Homotopical decompositions of simplicial and Vietoris Rips complexes
Wojciech Chacholski, Alvin Jin, Martina Scolamiero, Francesca Tombari
aa r X i v : . [ m a t h . A T ] F e b HOMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL ANDVIETORIS RIPS COMPLEXES
WOJCIECH CHACH ´OLSKI, ALVIN JIN, MARTINA SCOLAMIERO,AND FRANCESCA TOMBARI
Abstract.
Motivated by applications in Topological Data Analysis, we con-sider decompositions of a simplicial complex induced by a cover of its vertices.We study how the homotopy type of such decompositions approximates the ho-motopy of the simplicial complex itself. The difference between the simplicialcomplex and such an approximation is quantitatively measured by means ofthe so called obstruction complexes. Our general machinery is then specializedto clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes ofmetric gluings. For the latter we give metric conditions which allow to recoverthe first and zero-th homology of the gluing from the respective homologies ofthe components. Introduction
Homology is an example of an invariant that is both calculable and geometricallyinformative. These two features are key reasons why invariants derived from ho-mology are fundamental in Algebraic Topology in general and in Topological DataAnalysis (TDA) [8] in particular. Calculability is a consequence of the fact thathomology converts homotopy push-outs into Mayer-Vietoris exact sequences. De-composing a space into a homotopy push-out enables us to extract the homologiesof the decomposed space (global information) from the homologies of the spaces inthe push-out (local information).Ability of extracting global information from local is important. What is meantby local information however depends on the input and the description of consideredspaces. For example what is often understood as local information in TDA differsfrom the local information described above (push-out decomposition). In TDA theinput is typically a finite metric space. This information is then converted into spa-cial information and in this article we focus on the so called Vietoris-Rips construc-tion [19] for that purpose. Homologies extracted from this space give rise to invari-ants of the metric space used in TDA such as persistent homologies [7, 12, 14, 16, 17],bar-codes [23], stable ranks [10, 22], or persistent landscapes [6]. This conversionprocess, from metric into spacial information, does not in general transform thegluing of metric spaces [24] into homotopy push-outs and homotopy colimits ofsimplicial complexes. The aim of this paper is to understand how close such datadriven decompositions are to decompositions into homotopy push-outs. Our workwas inspired by [4] and [5], and grew out of realisation that analogous statementshold true for arbitrary simplicial complexes and not just Vietoris-Rips complexes.
Mathematics, KTH, S-10044 Stockholm, Sweden .
To get these general statements we use categorical techniques. This enables us toprove stronger results using arguments that for us are more transparent.The most general input for our investigation is a simplicial complex K anda cover X ∪ Y = K of its set of vertices. In this article we study the map K X ∪ K Y ⊂ K where K X and K Y are subcomplexes of K consisting of all thesesimplices of K which are subsets of X and Y respectively. The goal is to esti-mate the homotopy fibers of this inclusion. We do that in terms of obstructioncomplexes St( σ, X ∩ Y ) := { µ ⊂ X ∩ Y | ≤ | µ | and µ ∪ σ ∈ K } indexed bysimplices σ in K (see Definition 4.1). Our main result, Theorem 7.5, states thatthe homotopy fibers of K X ∪ K Y ⊂ K are in the same cellular class (see Para-graph 2.5, and [9, 13, 15]) as the obstruction complexes St( σ, X ∩ Y ) for all σ in K such that σ ∩ X = ∅ , σ ∩ Y = ∅ , and σ ∩ X ∩ Y = ∅ . For instance (see Corol-lary 7.6.1) if, for all such σ , the obstruction complex St( σ, X ∩ Y ) is contractible,then K X ∪ K Y ⊂ K is a weak equivalence and consequently K decomposes asa homotopy pushout hocolim( K X ← ֓ K X ∩ Y ֒ → K Y ) leading to a Mayer-Vietorisexact sequence. Another instance of our result (see Corollary 7.6.2) states that ifthese obstruction complexes have trivial homology in degrees not exceeding n , thenso do the homotopy fibers of K X ∪ K Y ⊂ K and consequently this map induces anisomorphism on homology in degrees not exceeding n , leading to a partial Mayer-Vietoris exact sequence. Yet another consequence (see Corollary 7.6.3) is that ifthese obstruction complexes have p -torsion homology for a prime p , then, for anyfield F of characteristic different than p , the inclusion K X ∪ K Y ⊂ K induces anisomorphism on homology with coefficients in F leading again to a Mayer-Vietorisexact sequence.In section 9 we specialise our theorem about the cellularity of the homotopy fibersof the inclusion K X ∪ K Y ⊂ K to the case when K is a clique complex and givesome conditions that imply the assumptions of the theorem in this case. Obtainedresults in principle generalize all the statements proven in [4, 5] for Vietoris-Ripscomplexes. The point we would like to make is that these statements are notabout Vietoris-Rips complexes but rather about these complexes being clique. Inparticular triangular inequality of the input metric space is not needed for ourstatements to hold. We then prove Theorem 11.5 for which it is essential thatconsidered complex is the Vietoris-Rips complex of the metric gluing of pseudo-metric spaces for which the triangular inequality is satisfied. This theorem gives 2connectedness of the relevant homotopy fibers and hence can be used to calculate H and H of the gluing in terms of H ’s and H ’s of the components and theintersection. 2. Small categories and simplicial sets
In this section we recall some elements of a convenient language for describing anddiscussing homotopical properties of small categories. The key role here is playedby the nerve construction [20, 21] that transforms small categories into simplicialsets. We refer the reader to [11, 18] for an overview of how to do homotopy theoryon simplicial sets. We consider the standard model structure on the category ofsimplicial sets where weak equivalences are given by the maps inducing bijectionson all the homotopy groups with respect to any choice of a base point.Here is a list of definitions and characterizations of various homotopical notionsfor small categories and some statements regarding these notions.
OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES 3 . Let C be a property of simplicial sets, such as being contractible, n -connected,having p -torsion integral reduced homology, or having trivial reduced homology insome degrees. By definition a small category I satisfies C if and only if its nerve N ( I ) satisfies C .2.2 . Let C be a property of maps of simplicial sets, such as being a weak equivalence,a homology isomorphism, or having n -connected homotopy fibers. By definition,a functor f : I → J between small categories satisfies C if and only if the map ofsimplicial sets N ( f ) : N ( I ) → N ( J ) satisfies C .2.3 . Functors f, g : I → J are homotopic if the maps N ( f ) , N ( g ) : N ( I ) → N ( J )are homotopic. For example, if there is a natural transformation φ : f ⇒ g between f and g , then f and g are homotopic.Assume I has a terminal object t . Then there is a unique natural transformationfrom the identity functor id : I → I to the constant functor t : I → I with value t . The identity functor is therefore homotopic to the constant functor, and conse-quently I is contractible. By a similar argument, a category with an initial objectis also contractible.2.4 . A commutative square of small categories is called a homotopy push-out (pull-back) if after applying the nerve construction the obtained commutative square ofsimplicial sets is a homotopy push-out (pull-back).2.5 . Recall that a collection C of simplicial sets is closed if it contains a nonemptysimplicial set and it is closed under weak equivalencies and homotopy colimitsindexed by arbitrary small contractible categories [9, Corollary 7.7]. Any closedcollection contains all contractible simplicial sets [9, Proposition 4.5]. If a closedcollection contains an empty simplicial set, then it contains all simplicial sets.The following are some examples of collections of simplicial sets that are closed:contractible simplicial sets, n -connected simplicial sets, connected simplicial setshaving p -torsion reduced integral homology, simplicial sets having trivial reducedhomology with some fixed coefficients up to a given degree, and more generally sim-plicial sets which are acyclic with respect to some (possibly not ordinary) homologytheory.Let C be a closed collection of simplicial sets and f : I → J be a functor betweensmall categories. We say that homotopy fibers of f satisfy C if the homotopy fibersof N ( f ) : N ( I ) → N ( J ), over any component in N ( J ), belong to C .2.6 . Let f : I → J be a functor between small categories. For an object j in J , thesymbol j ↑ f denotes the category whose objects are pairs ( i, α : j → f ( i )) consistingof an object i in I and a morphism α : j → f ( i ) in J . The set of morphisms in j ↑ f between ( i, α : j → f ( i )) and ( i ′ , α ′ : j → f ( i ′ )) is by definition the set ofmorphisms β : i → i ′ in I for which the following triangle commutes: jf ( i ) f ( i ′ ) ′ α α ′ f ( β ) The composition in j ↑ f is given by the composition in I .For an object j in J , the symbol f ↓ j denotes the category whose objects arepairs ( i, α : f ( i ) → j ) consisting of an object i in I and a morphism α : f ( i ) → j in WOJCIECH CHACH ´OLSKI, ALVIN JIN, MARTINA SCOLAMIERO, AND FRANCESCA TOMBARI J . The set of morphisms in f ↓ j between ( i, α : f ( i ) → j ) and ( i ′ , α ′ : f ( i ′ ) → j )is by definition the set of morphisms β : i → i ′ in I for which the following trianglecommutes: f ( i ) f ( i ′ ) j f ( β ) α α ′ The composition in f ↓ j is given by the composition in I .2.7. Theorem ([9, Theorem 9.1]) . Let C be a closed collection of simplicial setsand f : I → J be a functor between small categories. (1) If, for every j , f ↓ j satisfies C , then so do the homotopy fibers of f . (2) If, for every j , j ↑ f satisfies C , then so do the homotopy fibers of f . Depending on the choice of a closed collection, Theorem 2.7 leads to:2.8.
