψ ) is inde-pendent of the chosen approximations ι : Sd b ( K ) → Sd b − ( K ) of the identity.Proof. Suppose there are two approximations of the identity, namely, ι, ˜ ι : Sd b ( K ) → Sd b − ( K ) . Suppose also SimpD b ( ϕ, ψ ) and ^ SimpD b ( ϕ, ψ ) denote the invariants ob-tained from ι and ˜ ι , respectively.Consider an inclusion j : J → Sd b ( K ) of some subcomplex J ⊆ K . Since γ ϕ = π ϕ ◦ j and γ ψ = π ψ ◦ j are c -contiguous, there are h , h , . . . , h c : J → L such that π ϕ ◦ j = h and π ψ ◦ j = h c and ( h i − , h i ) is 1-contiguous. On the other hand, byRemark 2.8 and Remark 2.9, ( π ϕ ◦ j, e π ϕ ◦ j ) and ( π ψ ◦ j, e π ψ ◦ j ) are c -contiguouspairs. So we may obtain a contiguity chain e π ϕ ◦ j, h , h , . . . , h c , e π ψ ◦ j : J → L of length c + 2 . So ^ SimpD bc +2 ( ϕ, ψ ) ≤ SimpD bc ( ϕ, ψ ) . Proceeding similarly, onecan show that SimpD bc +2 ( ϕ, ψ ) ≤ ^ SimpD bc ( ϕ, ψ ) . Hence we have SimpD b ( ϕ, ψ ) = ^ SimpD b ( ϕ, ψ ) . (cid:3) Proposition 2.11.
There is a sequenceSimpD ( ϕ, ψ ) ≥ SimpD ( ϕ, ψ ) ≥ SimpD ( ϕ, ψ ) ≥ . . . Proof.
Consider a subcomplex J ⊆ K . Let λ : Sd ( J ) → J be an approximation ofthe identity on || J || . Then we have the following diagram in which the compositionsare 1-contiguous since α and β are approximations of the inclusions || J || → || K || . Sd ( J ) Sd b +1 ( K ) J Sd b ( K ) βλ ια Let us consider SimpD bc ( ϕ, ψ ) and suppose that γ ϕ and γ ψ are c -contiguous. Fromthe above diagram, we can construct c + 2 -contiguous simplicial maps ¯ γ ϕ and ¯ γ ψ such that they are 1-contiguous to γ ϕ and γ ψ , respectively. Hence we haveSimpD bc ( ϕ, ψ ) ≥ SimpD b +1 c +2 ( ϕ, ψ ) . ThereforeSimpD b ( ϕ, ψ ) = SimpD bc ( ϕ, ψ ) , for large c ≥ SimpD b +1 c +2 ( ϕ, ψ ) ≥ SimpD b +1 ( ϕ, ψ ) . (cid:3) Definition 2.12.
The stabilized value of the sequence in Proposition 3.5 will becalled simplicial distance between ϕ and ψ , and be denoted by SimpD ( ϕ, ψ ) . IMPLICIAL DISTANCE 5 Properties of Simplicial Distance
The first proposition shows the relation between the simplicial distance and thetopological complexity and the relation between the simplicial distance and thesimplicial complexity.
Proposition 3.1.
For a finite complex K , if pr i : K × K → K is the projection tothe i -th factor for i = 1 , , then we have the following (a) TC ( || K || ) = SimpD ( pr , pr ) . (b) SimpD ( pr , pr ) = SC ( K ) .Proof. (a) follows from Theorem 3.5 in [5] and Proposition 2.4 whereas (b) followsfrom Theorem 3.5 in [5] and (a). (cid:3) Corollary 3.2. If K and L are two finite complexes whose geometric realizationsare homotopy equivalent. Let pr i : K × K → K and f pr i : L × L → L be theprojections to the i -th factors for i = 1 , , then SimpD ( pr , pr ) = SimpD ( f pr , f pr ) . Corollary 3.3.
Let pr i : K × K → K be projections as in Proposition 3.1, then || K || is contractible iff SimpD ( pr , pr ) = 0 . Proposition 3.4.
