Vietoris-Rips Complexes of the Circle and the Torus

Transcription

1 UNIVERSIDAD DE LOS ANDES MASTER S THESIS Vietoris-Rips Compexes of the Circe and the Torus Author: Gustavo CHAPARRO SUMALAVE Advisor: Ph.D Andrés ÁNGEL A thesis submitted in fufiment of the requirements for the degree of Master in Mathematics in the Departmento de Matemáticas Facutad de Ciencias Universidad de os Andes Bogotá, Coombia October, 2016

2 i UNIVERSIDAD DE LOS ANDES Abstract Facuty of Science Department of Mathematics Master in Mathematics Vietoris-Rips Compexes of the Circe and the Torus by Gustavo CHAPARRO SUMALAVE In this thesis we study the detais of the computations by Adams and Adamaszek eading to the determination of the homotopy type of the Vietoris-Rips compexes of the circe. The resut shows that the Vietoris-Rips compexes are (up to homotopy) either contractibe, odd dimensiona spheres or wedge sums of spheres of the same even dimension. Different kinds of products on graphs and simpicia compexes are aso studied; this is done by introducing eements from discrete Morse theory for simpicia compexes. A reation between the product of cique compexes and the cique compex of a product is shown. The main contribution of the thesis is the determination (up to homotopy) of the Vietoris- Rips compexes of tori with the maximum metric using a the aforementioned resuts.

3 ii Acknowedgements I woud ike to thank my advisor Andrés Ánge for his support, guidance and patience during this thesis project. I aso want to express my deepest gratitude to a my oved ones, especiay for aways being there whenever I needed it.

4 iii List of Symbos Homotopy equivaence s Simpe homotopy equivaence K Geometric reaization of compex K ΣX Unreduced suspension of X Z n Integers greater than or equa to n κ X Wedge sum of κ copies of X K L Join of the simpicia compexes K and L M(G; n) Moore space for the group G and n Z 0 K (1) 1-skeeton of the simpicia compex K a b Adjacency vertices a and b VR < (X; r) Vietoris-Rips compex page 4 N (n, k) Nerve compex arcs S 1 page 4 N (n, k) Vietoris-Rips eveny spaced points page 11 VR < (X; r) Vietoris-Rips graph page 14 d Cockwise distance page 14 C(G) Cique compex of graph G page 14 wf(g) Winding fraction of graph G page 15 G 1 G 2 Boxtimes product of graphs G 1 and G 2 page 32 G 1 G 2 Ordered product of graphs G 1 and G 2 page 32 K 1 K 2 Categorica product compexes K 1 and K 2 page 32 K 1 K 2 Ordered product compexes K 1 and K 2 page 32

5 iv Contents Abstract Contents i iv Introduction 1 1 Vietoris-Rips Compex of a Circe Reduction to Eveny Spaced Arcs Homotopy Types of Nerves of Eveny Spaced Arcs Homotopy Equivaence Nerves and Ciques Cycic Graphs Vietoris-Rips Circe Generic Vaues Singuar Vaues Products of Graphs and Cique Compexes Basics of Discrete Morse Theory Products (Graphs, Compexes) Cique Compexes of Products The Vietoris Rips Compex of a Torus Products Vietoris-Rips Over Finite Sets Coimits and Products Vietoris-Rips Compexes of a Product Bibiography 45