Corollary.
Let f : I → J be a functor between small categories. (1) If, for every j , f ↓ j (respectively j ↑ f ) is contractible, then f is a weakequivalence. (2) If, for every j , f ↓ j (respectively j ↑ f ) is n -connected for some n ≥ , thenthe homotopy fibers of f are n -connected. Thus in this case f induces anisomorphism on homotopy groups in degrees , . . . , n and a surjection indegree n + 1 . (3) If, for every j , f ↓ j (respectively j ↑ f ) is connected and has p -torsionreduced integral homology in degrees not exceeding n ( n ≥ ), then thehomotopy fibers of f are connected and have p -torsion reduced integral ho-mology in degrees not exceeding n . Thus in this case, for primes q = p , f induces an isomorphism on H ∗ ( − , Z /q ) for ∗ ≤ n and a surjection on H n +1 ( − , Z /q ) . (4) If, for every j , f ↓ j (respectively j ↑ f ) is acyclic with respect to somehomology theory, then f is this homology isomorphism. Simplicial complexes and small categories . Fix a set U called a universe . A simplicial complex is a collection K offinite nonempty subsets of U that satisfies the following requirement: if σ ⊂ U is in K , then every non-empty subset of σ is also in K .Let X ⊂ U be a subset. The collection {{ x } | x ∈ X } , consisting of singletonsin X , is a simplicial complex denoted also by X , called the discrete simplicialcomplex on X . The collection { σ ⊂ X | ≤ | σ | < ∞} of all finite nonemptysubsets of X is also a simplicial complex denoted by ∆[ X ] and called the simplex on X . A simplicial complex is called a standard simplex if it is of the form ∆[ X ] forsome X ⊂ U . The simplex ∆[ ∅ ] is called the empty simplex or the empty simplicialcomplex.3.2 . Let K be a simplicial complex. An element σ in K is called a simplex of K of dimension | σ | −
1. The set of n -dimensional simplices in K is denoted by K n . An element x ∈ U is called a vertex of K if { x } is a simplex in K . Theassignment x
7→ { x } is a bijection between the set of vertices in K and the set ofits 0-dimensional simplices K . We use this bijection to identify these sets. Thuswe are going to refer to 0-dimensional simplices in K also as vertices. OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES 5 . If { K i } i ∈ I is a family of simplicial complexes, then both the intersection ∩ i ∈ I K i and the union ∪ i ∈ I K i are also simplicial complexes. If K is a simpli-cial complex and X ⊂ U is a subset, then the intersection K ∩ ∆[ X ] is a simplicialcomplex consisting of the elements of K that are subsets of X . This intersection iscalled the restriction of K to X and is denoted by K X .Note that ∆[ X ] ∩ ∆[ Y ] = ∆[ X ∩ Y ]. Thus the intersection of standard simplices(see 3.1) is again a standard simplex, which can possibly be empty.Let L and K be simplicial complexes. If L ⊂ K , then L is called a subcomplex of K . Being a subcomplex is a partial order relation on the collection of all simplicialcomplexes which gives this collection the structure of a lattice. The union is thejoin and the intersection is the meet.The collection ∪ ≤ i ≤ n K i is a subcomplex of K called the n -th skeleton of K and denoted by sk n K .3.4 . A map between two simplicial complexes K and L is by definition a func-tion φ : K → L for which there exists a function f : K → L such that φ ( σ ) = ∪ x ∈ σ f ( { x } ) for all σ in K . In particular φ ( { x } ) = f ( { x } ) for every vertex x in K .Thus f is uniquely determined by φ and we often use the symbol φ to denote f .If K and L are fixed, then φ is determined by f = φ . The inclusion L ⊂ K of asubcomplex is an example of a map.For any simplicial complex K , the inclusions K ⊂ K ⊂ ∆[ K ], between thediscrete simplicial complex K , K , and the simplex ∆[ K ] on K are maps ofsimplicial complexes. The induced functions on the set of vertices for these twoinclusions are given by the identity function id : K → K .3.5 . Classically, the geometrical realization is used to define and study homotopi-cal properties of simplicial complexes. For example, a commutative square of sim-plicial complexes is called a homotopy push-out (pull-back) if after applying therealization, the obtained commutative square of spaces is a homotopy push-out(pull-back). For instance two simplicial complexes K and L fit into the followingcommutative diagram of subcomplex inclusions: K ∩ L KL K ∪ L By applying the realization construction to this square, we obtain a commutativesquare of spaces which is a push-out and hence a homotopy push-out as the mapsinvolved are cofibrations.Since the realization of a simplicial complex K can be built from the realizationof its n -skeleton sk n K by attaching (possibly in many steps) cells of dimensionstrictly bigger than n , we get:3.6. Proposition.
Let n ≥ be a natural number. For every simplicial complex L such that sk n +1 K ⊂ L ⊂ K , the homotopy fibers of the inclusion L ⊂ K are n -connected. In particular, the map L ⊂ K induces an isomorphism on homotopyand integral homology groups in degrees , . . . , n and a surjection in degree n + 1 . There are situations however when another way of extracting homotopical prop-erties of simplicial complexes is more convenient. In the rest of this section, we
WOJCIECH CHACH ´OLSKI, ALVIN JIN, MARTINA SCOLAMIERO, AND FRANCESCA TOMBARI recall how one can retrieve and study such information by first transforming sim-plicial complexes into small categories and then using the nerve construction asexplained in Section 2.3.7 . Let K be a simplicial complex. The simplex category of K , denoted also bythe same symbol K , is by definition the inclusion poset of its simplices. Thus, theobjects of K are the simplices in K and the sets of morphisms are either empty orcontain only one element: | mor K ( σ, τ ) | = ( σ ⊂ τ φ : K → L is a map of simplicial complexes, then the assignment σ φ ( σ ) is afunctor of simplex categories. We denote this functor also by the symbol φ : K → L .Not all functors between K and L are of such a form.The geometrical realization of a simplicial complex is weakly equivalent to therealization of the nerve of this simplicial complex. Thus to describe homotopicalproperties of simplicial complexes we can either use their geometrical realizationsor the nerves of their simplex categories.3.8 . Let K be a simplicial complex and σ be its simplex. Define the star of σ tobe St( σ ) := { µ ∈ K | σ ∪ µ ∈ K } . Note that St( σ ) is a subcomplex of K . The starof any simplex is contractible. More generally:3.9. Proposition.
Let σ be a simplex in K . Then, for any proper subset S ( σ ,the collection L := { µ ∈ K | µ ∩ S = ∅ and σ ∪ µ ∈ K } is a contractible simplicialcomplex (note that if S = ∅ , then L = St( σ ) ).Proof. For all µ in L , the inclusions µ ֒ → µ ∪ ( σ \ S ) ← ֓ σ \ S form naturaltransformations between: • the identity functor id : L → L , µ µ , • the constant functor L → L , µ σ \ S , • and L → L , given by µ µ ∪ ( σ \ S ).The identity functor id : L → L is therefore homotopic to the constant functor andconsequently L is contractible. (cid:3) . Let K be a simplicial complex. It’s simplex τ is called central if K = St( τ ),i.e., if for any simplex σ in K , the set σ ∪ τ is also a simplex in K . For example,if X ⊂ U is non empty, then all simplices in ∆[ X ] (see 3.1) are central. If τ is acentral simplex in K , then so is any subset τ ′ ⊂ τ . According to Proposition 3.9,a simplicial complex that has a central simplex is contractible.3.11 . Let K and L be simplicial complexes. If K ∩ L = ∅ , then we define their join K ∗ L to be the simplicial complex consisting of all subsets of U of the form σ ∪ τ where σ is in K and τ is in L . The join is only defined for disjoint simplicialcomplexes. The set of vertices ( K ∗ L ) is given by the (disjoint) union K ∪ L .Note that K ∗ ∆[ ∅ ] = K . If σ ∈ L is central in L , then it is central in K ∗ L .Thus, for any non-empty subset X ⊂ U \ K , the join K ∗ ∆[ X ] is contractible.Furthermore the join commutes with unions and intersections: if ( K ∪ K ) ∩ L = ∅ ,then ( K ∪ K ) ∗ L = ( K ∗ L ) ∪ ( K ∗ L ) and ( K ∩ K ) ∗ L = ( K ∗ L ) ∩ ( K ∗ L ). Thiscan be used to show that, for any choice of a base-points in K and L , the join K ∗ L has the homotopy type of the suspension of the smash Σ( | K | ∧ | L ]). In particularif K is n -connected and L is m -connected, then K ∗ L is n + m + 1-connected. OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES 7 One outside point
In this section we recall how the homotopy type of a simplical complex changeswhen a vertex is added. We start with defining subcomplexes that play an impor-tant role in describing such changes. These complexes are essentially used through-out the entire paper.4.1.