SimpD ( ϕ, ψ ) = SimpD ( ψ, ϕ ) . (cid:3) Theorem 3.5. If ϕ ∼ ψ , then SimpD ( ϕ, ψ ) = 0 .Proof. If ϕ ∼ ψ , then there is some non-negative integer c such that ϕ ∼ c ψ , that is,there exists H : K → L such that H = ϕ , H c = ψ and ( H i − , H i ) is 1-contiguousfor each i . If we could show that SimpD bc ( ϕ, ψ ) = 0 , then it immediately followsthat SimpD ( ϕ, ψ ) = 0 .Consider the following simplicial maps γ ϕ : K j −→ Sd b ( K ) ι ֒ −→ K ϕ −→ L and γ ψ : K j −→ Sd b ( K ) ι ֒ −→ K ψ −→ L where ϕ and ψ are c -contiguous. Define G : K j −→ Sd b ( K ) ι ֒ −→ K H −→ L We have G = H ◦ ι ◦ j = ϕ ◦ ι ◦ j and G c = H c ◦ ι ◦ j = ψ ◦ ι ◦ j , and ( G i − , G i ) is 1-contiguous for each i by Remark 2.7. So γ ϕ ∼ c γ ψ . Hence SimpD ( ϕ, ψ ) = 0 . (cid:3) It is known that if f, g : || K || → || L || are homotopic then their simplicial approx-imations ϕ f , ϕ g : Sd N ( K ) → L are in the same contiguity class for some integer N (see for instance [8]). By Proposition 3.5, SimpD ( ϕ f , ϕ g ) = 0 . Notice that thisresult is concordant with the result (2) in Subsection 2.1 in [6]. AYSE BORAT
Theorem 3.6. If ϕ ∼ ϕ ′ and ψ ∼ ψ ′ , then SimpD ( ϕ, ψ ) = SimpD ( ϕ ′ , ψ ′ ) .Proof. First we will show that SimpD ( ϕ, ψ ) ≤ SimpD ( ϕ ′ , ψ ′ ) . Since ϕ ∼ ϕ ′ and ψ ∼ ψ ′ , we have ϕ ∼ c ϕ ′ and ψ ∼ d ψ ′ for some c, d ∈ Z ≥ .Suppose SimpD ( ϕ ′ , ψ ′ ) = k . So k = SimpD ba ( ϕ ′ , ψ ′ ) for some large a, b ∈ Z ≥ . By definition of ( b, a ) -simplicial distance, there are k + 1 subcomplexes J i ’s suchthat for each i γ ′ ϕ : J i j −→ Sd b ( K ) ι ֒ −→ K ϕ ′ −→ L and γ ′ ψ : J i j −→ Sd b ( K ) ι ֒ −→ K ψ ′ −→ L are a -contiguous. Hence on each J i , ϕ ◦ ι ◦ j ∼ c ϕ ′ ◦ ι ◦ j = γ ϕ ′ ∼ a γ ψ ′ = ψ ′ ◦ ι ◦ j ∼ d ψ ◦ ι ◦ j It follows that γ ϕ ∼ c + a + d γ ψ where γ ϕ = ϕ ◦ ι ◦ j and γ ψ = ψ ◦ ι ◦ j . In otherwords, there exists a simplicial map G j : J i → L such that G = γ ϕ , G c + a + d = γ ψ and ( G j − , G j ) is 1-contiguous for each j . More precisely, G j is given as follows. G i := ¯ H j ◦ ι ◦ j , ≤ j ≤ c ¯ F j − c , c ≤ j ≤ c + a ¯ B j − c − a ◦ ι ◦ j , c + a ≤ j ≤ c + a + d where ¯ H j is the family of maps of length c (between ϕ and ϕ ′ ), ¯ F j is the family ofmaps of length a (between γ ϕ ′ and γ ψ ′ ) and ¯ B j is the family of maps of length d (between ψ and ψ ′ ).Therefore SimpD bc + a + d ( ϕ, ψ ) ≤ k which implies that SimpD ( ϕ, ψ ) ≤ k . The otherway around can be proved similarly and the required equality holds. (cid:3) The following theorem shows how simplicial distance behave under the composition.
Theorem 3.7. If ϕ, ψ : K → L and α, ¯ α : L → M are simplicial maps and α ∼ a ¯ α for some a ∈ Z ≥ , thenSimpD ( α ◦ ϕ, ¯ α ◦ ψ ) ≤ SimpD ( ϕ, ψ ) . Proof.