6 1 Introduction The main object of study of this thesis is the Vietoris-Rips compex, denoted by VR < (X; r). Given a metric space X and a fixed rea number r > 0, the simpicia compex VR < (X; r) has vertex set X and a finite subset of X of size k + 1 determines a face of dimension k if the diameter of the set is stricty smaer than r. The set X can be either finite or infinite. When X is finite, the number of simpices of VR < (X; r) grows exponentiay as X increases in size. The first thing we want to know about these objects is whether or not they keep some (topoogica) information about the space X we started with. For instance, we may suspect that taking sufficienty many points of our possiby infinite space X and r sma enough, the simpicia compex associated to this input wi preserve, at the very east, the homoogy of the space. Something a itte bit stronger hods. The first resut in this direction that we are aware of goes back to [Hausmann, 1995]. In that paper, Hausmann proves, among other things, that if the metric space is a compact Riemannian manifod and the distance is taken with respect to the infimum of the enghts of curves connecting points, then, for r sufficienty sma 1, there is a homotopy equivaence between the geometric reaization of VR < (X; r) and X. In particuar, this says that a combinatoria object, VR < (X; r), preserves a the information about X up to homotopy. Hausmann aso modifies the concept of deformation retraction (defining crushing) from X to A X; he adds a condition on the homotopy taking the distance into account (intuitivey, not aowing to increase distances between points as the space retracts). He says that a space is crushabe if there is a crushing from the space to a point. A few things to have in mind is that every normed space is crushabe, every crushabe space is contractibe and that not every contractibe space is crushabe. He showed that whenever there is a crushing from X to a subspace A of itsef, the map VR(A; r) VR(X; r) induced by the incusion is a homotopy equivaence. Hausmann uses this to define a cohomoogy theory on compact metric spaces and eaves open questions about the kinds of spaces that arise as Vietoris-Rips compexes of manifods. Three of these questions stand out. The first asks about the behaviour of our simpicia compex a r gets arger, since we aready know what happens in the opposite case. The second question is aso natura and asks the foowing: Once we know we can recover the space taking a the points of X, is it possibe to do this taking ony finitey many points? In other words, is it possibe to find a finite subspace of X and an appropriate r 1 This works, for exampe, when r is taken ess than the convexity radius of the space.

7 Contents 2 so that the Vietoris-Rips compex associated to this subspace is homotopy equivaent to X? Lasty, it conjectures that the connectivity of the Vietoris-Rips compexes is a non-decreasing function of the parameter r unti it is greater than the diameter of the space, in which case the compex is contractibe. Just a few years after the paper of Hausmann, a positive answer is given to the second question in genera. In [Latschev, 2001] it is proved the stronger fact that for a cosed Riemannian manifod X, there is a positive number ε such that for any 0 < ε < ε there is a δ > 0 such that the geometric reaization of any VR(Y ; ε), where Y is a metric space with Gromov-Hausdorff distance to X ess than δ, is homotopy equivaent to X. So, for instance, a finite ε-dense subset of Y can be taken as representative of X. Despite this, the question about the spaces that arise as Vietoris-Rips compexes sti remains, and itte is known about it. Nevertheess, recenty, it has been possibe to determine a the Vietoris-Rips compexes (for any r, sma and arge) for the circe thought of as R/Z. In [Adamaszek and Adams, 2015], it is shown that the compexes behave in the expected way for r sma, resembing the circe. It is aso shown that the homotopy type of each compex ies in one of the foowing categories: a point, odd dimensiona spheres, and wedge sums of spheres of the same even dimension. (This, in particuar, gives positive answers to Hausmann s questions for the circe.) The main contribution of this thesis is the computation of the homotopy type of the Vietoris-Rips compexes of tori with the maximum metric. We use a resut of [Larrión, Pizaña, and Viarroe-Fores, 2013] to ink points in tori with this metric and points in products of Vietoris-Rips compexes. Henry Adams in his Research Statement mentions that this extension of the resut has aready been obtained, but, as far as we know, it has not been pubished anywhere, hence we provide a (possiby different) proof here. Here is a brief outine of this document. In Chapter 1, we study the detais of the proof that determines the homotopy type of the Vietoris-Rips compexes for the circe. First, the case when X is a finite subset of S 1 is considered. It is shown that it can even be reduced to the study of eveny spaced finite subsets. The homotopy type of these spaces is determined and it can be aready foreseen the sort of spaces we expect for arbitrary subsets of the circe. Later, it is noted that the 1-skeeton of these simpicia compexes keep a the information of the compex up to homotopy. A numeric invariant is associated to these graphs aowing to contro the homotopy type of the Vietoris- Rips compexes. Finay, it is found that the behaviour of the compexes changes according to the vaues of r taken; two different paths are necessary and are carried at the end of the chapter. In Chapter 2, we study two different notions of products for simpicia compexes and for graphs from the topoogica viewpoint. A few eements of discrete Morse theory for simpicia compexes are needed and are deveoped within the chapter. The main resut here wi connect the topoogy of the cique compex of a specia kind of product on graphs and the product of the cique compexes.