Definition.
Let K be a simplicial complex and A ⊂ U be a subset. For asimplex σ in K , define the obstruction complex :St( σ, A ) := { µ ⊂ A | < | µ | and µ ∪ σ ∈ K } = K A ∩ St( σ )If µ belongs to St( σ, A ), then so does any of its non empty finite subsets. ThusSt( σ, A ) is a simplicial complex. It is a subcomplex of K A . Note that the complexSt( σ, A ) may be empty. If τ ⊂ σ , then St( σ, A ) ⊂ St( τ, A ).Fix a vertex v in K . Any simplex in K either contains v or it does not. Thismeans K = K K \{ v } ∪ St( v ) and hence we have a homotopy push-out square: K K \{ v } ∩ St( v ) St( v ) K K \{ v } K By definition K K \{ v } ∩ St( v ) = St( v, K \{ v } ). Proposition 3.9 gives contractibilityof St( v ). The simplicial complex K fits therefore into the following homotopy cofibersequence: St( v, K \ { v } ) ֒ → K K \{ v } ֒ → K Here are some basic consequences of this fact:4.2.
Corollary.
Let v be a vertex in a simplical complex K . (1) If St( v, K \ { v } ) is contractible, then K K \{ v } ⊂ K is a weak equivalence. (2) If St( v, K \ { v } ) is n -connected for a natural number n ≥ , then themap K K \{ v } ⊂ K induces an isomorphism on homotopy groups in degrees , . . . , n and a surjection in degree n + 1 . (3) If St( v, K \ { v } ) is connected and has p -torsion reduced integral homologyin degrees not exceeding n ( n ≥ ), then for a prime q not dividing p , K K \{ v } ⊂ K induces an isomorphism on H ∗ ( − , Z /q ) for ∗ ≤ n and asurjection on H n +1 ( − , Z /q ) . (4) If St( v, K \ { v } ) is acyclic with respect to some homology theory, then K K \{ v } ⊂ K is this homology isomorphism. Two outside points
Let us fix two distinct vertices v and v in a simplicial complex K . Note that( K \ { v } ) ∪ ( K \ { v } ) = K . In this section we are going to investigate theinclusion K K \{ v } ∪ K K \{ v } ⊂ K .There are two possibilities. First, { v , v } is not a simplex in K . In this case K K \{ v } ∪ K K \{ v } = K .Assume { v , v } is a simplex in K . Then: K = K K \{ v } ∪ K K \{ v } ∪ St( v , v ) WOJCIECH CHACH ´OLSKI, ALVIN JIN, MARTINA SCOLAMIERO, AND FRANCESCA TOMBARI
Consequently there is a homotopy push-out square (see 3.5): (cid:0) K K \{ v } ∪ K K \{ v } (cid:1) ∩ St( v , v ) St( v , v ) K K \{ v } ∪ K K \{ v } K Since the star complex St( v , v ) is contractible (see Proposition 3.9), K is thereforeweakly equivalent to the homotopy cofiber of the map: (cid:0) K K \{ v } ∪ K K \{ v } (cid:1) ∩ St( v , v ) ֒ → K K \{ v } ∪ K K \{ v } The complex (cid:0) K K \{ v } ∪ K K \{ v } (cid:1) ∩ St( v , v ) fits into the following homotopypush-out square: K K \{ v } ∩ K K \{ v } ∩ St( v , v ) K K \{ v } ∩ St( v , v ) K K \{ v } ∩ St( v , v ) (cid:0) K K \{ v } ∪ K K \{ v } (cid:1) ∩ St( v , v )Let us identify the complexes in this square: • K K \{ v } ∩ St( v , v ) = { µ ∈ K | µ ∩ { v } = ∅ and { v , v } ∪ µ ∈ K } andthus according to Proposition 3.9 this complex is contractible; • by the same argument K K \{ v } ∩ St( v , v ) is also contractible; • K K \{ v } ∩ K K \{ v } ∩ St( v , v ) = K K \{ v ,v } ∩ St ( v , v ) == St ( { v , v } , K \ { v , v } )It follows that (cid:0) K K \{ v } ∪ K K \{ v } (cid:1) ∩ St( v , v ) has the homotopy type of thesuspension of the obstruction complex St := St ( { v , v } , K \ { v , v } ) and hencewe have a homotopy cofiber sequence of the form:ΣSt → K K \{ v } ∪ K K \{ v } ֒ → K n + 1 outside points Homotopy cofiber sequences described in Sections 4 and 5 are particular casesof a more general statement regarding an arbitrary number of outside points. Theaim of this section is to present this generalization.Let us fix a set σ = { v , v , . . . , v n } ⊂ K of n + 1 distinct vertices in asimplcial complex K which may not necessarily be a simplex in K . Note that S v ∈ σ ( K \ { v } ) = K . In this section we are going to investigate the inclusion (cid:0)S v ∈ σ K K \{ v } (cid:1) ⊂ K There are two possibilities. First, σ is not a simplex in K . In this case S v ∈ σ K K \{ v } = K .Assume σ is a simplex in K . Then: K = [ v ∈ σ K K \{ v } ! ∪ St( σ ) OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES 9
Consequently there is a homotopy push-out square (see 3.5): (cid:0)S v ∈ σ K K \{ v } (cid:1) ∩ St( σ ) St( σ ) S v ∈ σ K K \{ v } K Since the star complex St( σ ) is contractible (see Proposition 3.9), K is thereforeweakly equivalent to the homotopy cofiber of the map: [ v ∈ σ K K \{ v } ! ∩ St( σ ) ֒ → [ v ∈ σ K K \{ v } Next we identify the homotopy type of (cid:0)S v ∈ σ K K \{ v } (cid:1) ∩ St( σ ):6.1. Proposition.
Let σ be a simplex of dimension n in a simplicial complex K .Then (cid:0)S v ∈ σ K K \{ v } (cid:1) ∩ St( σ ) has the homotopy type of the n -th suspension of theobstruction complex Σ n St( σ, K \ σ ) .Proof. Consider the inclusion poset of all subsets τ ⊂ σ . For any such subset τ ⊂ σ ,define: F ( τ ) := (T v ∈ τ K K \{ v } = K K \ τ if τ = ∅ S v ∈ σ K K \{ v } if τ = ∅ Note that if τ ′ ⊂ τ ⊂ σ , then F ( τ ) ⊂ F ( τ ′ ). Thus by assigning to the inclusion τ ′ ⊂ τ the map F ( τ ) ⊂ F ( τ ′ ), we obtain a contra-variant functor indexed by theinclusion poset of all subsets of σ . For example in the case σ = { v , v , v } , thiscontra-variant functor describes a commutative cube: K K \{ v ,v ,v } K K \{ v ,v } K K \{ v ,v } K K \{ v } K K \{ v ,v } K K \{ v } K K \{ v } K K \{ v } ∪ K K \{ v } ∪ K K \{ v } For arbitrary n , the functor F describes a commutative cube of dimension n + 1. This cube is both co-cartesian and strongly cartesian ([]). It is thereforealso a homotopy co-cartesian. By intersecting with St( σ ), we obtain a new cube τ F ( τ ) ∩ St( σ ). The properties of being co-cartesian and strongly cartesian arepreserved by taking such interection. Consequently (cid:0)S v ∈ σ K K \{ v } (cid:1) ∩ St( σ ) has thehomotopy type of hocolim ∅6 = τ ⊂ σ (cid:0) K K \ τ ∩ St( σ ) (cid:1) .For any proper subset ∅ 6 = τ ( σ , we have an equality: K K \ τ ∩ St( σ ) = { µ | µ ∩ τ = ∅ and σ ∪ µ ∈ K } We can then use Proposition 3.9 to conclude that K K \ τ ∩ St( σ ) is contractible if ∅ 6 = τ ( σ . Thus all the spaces in the cube τ F ( τ ) ∩ St( σ ), except for the initialand the terminal, are contractible. That implies that the terminal space F ( ∅ ) ∩ St( σ ) = (cid:0)S v ∈ σ K K \{ v } (cid:1) ∩ St( σ ) is homotopy equivalent to the n -th suspension ofthe initial space: Σ n ( F ( σ ) ∩ St( σ )) = Σ n (cid:0) K K \ σ ∩ St( σ ) (cid:1) = Σ n St( σ, K \ σ ). (cid:3) We finish this section with summarising the consequences of the discussion lead-ing to Proposition 6.1 and the proposition itself:6.2.
Corollary.