Let SimpD ( ϕ, ψ ) = k . Then for some large b, ck = SimpD bc ( ϕ, ψ ) . So there exist subcomplexes J , J , . . . , J k covering of Sd b ( K ) such that γ ϕ : J i j −→ Sd b ( K ) ι ֒ −→ K ϕ −→ L IMPLICIAL DISTANCE 7 and γ ψ : J i j −→ Sd b ( K ) ι ֒ −→ K ψ −→ L are c -contiguous for each i .Define α ◦ γ ϕ : J i j −→ Sd b ( K ) ι ֒ −→ K ϕ −→ L α −→ M and ¯ α ◦ γ ψ : J i j −→ Sd b ( K ) ι ֒ −→ K ψ −→ L ¯ α −→ M In the next argument we will show that these maps are max { a, c } -contiguous.Since γ ϕ ∼ c γ ψ , for each i there exists H j : J i → L such that H = γ ϕ , H c = γ ψ and ( H s − , H s ) is 1-contiguous for s = 1 , , . . . , c .Since α ∼ a ¯ α , there exist F j : L → M such that F = α , F a = ¯ α and ( F s − , F s ) is1-contiguous for s = 1 , , . . . , a .Define f F j : L → M by f F j = ( F j , if ≤ j ≤ aF a , if a ≤ j ≤ max { a, c } Define f H j : J i → L by f H j = ( H j , if ≤ j ≤ cH c , if c ≤ j ≤ max { a, c } Define G j = f F j ◦ f H j : J i → L → M so that G = α ◦ γ ϕ and G max { a,c } = ¯ α ◦ γ ψ .Moreover by Remark 2.7, each ( G j − , G j ) is 1-contiguous for j = 1 , , . . . , max { a, c } .This gives the max { a, c } -contiguity between α ◦ γ ϕ and ¯ α ◦ γ ψ . Hence SimpD b max { a,c } ( α ◦ ϕ, ¯ α ◦ ψ ) ≤ k . Therefore SimpD ( α ◦ ϕ, ¯ α ◦ ψ ) ≤ k . (cid:3) Corollary 3.8.
Let ϕ, ψ : K → L be simplicial maps and let α : L → M be asimplicial map with a left strong homotopy type inverse (i.e., β ◦ α ∼ id L for some β : M → L ). Then SimpD ( α ◦ ϕ, α ◦ ψ ) = SimpD ( ϕ, ψ ) .Proof. By Theorem 3.7, we have SimpD ( α ◦ ϕ, α ◦ ψ ) ≤ SimpD ( ϕ, ψ ) . We will showthe other way around. Let SimpD ( α ◦ ϕ, α ◦ ψ ) = k . Then for some large b, c ∈ Z ≥ , k = SimpD bc ( α ◦ ϕ, α ◦ ψ ) . So there exists J , J , . . . , J k covering Sd b ( K ) such that γ αϕ : J i j −→ Sd b ( K ) ι ֒ −→ K ϕ −→ L α −→ M and γ αψ : J i j −→ Sd b ( K ) ι ֒ −→ K ψ −→ L α −→ M are c -contiguous for each i . Thus AYSE BORAT α ◦ ( ϕ ◦ ι ) ◦ j ∼ c α ◦ ( ψ ◦ ι ) ◦ jβ ◦ α ◦ ( ϕ ◦ ι ) ◦ j ∼ c β ◦ α ◦ ( ψ ◦ ι ) ◦ j. On the other hand, since β ◦ α ∼ id L , for some ¯ c ∈ Z ≥ we have β ◦ α ∼ ¯ c id L . Henceid L ◦ ( ϕ ◦ ι ) ◦ j ∼ ¯ c β ◦ α ◦ ( ϕ ◦ ι ) ◦ j ∼ c β ◦ α ◦ ( ψ ◦ ι ) ◦ j ∼ ¯ c id L ◦ ( ψ ◦ ι ) ◦ j Thus id L ◦ ( ϕ ◦ ι ) ◦ j ∼ c + c id L ◦ ( ψ ◦ ι ) ◦ j. ( ϕ ◦ ι ) ◦ j ∼ ( ψ ◦ ι ) ◦ j It follows that SimpD bc ( ϕ, ψ ) ≤ k . Therefore SimpD ( ϕ, ψ ) ≤ k . (cid:3) References [1] A. Borat, Higher dimensional simplicial complexity, to appear in New York Journal ofMathematics.[2] A. Borat, T. Vergili, Higher homotopic distance, submitted. ArXiv: 1907.08384.[3] M. Farber, Topological complexity of motion planning, Discrete and Computational Geome-try 29 (2003), 211-221.[4] D. Fernández-Ternero, E. Minuz, E. Macias-Virgós, J. A. Vilches, Simplicial Lusternik-Schnirelmann category, Publicacions Matematiques, Volume 63, Number 1 (2019), 265-293.[5] J. Gonzalez, Simplicial Complexity: Piecewise Linear Motion Planning in Robotics, NewYork Journal of Mathematics 24 (2018), 279-292.[6] E. Maciás-Virgós, D. Mosquera-Lois, Homotopic distance between maps, Preprint. ArXiv:1810.12591.[7] E. Macias-Virgos, D. Mosquera-Lois, Homotopic distance between functors. Preprint. ArXiv:1902.06322.[8] E.H. Spanier, Algebraic Topology. McGraw-Hill Book Co., New York-Toronto, Ony.-London,1966.
Ayse Borat Bursa Technical University, Faculty of Engineering and Natural Sci-ences, Department of Mathematics, Bursa, Turkey
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