Let σ ⊂ K be a subset consisting of n + 1 distinct vertices in asimplicial complex K . (1) If σ is not a simplex in K , then S v ∈ σ K K \{ v } = K . (2) Assume σ is a simplex in K . (a) Then there is a homotopy cofiber sequence: Σ n St( σ, K \ σ ) → [ v ∈ σ K K \{ v } ֒ → K (b) If St( σ, K \ σ ) = ∅ , then there is a homotopy cofiber sequence (here S − = ∅ ): S n − → [ v ∈ σ K K \{ v } ֒ → K (c) If St( σ, K \ σ ) = ∅ , then the homotopy fibers of S v ∈ σ K K \{ v } ֒ → K are m ≥ connected if and only if H i (St( σ, K \ σ ) , Z ) = 0 for i ≤ m − n . (d) If St( σ, K \ σ ) = ∅ , then S v ∈ σ K K \{ v } ֒ → K is a weak equivalence ifand only if H i (St( σ, K \ σ ) , Z ) = 0 for all i . Push-out decompositions I.
In this section our starting assumption is:7.1 (
Starting input I). K is a simplicial complex, X ∪ Y = K is a cover of itsset of vertices, and A := X ∩ Y . By restricting K to X and Y , and taking the union of these restrictions weobtain a subcomplex K X ∪ K Y ⊂ K . Since K X ∩ K Y = K A , this subcomplex fitsinto the following homotopy push-out square: K A K X K Y K X ∪ K Y This push-out can be then used to extract various homotopical properties of theunion K X ∪ K Y from the properties of K X , K Y and K A . For example, if K X , K Y and K A belong to a closed collection (see 2.5), then so does K X ∪ K Y . If K A is contractible, then K X ∪ K Y has the homotopy type of the wedge of K X and K Y , and its reduced homology is the sum of the reduced homologies of K X and K Y . More generally, there is a Mayer-Vietoris sequence connecting homologies of K X ∪ K Y with those of K X , K Y and K A .A fundamental question discussed in this article is: under what circumstancesthe inclusion K X ∪ K Y ⊂ K is a weak equivalence, or homology isomorphism, orhas highly connected homotopy fibers etc? Such circumstances would enable usto express various homotopical properties of K in terms of the properties of itsrestrictions K X , K Y and K A . OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES11
Definition.
Under the starting assumption 7.1, define P to be the subposetof K given by: P := { σ ∈ K | σ ⊂ X or σ ⊂ Y or σ ∩ A = ∅} We are going to be more interested in the set of simplices of K that do notbelong to P , which explicitly can be described as: K \ P = { σ ∈ K | σ ∩ X = ∅ and σ ∩ Y = ∅ and σ ∩ A = ∅} The poset P may not be the simplex category of any simplicial complex. Thereare two poset inclusions that we denote by f and g : K X ∪ K Y P K f g
Our first general observation is:7.3.
Proposition.
The functor f : K X ∪ K Y ֒ → P is a weak equivalence.Proof. We are going to show that, for every σ in P , f ↓ σ is contractible.First assume σ ⊂ X or σ ⊂ Y . Then the object ( σ, id : σ → σ ) is terminal in f ↓ σ and consequently this category is contractible.Assume σ ∩ A = ∅ . Then, for any object ( τ, τ ⊂ σ ) in f ↓ σ , the subsets τ , τ ∪ ( σ ∩ A ), and σ ∩ A of σ are simplices that belong to K X ∪ K Y . We can thenform the following commutative diagram in P where the top horizontal arrowsrepresent morphisms in K X ∪ K Y : τ τ ∪ ( σ ∩ A ) σ ∩ Aσ These horizontal morphisms form natural transformations between: • the identity functor id : f ↓ σ → f ↓ σ , ( τ, τ ⊂ σ ) ( τ, τ ⊂ σ ), • the constant functor f ↓ σ → f ↓ σ , ( τ, τ ⊂ σ ) ( σ ∩ A, σ ∩ A ⊂ σ ), • and f ↓ σ → f ↓ σ given by ( τ, τ ⊂ σ ) ( τ ∪ ( σ ∩ A ) , τ ∪ ( σ ∩ A ) ⊂ σ ).The identity functor id : f ↓ σ → f ↓ σ is therefore homotopic to the constantfunctor. This can happen only if f ↓ σ is a contractible category. (cid:3) According to Proposition 7.3, the homotopy fibers of g : P ⊂ K and the inclusion K X ∪ K Y ⊂ K are weakly equivalent. To understand these homotopy fibers, weare going to focus on the categories σ ↑ g and then utilise Corollary 2.8. Thefunctor σ σ ↑ g fits into the following diagram of natural transformations betweenfunctors indexed by K op with small categories as values: σ St( σ, A ) σ ↑ g St( σ ) ψ σ φ σ where: • ψ σ : St( σ, A ) → σ ↑ g assigns to µ in St( σ, A ) the object in σ ↑ g given bythe pair ψ σ ( µ ) := ( µ ∪ σ, σ ⊂ µ ∪ σ ). • φ σ : σ ↑ g → St( σ ) assigns to ( τ, σ ⊂ τ ) the simplex τ in St( σ ).These natural transformations satisfy the following properties: Proposition.
Let σ be a simplex in K . (1) If σ is in P , then σ ↑ g is contractible and φ σ : σ ↑ g → St( σ ) is a weakequivalence. (2) If σ is in K \ P , then ψ σ : St( σ, A ) → σ ↑ g is a weak equivalence.Proof. If σ is in P , then ( σ, id : σ → σ ) is an initial object in σ ↑ g and hence thiscategory is contractible. That proves (1).Assume σ is not in P , which is equivalent to σ ∩ Y = ∅ and σ ∩ X = ∅ and σ ∩ A = ∅ . Let ( τ, σ ⊂ τ ) be an object in σ ↑ g . Define α σ ( τ, σ ⊂ τ ) := τ ∩ A .Since σ ∩ Y = ∅ and σ ∩ X = ∅ , then τ ∩ Y = ∅ and τ ∩ X = ∅ . This togetherwith the fact that τ belongs to P implies α σ ( τ, σ ⊂ τ ) = τ ∩ A = ∅ . Furthermore( τ ∩ A ) ∪ σ ⊂ τ ∈ P ⊂ K . Thus α σ defines a functor α σ : σ ↑ g → St( σ, A ). Note: α σ ψ σ ( µ ) = α σ ( µ ∪ σ, σ ⊂ µ ∪ σ ) = ( µ ∪ σ ) ∩ A Since σ ∩ A = ∅ and µ ⊂ A , we get α σ ψ σ ( µ ) = ( µ ∪ σ ) ∩ A = µ . The composition α σ ψ σ is therefore the identity functor.Note further: ψ σ α σ ( τ, σ ⊂ τ ) = ψ σ ( τ ∩ A ) = (( τ ∩ A ) ∪ σ, σ ⊂ ( τ ∩ A ) ∪ σ )Since σ ⊂ τ , we have a commutative diagram: σ ( τ ∩ A ) ∪ σ τ The bottom horizontal morphisms form a natural transformation between: • the composition ψ σ α σ : σ ↑ g → σ ↑ g and • the identity functor id : σ ↑ g → σ ↑ g .The functor ψ σ : St( σ, A ) → σ ↑ g has therefore a homotopy inverse and hence is aweak equivalence which proves (2). (cid:3) We use Corollary 2.8 and Proposition 7.4 to obtain our main statement describingproperties of the homotopy fibers of the inclusion K X ∪ K Y ⊂ K :7.5. Theorem.
Notation as in 7.1 and Definition 7.2. Let C be a closed collectionof simplicial sets (see 2.5). Assume that, for every σ in K \ P , the obstructioncomplex St( σ, A ) (see 4.1) satisfies C . Then the homotopy fibers of the inclusion K X ∪ K Y ⊂ K also satisfy C . The following are some particular cases of the above theorem specialized todifferent closed collections of simplicial sets.7.6.
Corollary.
Notation as in 7.1 and 7.2. Let n be a natural number. (1) If, for every σ in K \ P (see 7.2), the simplicial complex St( σ, A ) (see 4.1)is contractible, then K X ∪ K Y ⊂ K is a weak equivalence. (2) If, for every σ in K \ P , the simplicial complex St( σ, A ) is n -connected, thenthe homotopy fibers of K X ∪ K Y ⊂ K are n -connected and this map inducesan isomorphism on homotopy groups in degrees , . . . , n and a surjection indegree n + 1 . OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES13 (3)
Let p be a prime number. If, for every σ in K \ P , the simplicial complex St( σ, A ) is connected and has p -torsion reduced integral homology in degreesnot exceeding n , then the homotopy fibers of K X ∪ K Y ⊂ K are connectedand have p -torsion reduced integral homology in degrees not exceeding n .Thus in this case, for prime q = p , K X ∪ K Y ⊂ K induces an isomorphismon H ∗ ( − , Z /q ) for ∗ ≤ n and a surjection on H n +1 ( − , Z /q ) . (4) If, for every σ in K \ P , the simplicial complex St( σ, A ) is acyclic withrespect to some homology theory, then K X ∪ K Y ⊂ K is this homologyisomorphism. We remark that Corollary 7.6.1 is a generalization of [5, Theorem 2] to abstractsimplicial complexes.Requirements for obtaining n -connected fibers can be weakened:7.7. Proposition.
Notation as in 7.1 and 7.2. Let n be a natural number. If St( σ, A ) is n -connected for every σ in (sk n +1 K ) \ P , then the homotopy fibers of K X ∪ K Y ⊂ K are n -connected.Proof. Consider the following poset inclusions: K X ∪ K Y P P ∪ sk n +1 K K f g g g According to Proposition 7.3, f is a weak equivalence. The homotopy fibers of g are n -connected by Proposition 3.6. Thus if the homotopy fibers of g are n -connected, then so are the homotopy fibers of the inclusion K X ∪ K Y ⊂ K . Toshow that the homotopy fibers of g are n -connected it is enough to show that thecategories σ ↑ g are n -connected for every σ in P ∪ sk n +1 K . Proposition 7.4 givesthat σ ↑ g is contractible if σ is in P , and is weakly equivalent to St( σ, A ) if σ is insk n +1 K \ P . By the assumption St( σ, A ) are therefore n -connected. (cid:3) Push-out decompositions II.
Theorem 7.5 states that the homotopy fibers of the inclusion K X ∪ K Y ⊂ K belong to the smallest closed collection containing all the complexes St( σ, A ) for σ in K \ P . Recall that if a closed collection contains an empty simplicial set, thenit contains all simplicial sets, in which case Theorem 7.5 has no content. ThusSt( σ, A ) being non empty, for all σ in K \ P , is an absolute minimum requirementfor Theorem 7.5 to have any content. In most of our statements that follow, theassumptions we make have much stronger global non emptiness consequences of theform: \ σ ∈ K \ P St( σ, A ) = ∅ \ σ ∈ K n +1 \ P St( σ, A ) = ∅ \ σ ∈ (sk n +1 K ) \ P St( σ, A ) = ∅ Here is a consequence of having one of these intersections non-empty:8.1.
Proposition.
Notation as in 7.1 and 7.2. Assume: \ σ ∈ K \ P St( σ, A ) = ∅ Then the homotopy fibers of K X ∪ K Y ⊂ K are connected. Proof.
Let v be a vertex in T σ ∈ K \ P St( σ, A ). Observe that sk ( K ) is a disjointunion of K \ P and sk ( K ) ∩ P . This can fail for sk n ( K ) if n >
1. For every τ insk ( K ), define: φ ( τ ) := ( τ ∪ v if τ ∈ K \ Pτ if τ ∈ sk ( K ) ∩ P If τ ( τ ′ in sk ( K ), then τ is in P and hence τ = φ ( τ ) ⊂ φ ( τ ′ ). In this way we obtaina functor φ : sk ( K ) → P . The inclusion τ ⊂ φ ( τ ), is a natural transformationbetween the skeleton inclusion sk ( K ) ⊂ K and the composition:sk ( K ) P K φ g
Thus these two functors from sk ( K ) to K are homotopic. The statement of theproposition is then a consequence of Proposition 3.6. (cid:3) Proposition 8.1 does not generalise to n >
0. Non-emptiness of the intersection T σ ∈ (sk n +1 K ) \ P St( σ, A ) does not imply that the homotopy fibers of K X ∪ K Y ⊂ K are n -connected. For an easy example see 10.4. To guarantee n -connectedness ofthese homotopy fibers we need additional restrictions. For example in the followingcorollary the assumptions imply that St( σ, A ) does not depend on σ in (sk n +1 K ) \ P :8.2. Corollary.
Notation as in 7.1 and 7.2. Let n be a natural number. Assumethat one of the following conditions is satisfied: (1) There is an n -connected simplicial complex L such that, for every simplex σ in (sk n +1 K ) \ P , St( σ, A ) = L . (2) The complex K A is n -connected and, for every simplex σ in (sk n +1 K ) \ P , St( σ, A ) = K A . (3) The set A is non empty. Furthermore, for every simplex σ in (sk n +1 K ) \ P and every finite subset µ in A , the union σ ∪ µ is a simplex in K . (4) A = { v } and, for every simplex σ in (sk n +1 K ) \ P , the union σ ∪ { v } isalso a simplex in K .Then the homotopy fibers of K X ∪ K Y ⊂ K are n -connected.Proof. The corollary under the assumption (1) is a direct consequence of Propo-sition 7.7. The assumption (2) is a particular case of (1) with L = K A . Theassumption (3) is a particular case of (1) with L = ∆[ A ]. Finally, the assumption(4) is a particular case of (3). (cid:3) Here is another example of a statement whose assumption, referred to as “oneentry point”, has a global nonemptiness consequence:8.3.
Corollary.
Notation as in 7.1 and 7.2. Let n be a natural number. Assumethere is an element v in A with the following property. For every simplex τ in K such that τ ∩ ( X \ A ) = ∅ , τ ∩ ( Y \ A ) = ∅ , and | τ ∩ ( K \ A ) | ≤ n + 2 , theunion τ ∪ { v } is also a simplex in K . Then, for every simplex σ in (sk n +1 K ) \ P ,the element v is a central vertex (see 3.10) in St( σ, A ) . Furthermore the homotopyfibers of K X ∪ K Y ⊂ K are n -connected.Proof. Let σ be a simplex in (sk n +1 K ) \ P . If µ belongs to St( σ, A ) then, byapplying the assumption of the corollary to τ = σ ∪ µ , we obtain that σ ∪ µ ∪ { v } is a simplex in K and hence µ ∪ { v } is a simplex in St( σ, A ). This means that v iscentral in St( σ, A ) (see 3.10). Consequently St( σ, A ) is contractible (see 3.10) andthe corollary follows from Proposition 7.7. (cid:3) OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES15 Clique complexes
Recall that a simplicial complex K is called clique if it satisfies the followingcondition: a set σ of size at least 2 is a simplex in K if and only if all the twoelement subsets of σ are simplices in K . Thus a clique complex is determined byits sets of vertices and edges.If K is clique, then the complexes St( σ, A ) satisfy the following properties:9.1. Proposition.
Notation as in 7.1. Assume K is clique. Then: (1) For all σ in K , St( σ, A ) is clique. (2) If τ and σ are simplices in K such that τ ∪ σ is also a simplex in K , then St( τ ∪ σ, A ) = St( τ, A ) ∩ St( σ, A ) . (3) If σ is a simplex in K and σ = τ ∪ · · · ∪ τ n , then St( σ, A ) = T ni =1 St( τ i , A ) . (4) For every simplex σ in K , St( σ, A ) = T x ∈ σ St( { x } , A ) .Proof. Let µ be a subset of A such that, for every two element subset τ of µ , theset τ ∪ σ is a simplex in K , i.e., τ is in St( σ, A ). Then, since K is clique, µ ∪ σ is also a simplex in K . Consequently µ belongs to St( σ, A ) and hence St( σ, A ) isclique. That proves (1).To prove (2), first note that the inclusion St( τ ∪ σ, A ) ⊂ St( τ, A ) ∩ St( σ, A ) holdseven without the clique assumption. Let µ belong to both St( τ, A ) and St( σ, A ).This means that µ ∪ τ and µ ∪ σ are simplices in K . Since every 2 element subset of µ ∪ τ ∪ σ is a subset of either µ ∪ τ or µ ∪ σ or τ ∪ σ , by the assumption it is an edge in K . By the clique assumption, µ ∪ τ ∪ σ is then also a simplex in K and consequently µ is in St( τ ∪ σ, A ). This shows the other inclusion St( τ ∪ σ, A ) ⊃ St( τ, A ) ∩ St( σ, A )proving (2).Statements (3) and (4) follow from (2). (cid:3)
Recall that an intersection of standard simplices is again a standard simplex(see 3.3). This observation together with Propositions 7.7 and 9.1 gives:9.2.
Corollary.
Notation as in 7.1 and 7.2. Assume K is clique and, for everyedge τ in K \ P , the complex St( τ, A ) is a standard simplex. If, for all simplices σ in (sk n +1 K ) \ P , the complex St( σ, A ) is non-empty, then the homotopy fibers ofthe inclusion K X ∪ K Y ֒ → K are n -connected. Since clique complexes are determined by their edges, one can wonder if, for suchcomplexes, the conclusions of Corollaries 8.2 and 8.3 would still hold true if theirassumptions are verified only for low dimensional simplices. Here is an analogue ofCorollary 8.2 for clique complexes.9.3.
Proposition.
Notation as in 7.1 and 7.2. Let n be a natural number. Assume K is clique and that one of the following conditions is satisfied: (1) There is an n -connected simplicial complex L such that, for every edge τ in K \ P , St( τ, A ) = L . (2) The complex K A is n -connected and, for every edge τ in K \ P and everyelement v in A , the set τ ∪ { v } is a simplex in K .Then the homotopy fibers of the inclusion K X ∪ K Y ֒ → K are n -connected.Proof. The assumption (1) together with Proposition 9.1.(4) implies the assumption(1) of Corollary 8.2, proving the proposition in this case.Let τ be an edge in K \ P and µ be a simplex in K A . Assume (2). Thisassumption implies that any two element subset of τ ∪ µ is a simplex in K . Since K is clique, the set τ ∪ µ is a simplex in K and consequently µ is a simplex inSt( τ, A ). Thus for any τ in K \ P , there is an inclusion K A ⊂ St( τ, A ), and hence K A = St( τ, A ) for any such τ . The assumption (2) implies therefore the assumption(1) with L = K A . (cid:3) Corollary.
Notation as in 7.1 and 7.2. Assume K is clique and that one ofthe following conditions is satisfied: (1) The set A is non empty. Furthermore, for every edge τ in K \ P and everysubset µ in A such that | µ | ≤ , the union τ ∪ µ is a simplex in K . (2) A = { v } and, for every edge τ in K \ P , the set τ ∪ { v } is also a simplexin K .Then the inclusion K X ∪ K Y ֒ → K is a weak equivalence.Proof. Assume (1). If K \ P is empty, then so is K \ P , and hence P = K . Inthis case the corollary follows from Proposition 7.3. Assume K \ P is non-empty.Since any two element subset of A is a simplex in K and K is clique, then allfinite non-empty subsets of A belong to K and hence K A = ∆[ A ]. In this case theassumption (1) is a particular case of the condition (2) in Proposition 9.3 for all n as ∆[ A ] is contractible.Finally note that the assumption (2) is a particular case of (1). (cid:3) The following is an analogue of Corollary 8.3 which is also referred to as “oneentry point”.9.5.
Proposition.
Notation as in 7.1 and 7.2. Assume K is clique and that oneof the following conditions is satisfied: (1) There is a vertex v in T τ ∈ K \ P St( τ, A ) such that, for every edge τ in K \ P and every vertex w in St( τ, A ) , { v, w } is a simplex in K . (2) There is a vertex v in T τ ∈ K \ P St( τ, A ) such that, for every edge τ in K \ P , v is a central vertex of St( τ, A ) (see 3.10). (3) There is an element v in A with the following property. For every simplex τ in K such that | τ ∩ ( X \ A ) | = 1 , | τ ∩ ( Y \ A ) | = 1 , and | τ ∩ A | ≤ , theunion τ ∪ { v } is also a simplex in K .Then the inclusion K X ∪ K Y ֒ → K is a weak equivalence.Proof. Assume (1). Let σ be a simplex in K \ P . Choose a cover σ = τ ∪ · · · ∪ τ n where τ i is an edge in K \ P for all i . Then according to Proposition 9.1, St( σ, A ) = T ni =1 St( τ i , A ). Let w be a vertex in St( σ, A ). Then it is also a vertex in St( τ i , A )for all i . By the assumption { v, w } is then a simplex in K . Thus all the 2 elementsubsets of σ ∪{ v, w } are simplices in K and hence { v, w } is a simplex in F ( σ, A ). Asthis happens for all vertices w in St( σ, A ), since St( σ, A ) is clique, for every simplex µ in St( σ, A ), the set µ ∪ { v } is also a simplex in St( σ, A ). The vertex v is thereforecentral in St( σ, A ) and consequently St( σ, A ) is contractible. The proposition underassumption (1) follows then from Corollary 7.6.(1).Condition (2) is a particular case of (1).Assume (3). Let τ be an edge in K \ P . Condition (3) applied to the simplex τ gives that τ ∪ { v } is a simplex in K and hence v is a vertex in St( τ, A ). Let w be a vertex in St( τ, A ). Condition (3) applied to the simplex τ ∪ { w } gives that { v, w } ⊂ τ ∪ { v, w } are simplces in K . We can conclude (3) implies (1). (cid:3) OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES17
We finish this section with a statement referred to as “two entry points”. Thishas been inspired by [5, Theorem 3], in which the gluing of two metric graphs alonga path is considered. While in that case the two entry points are the endpoints ofthe path the graphs are glued along, in our framework they have to satisfy one ofthe listed properties. In both cases however these couple of points determine theweak equivalence stated.9.6.
Proposition.
Notation as in 7.1 and 7.2. Assume K is clique and there aretwo elements a X and a Y in A with the following properties: • For every edge τ in K such that | τ ∩ A | = 1 and | τ ∩ ( X \ A ) | = 1 , the set τ ∪ { a X } is a simplex in K . • For every edge τ in K such that | τ ∩ A | = 1 and | τ ∩ ( Y \ A ) | = 1 , the set τ ∪ { a Y } is a simplex in K . • For every edge τ in K \ P , the set τ ∪ { a X , a Y } is a simplex in K .Then, for every σ in K \ P , the set { a X , a Y } is a central simplex (see 3.10)in St( σ, A ) , and the inclusion K X ∪ K Y ⊂ K is a weak equivalence.Proof. Let σ be a simplex in K \ P . Any vertex v in σ is a vertex of an edge τ ⊂ σ that belongs to K \ P . According to the assumption, the sets { v, a X , a Y } ⊂ τ ∪ { a X , a Y } are simplices in K . This, together with the clique assumption on K , imply σ ∪ { a X , a Y } is a simplex in K . Consequently { a X , a Y } is a simplex inSt( σ, A ).Let µ be a simplex in St( σ, A ). To prove the proposition, we need to show theset µ ∪ { a A , a Y } is a simplex in F ( σ, A ) or equivalently σ ∪ µ ∪ { a X , a Y } is a simplexin K . Let x be an arbitrary element in σ ∩ X , y an arbitrary element in σ ∩ Y , and v an arbitrary element in µ . The sets { x, v } , { y, v } , and { x, y } are simplices in K .Thus according to the assumptions so are { x, v, a X } , { y, v, a Y } , and { x, y, a X , a Y } .Consequently the two element sets { x, a X } , { v, a X } , { y, a X } , { x, a Y } , { v, a Y } , { y, a Y } , { a X , a Y } , { x, y } are simplices in K . Since all the two element subsets of σ ∪ µ ∪ { a X , a Y } are of such a form and K is clique, σ ∪ µ ∪ { a X , a Y } is a simplexin K . (cid:3) Vietoris-Rips complexes for distances
Let Z be a subset of the universe U (see 3.1). A function d : Z × Z → [0 , ∞ ] iscalled a distance if it is symmetric d ( x, y ) = d ( y, x ) and reflexive d ( x, x ) = 0 forall x and y in Z . A pair ( Z, d ) is called a distance space. A distance space (
Z, d ) issometimes denoted simply by Z , if d is understood from the context, or by d , if Z is understood from the context.Let ( Z, d ) be a distance space. The diameter of a non empty and finite subset σ ⊂ Z is by definition diam( σ ) := max { d ( x, y ) | x, y ∈ σ } .A subset X ⊂ Z together with the distance function given by the restriction of d to X is called a subspace of ( Z, d ).Let (
Z, d ) be a distance space and r be in [0 , ∞ ). By definition, the Vietoris-Rips complex VR r ( Z ) consists of these non-empty finite subsets σ ⊂ Z for whichdiam( σ ) ≤ r (explicitly: d ( x, y ) ≤ r for all x and y in σ ). Vietoris-Rips complexesare examples of clique complexes (see Section 9).Let X be a subspace of ( Z, d ). Then the Vietoris-Rips complex VR r ( X ) coincideswith the restriction VR r ( Z ) X (see 3.3).Our starting assumption in this section is: Starting input II). ( Z, d ) is a distance space, X ∪ Y = Z is a cover of Z ,and A := X ∩ Y . In the rest of this section we are going to reformulate in terms of the distance d on Z some of the statements given in the previous sections regarding the homotopyproperties of the inclusion VR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) for various r in [0 , ∞ ).Here is a direct restatement of Proposition 9.3:10.2. Proposition.
Notation as in 10.1. Let r be an element in [0 , ∞ ) and n be anatural number. Assume that one of the following conditions is satisfied: (1) There is a subset L ⊂ A such that VR r ( L ) is n -connected and, for every x in X \ A and every y in Y \ A with d ( x, y ) ≤ r , there is an equality { v ∈ A | d ( x, v ) ≤ r and d ( y, v ) ≤ r } = L , in particular the set on the leftdoes not depend on x and y . (2) The complex VR r ( A ) is n -connected and, for all x in X \ A , y in Y \ A ,and v in A , if d ( x, y ) ≤ r , then both d ( x, v ) ≤ r and d ( v, y ) ≤ r .Then the homotopy fibers of the inclusion VR r ( X ) ∪ VR r ( Y ) ⊂ VR r ( Z ) are n -connected. The assumption (2) of Proposition 10.2 can be restated as: (connectivity condi-tion) VR r ( A ) is n -connected, and (intersection condition) if VR r ( Z ) \ P (see 7.2)is non-empty, then A = \ σ ∈ VR r ( Z ) \ P St( σ, A ) What if the intersection above does not contain all the points of A (the intersectioncondition is not satisfied)? For example consider Z = { x, a , a , a , a , y } with thedistance function depicted by the following diagram, where the dotted lines indicatedistance 2 and the continuous lines indicate distance 1: a a x ya a Let X = { x, a , a , a , a } and Y = { a , a , a , a , y } . Choose r = 1. In this caseVR ( Z ) \ P consists of only one edge { x, y } and St( { x, y } , A ) = ∆[ { a , a } ] ∪ ∆[ { a , a } ]. Thus the condition (2) of Proposition 10.2 is not satisfied. However,since the complex St( { x, y } , A ) is contractible, according to Corollary 7.6.(1), theinclusion VR ( X ) ∪ VR ( Y ) ⊂ VR ( Z ) is a weak equivalence.The assumption (1) of Proposition 10.2 can be restated as: (connectivity con-dition) for every τ in VR r ( Z ) \ P , the complex St( τ, A ) is n -connected, and (in-dependence condition) for all pairs of edges τ and τ in VR r ( Z ) \ P , there is anequality St( τ , A ) = St( τ , A ). What if the independence condition is not satisfied?For example consider a distance space Z = { x , x , a , a , a , a , y } with the dis-tance function depicted by the following diagram, where the dotted lines indicate OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES19 distance 2 and the continuous lines indicate distance 1: x a a yx a a Let X = { x , x , a , a , a , a } and Y = { a , a , a , a , y } . Choose r = 1. In thiscase VR ( Z ) \ P consists of two edges { x , y } and { x , y } . Note that St( { x , y } , A ) =∆[ { a , a } ] ∪ ∆[ { a , a } ] and St( { x , y } , A ) = ∆[ { a , a } ]. Thus the independencecondition does not hold in this case. However, since the obstruction complexesSt( { x , y } , A ), St( { x , y } , A ) and St( { x , x , y } , A ) = St( { x , y } , A ) ∩ St( { x , y } , A ) =∆[ { a , a } ] are contractible, the inclusion VR ( X ) ∪ VR ( Y ) ⊂ VR ( Z ) is a weakequivalence by Corollary 7.6.Consider a relaxation of the intersection condition in the assumption (2) ofProposition 10.2.10.3 ( Assumption I).
Notation as in 10.1. Let r be an element in [0 , ∞ ) . Thereexists an element v in A satisfying the following property. For all x in X \ A and y in Y \ A , if d ( x, y ) ≤ r , then d ( x, v ) ≤ r and d ( y, v ) ≤ r . . The assumption 10.3 is equivalent to non-emptiness of the following intersec-tion, where n ≥ \ σ ∈ (sk n +1 VR r ( Z )) \ P St( σ, A ) = \ σ ∈ VR r ( Z ) \ P St( σ, A ) = ∅ According to Proposition 8.1, Assumption 10.3 implies connectedness of the homo-topy fibers of VR r ( X ) ∪ VR r ( Y ) ⊂ VR r ( Z ). This assumption however does not im-ply n -connectedness of these homotopy fibers. For example consider Z = { x, a, b, y } with the distance function depicted by the following diagram where the dotted lineindicates distance 2 and the continuous lines indicate distance 1: ax yb Let X = { x, a, b } and Y = { a, b, y } . Then VR ( Z ) is contractible but VR ( X ) ∪ VR ( Y ) has the homotopy type of a circle. Thus in this case the homotopy fiber ofthe inclusion VR r ( X ) ∪ VR r ( Y ) ⊂ VR r ( Z ) is not 1-connected. Note further thatthe complex St( { x, y } , A ) consists of two vertices a and b with no edges.To assure VR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) is a weak equivalence assumption 10.3is not enough and we need additional requirements. For example the following isan analogue of Corollary 9.4. Proposition.
Notation as in 10.1. Assume 10.3. Let v be an element in A given by this assumption. In addition assume that one of the following conditionsis satisfied: (1) For every x in X \ A and y in Y \ A such that d ( x, y ) ≤ r , if w in A satisfies d ( w, x ) ≤ r and d ( w, y ) ≤ r , then d ( v, w ) ≤ r . (2) diam( A ) ≤ r .(3) A = { v } . Then the inclusion VR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) is a weak equivalence.Proof. If Assumption 10.3 and the condition 1 hold, then so does the assumption1 of Proposition 9.5. Furthermore the condition 3 implies 2 and the condition 2implies 1. Thus this proposition is a consequence of Proposition 7.1. (cid:3)
The intersection condition of the assumption (2) in Proposition 10.2 requires achoice of a parameter r . The following is its universal version where no parameteris required:10.6 ( Assumption II).
Notation as in 10.1. The set A is non empty and for every x in X \ A , y in Y \ A , and v in A , the following inequalities hold d ( x, y ) ≥ d ( x, v ) and d ( x, y ) ≥ d ( y, v ) . Assumption 10.6 has an intuitive interpretation in terms of angles when Z is asubspace of the Euclidean space. In such a setting this condition means that everytriangle xvy with vertices x in X \ A , y in Y \ A and v in A , the angle at v mustbe at least 60 ◦ . We therefore refer to this assumption as the 60 ◦ angle condition.10.7. Proposition.
Notation as in 10.1. Assume 10.6. Assume in addition that,for every x in X \ A and y in Y \ A , the following inequality holds d ( x, y ) ≥ diam( A ) .Then VR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) is a weak equivalence for all r in [0 , ∞ ) .Proof. We already know that the proposition holds if r ≥ diam( A ). Assume r < diam( A ). We claim that in this case VR r ( Z ) = P (see 7.2). If not, there are x in X \ A and y in Y \ A such that d ( x, y ) ≤ r . The assumption would then lead to thefollowing contradictory inequalities r ≥ d ( x, y ) ≥ diam( A ) > r . Thus in this caseVR r ( Z ) = P and the proposition follows from Proposition 7.3. (cid:3) Metric gluings
A distance d on Z is called a pseudometric if it satisfies the triangle inequality: d ( x, z ) ≤ d ( x, y ) + d ( y, z ) for all x, y, z in Z .11.1 . Notation as in 10.1. Assume that the distance d on Z is a pseudometric. Let x be in X \ A and y be in Y \ A . For all a in A , by the triangular inequality, d ( x, y ) ≤ d ( x, a ) + d ( a, y ), and hence: d ( x, y ) ≤ inf { d ( x, a ) + d ( a, y ) | a ∈ A } The pseudometric space (
Z, d ) is called metric gluing if the above inequality isan equality for all x in X \ A and y in Y \ A .If A is finite, then the pseudometric ( Z, d ) is a metric gluing if and only if, forevery x in X \ A and y in Y \ A , there is a in A such that d ( x, y ) = d ( x, a ) + d ( a, y ).If d X is a pseudometric on X and d Y is a pseudometric on Y such that d X ( a, b ) = d Y ( a, b ) for all a and b in A , then the following function defines a pseudometric on OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES21 Z which is a metric gluing: d Z ( z, z ′ ) = d X ( z, z ′ ) if z, z ′ ∈ Xd Y ( z, z ′ ) if z, z ′ ∈ Y inf { d ( z, a ) + d ( z ′ , a ) | a ∈ A } if z ∈ X \ A and z ′ ∈ Y \ A If (
Z, d ) is a metric gluing and A is finite, then, for any edge σ = { x, y } inVR r ( Z ) \ P , there is a in A such that r ≥ d ( x, y ) = d ( x, a ) + d ( a, y ). Thus in thiscase the obstruction complex St( τ, A ) is non-empty as it contains the vertex a . Toassure contractibility of St( τ, A ) we need additional assumptions, for example:11.2 ( Simplex assumption).
Notation as in 10.1 and 7.2. Let r be in [0 , ∞ ). Forany vertex v in an edge σ in VR r ( Z ) \ P , if a and b are elements in A such that d ( a, v ) ≤ r and d ( v, b ) ≤ r , then d ( a, b ) ≤ r .The simplex assumption can be reformulated as follows: for any vertex v in asimplex σ in VR r ( Z ) \ P , the complex St( v, A ) is a standard simplex (see 3.1).Since the intersection of standard simplices is again a standard simplex, underAssumption 11.2, an obstruction complex St( σ, A ), for an arbitrary simplex σ inVR r ( Z ) \ P , is contractible if and only if it is non empty. This, together with thediscussion at the end of 11.1 and Corollary 9.2 gives:11.3. Proposition.
Notation as in 10.1 and 7.2. Let r be in [0 , ∞ ) . Assume A is finite and ( Z, d ) is a metric gluing that satisfies the simplex assumption 11.2.Then, for any edge σ in VR r ( Z ) \ P , the obstruction complex St( σ, A ) is contractible.The homotopy fibers of VR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) are connected and this mapinduces an isomorphism on π and a surjection on π . The assumptions of Proposition 11.3 are not enough to guarantee the non-emptiness of the obstruction complexes for simplices in VR r ( Z ) \ P of dimension2 and higher. For example consider Z = { x , x , a , a , y } with the distance func-tion depicted by the following diagram, where the dotted lines indicate distance 4,the dashed lines indicate distance 3, the squiggly lines indicate distance 2 and thecontinuous lines indicate distance 1: x a yx a Let r = 3, X = { x , x , a , a } , Y = { y, a , a } , and A = { a , a } . Then Z is a met-ric gluing. Note that St( y, A ) = ∆[ { a , a } ], St( x , A ) = St( { x , y } , A ) = ∆[ { a } ],St( x , A ) = St( { x , y } , A ) = ∆[ { a } ], and St( { x , x , y } , A ) is empty. Furthermore A and Y have diameter not exceeding 3. Consequently VR ( Y ) and VR ( A ) arecontractible. The complex VR ( X ) has the homotopy type of the circle S and sodoes VR ( X ) ∪ VR ( Y ). The entire complex VR ( Z ) is however contractible. Theinclusion VR ( X ) ∪ VR ( Y ) ⊂ VR ( Z ) induces therefore a surjection on π but not an isomorphism. This example should be compared with Proposition 10.5 underthe condition 2.To assure isomorphism on π , the simplex assumption 11.2 should be strength-ened.11.4 ( Strong simplex assumption).
Notation as in 10.1 and 7.2. Let r be in[0 , ∞ ). For any vertex v in an edge σ in VR r ( Z ) \ P , if a and b are elements in A such that d ( a, v ) ≤ r and d ( v, b ) ≤ r , then 2 d ( a, b ) ≤ d ( a, v ) + d ( v, b ).Note that the strong simplex assumption 11.4 implies the simplex assump-tion 11.2.11.5. Theorem.
Notation as in 10.1 and 7.2. Let r be in [0 , ∞ ) . Assume A isfinite and ( Z, d ) is a metric gluing that satisfies the strong simplex assumption 11.4.Then, for any simplex σ in VR r ( Z ) \ P such that either | σ ∩ X | = 1 or | σ ∩ Y | =1 , the obstruction complex St( σ, A ) is contractible. The homotopy fibers of theinclusion VR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) are simply connected and this map inducesan isomorphism on π and π and a surjection on π .Proof. Since the strong simplex assumption 11.4 is satisfied, then so is the simplexassumption 11.2 and consequently any obstruction complex St( σ, A ) is a simplex.Thus St( σ, A ) is contractible if and only if it is non empty.We are going to show by induction on the dimension of a simplex a more generalstatement:
Under the assumption of Theorem 11.5, for every simplex σ in VR r ( Z ) \ P forwhich σ ∩ X = { x , . . . , x n } and σ ∩ Y = { y } , if ( a , . . . , a n ) is a sequence in A such that d ( x i , y ) = d ( x i , a i ) + d ( a i , y ) for every i , then there is l for which a l is in St( σ, A ) ( d ( x i , a l ) ≤ r for all i ). If σ = { x, y } is such an edge, then the statement is clear.Let n > n . Let σ be in VR r ( Z ) \ P be such that σ ∩ X = { x , . . . , x n } and σ ∩ Y = { y } . Choose a sequence ( a , . . . , a n ) in A such that d ( x i , y ) = d ( x i , a i )+ d ( y, a i ) for every i .By the inductive assumption, for every j = 1 , . . . , n , the statement is true for τ j = σ j \ { x j } and the sequence ( a , . . . , b a j , . . . , a n ) obtained from ( a , . . . , a n ) byremoving its j -th element. Thus for every j = 1 , . . . , n , there is a s ( j ) such that s ( j ) = j and d ( x i , a s ( j ) ) ≤ r for all i = j . If, for some j , d ( x j , a s ( j ) ) ≤ r , then a s ( j ) would be a vertex in St( σ, A ), proving the statement. Assume d ( x j , a s ( j ) ) > r forall j . If j = j ′ , then d ( x j , a s ( j ′ ) ) ≤ r and d ( x j , a s ( j ) ) > r , and hence a s ( j ) = a s ( j ′ ) .It follows that s is a permutation of the set { , . . . , n } . This together with thestrong simplex assumption leads to a contradictory inequality: nr < n X i =1 d ( x i , a s ( i ) ) ≤ n X i =1 (cid:0) d ( x i , a i ) + d ( a i , a s ( i ) ) (cid:1) ≤≤ n X i =1 (cid:18) d ( x i , a i ) + 12 d ( a i , y ) + 12 d ( y, a s ( i ) ) (cid:19) ≤≤ n X i =1 ( d ( x i , a i ) + d ( y, a i )) ≤ nr OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES23
Note that, for any simplex σ in sk VR( Z ) \ P , either | σ ∩ X | = 1 or | σ ∩ Y | = 1.Thus, for any such simplex, the obstruction complex St( σ, A ) is contractible. Wecan then use Proposition 7.7 to conclude that the homotopy fibers of the inclusionVR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) are simply connected. (cid:3) The conclusion of Theorem 11.5 is sharp. Its assumptions are not enough toassure that the homotopy fibers of the map VR r ( X ) ∪ VR r ( Y ) ֒ → VR r ( Z ) are 2connected. We finish this section with an example illustrating this fact.11.6 . Let Z = { x , x , a , a , a , a , y , y } , X = { x , x , a , a , a , a } and Y = { y , y , a , a , a , a } . Consider the distance function d on Z described bythe following table: x a a a a y y x x a a a a y d satisfies the triangular inequality and hence ( Z, d ) is a metric space.Furthermore d ( x i , y j ) = d ( x i , a ij )+ d ( a ij , y j ) for any i and j . Thus ( Z, d ) is a metricgluing of X and Y . The metric space ( Z, d ) can be represented by the followingdiagram, where the continuous lines or no line indicate distance 8 or smaller andthe dotted lines indicate distance 9: a y a x a y x a By direct calculation one checks that, for r = 8 and Z = X ∪ Y , the metricspace ( Z, d ) satisfies the strong simplex assumption 11.4. However, the com-plex VR ( X ) ∪ VR ( Y ) is contractible and VR ( Z ) is weakly equivaent to S (3-dimensional sphere).The homotopy fiber of VR ( X ) ∪ VR ( Y ) ⊂ VR ( Z ) istherefore weakly equivalent to the loops space Ω S and hence is not 2 connected.Here are steps that one might use to see that VR ( Z ) is weakly equivalent to S . Consider the simplex { x , x } in VR ( Z ). According to Corollary 6.2, there isa homotopy cofiber sequence:ΣSt ( { x , x } , VR ( Z \ { x , x } ) → VR ( Z \ { x } ) ∪ VR ( Z \ { x } ) → VR ( Z )The complexes in this sequence have the following homotopy types: • St ( { x , x } , VR ( Z \ { x , x } ) is weakly equivalent to the circle S ; • VR ( Z \ { x } ), VR ( Z \ { x } ), VR ( Z \ { x .x } ) are contractible; • the above implies that VR ( Z \ { x } ) ∪ VR ( Z \ { x } ) is also contractible; • we can then use the cofiber sequence above to conclude VR ( Z ) is weaklyequivalent to Σ S ≃ S as claimed.12. Vietoris-Rips of 9 points on a circle
In [1, 2, 3] a lot of techniques were introduced aiming at describing homotopytypes of certain Vietoris-Rips complexes, particularly for metric graphs built frompoints on a circle. In this section we showcase how our techniques can be used todescribe the homotopy type of one of such examples. Consider a metric space givenby 9 points, Z = { z i } i with the following distances between them: z z z z z z z z z z z z z z z z z z z z z z z z z We are going to illustrate how to use our techniques to prove that VR ( Z ) is weaklyequivalent to the wedge S ∨ S of two 2-dimensional spheres, a result alreadypresent in the mentioned work of Adamaszek et al. Set X := { z , z , z , z , z , z } and Y := { z , z , z , z , z , z } . Note that X ∪ Y = Z and A := X ∩ Y = { z , z , z } .Note that VR ( X ∩ Y ) is contractible, and thus VR ( X ) ∪ VR ( Y ) is homotopyequivalent to the wedge VR ( X ) ∨ VR ( Y ). Furthermore we claim that all theobstruction complexes St( σ, A ) are contractible for all simplices σ in VR ( Z ) suchthat σ ∩ X = ∅ , σ ∩ Y = ∅ , and σ ∩ X ∩ Y = ∅ . For example if σ = { x , x } then St( σ, A ) = ∆[ x , x ] which is contractible. According to Corollary 7.6, theinclusion V R ( X ) ∪ V R ( Y ) ⊂ V R ( Z ) is therefore a weak equivalence and con-sequently V R ( Z ) has the homotopy type of the wedge VR ( X ) ∨ VR ( Y ). Themetric spaces X and Y are isometric, and hence the corresponding Vietoris-Ripscomplexes are isomorphic. It remains to show that V R ( X ) has the homotopy typeof S . Consider X ′ = { z , z } and X ′′ = { z , z , z , z } . Note that X = X ′ ` X ′′ ,VR ( X ′ ) has the homotopy type of S and VR ( X ′′ ) has the homotopy type of S . Finally note that, for all simplices σ in VR ( X ′ ) and µ in VR ( X ′′ ), the union OMOTOPICAL DECOMPOSITIONS OF SIMPLICIAL AND VIETORIS RIPS COMPLEXES25 σ ∪ µ is a simplex in V R ( X ). This implies that V R ( X ) is the join of VR ( X ′ ) andVR ( X ′′ ) (see Paragraph 3.11) and hence it is weakly equivalent to Σ( S ∧ S ) ≃ S . Acknowledgments.
A. Jin and F. Tombari were supported by the Wallenberg AI,Autonomous System and Software Program (WASP) funded by Knut and AliceWallenberg Foundation. W. Chach´olski was partially supported by VR and WASP.M. Scolamiero was partially supported by Brummer & Partners MathDataLab andWASP.
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