8 Contents 3 Finay, in Chapter 3, we determine up to homotopy the Vietoris-Rips compexes of tori with the maximum metric. The resut we get is that these spaces are products of the Vietoris-Rips compexes of the factors. We do this by using our knowedge of the homotopy type of the Vietoris-Rips compexes studied in the first chapter and an observation about the products on graphs defined in the second chapter.

9 4 Chapter 1 Vietoris-Rips Compex of a Circe In this chapter we study the detais of the computation of the Vietoris-Rips compexes of the circe S 1 (thought of as R/Z) given in [Adamaszek and Adams, 2015; Adamaszek et a., 2016]. Definition 1. Let (X, d) be a metric space and r > 0. The simpicia compex VR < (X; r) has vertex set X and the k-simpices are subsets {x 0,..., x k } of X of size k + 1 such that d(x i, x j ) < r for a i, j. The simpicia compex VR (X; r) is defined anaogousy. Theorem 2 (Adamaszek, Adams). If X S 1 is dense (in particuar when X = S 1 ) and 0 < r < 1 2, then VR < (X; r) S 2+1 for < r + 1, = 0, 1, Moreover, if 2+1 < r r , then the incusion VR < (X; r) VR < (X; r ) is a homotopy equivaence. Theorem 3 (Adamaszek, Adams). For 0 r < 1 2, there is a homotopy equivaence ( VR S 1 ; r ) { c S 2 if r = +1 S 2+1 if 2+1 < r < , = 0, 1, 2,..., where c is the cardinaity of the continuum. Moreover, if 2+1 < r r , then the incusion VR (X; r) VR (X; r ) is a homotopy equivaence. In order to prove Theorem 2 and Theorem 3, the path we take starts by determining the homotopy types of the Čech compex of arcs in S 1 as in [Adamaszek et a., 2016]. Definition 4. For n Z 1, k Z 0, et N (n, k) be the simpicia compexes with set of vertices {0, 1,..., n 1} and maxima faces {i, i + 1,..., i + k} (where everything is taken moduo n) for i = 0, 1,..., n 1. This compex N (n, k) is not arbitrary: it can be seen as Čech compex associated to the cover {[ i U n,k = n, i + k ] } S 1 : 0 i n 1 (1.1) n

10 Chapter 1. Vietoris-Rips Compex of a Circe 5 FIGURE 1.1: Visua representation N (4, 2) = S 2. of S 1. This means that we are going to compute the homotopy type of the nerve compex of these eveny spaced arcs over S 1. We note that N (n, 0) is simpy a set of n points with the discrete topoogy, and it wi be convenient to express this as n 1 S 0. Aso, note that N (n, k) S 1 if 1 k < n 2 as a consequence of the Nerve Lemma [Kozov, 2008, Thm 15.21]. Reca that the Nerve Lemma says that whenever the nonempty intersections of the cover considered are contractibe (and this is the case here by the restriction on k), then the compex is homotopy equivaent to the union of the eements of the cover, namey, S 1. Another easy case where we can directy determine N (n, k) is when k = n 2. N (n, n 2) gives a simpicia compex with n vertices and such that every subset of n 1 vertices is a maxima face, so it is the boundary of the (n 1)-simpex, that is, the (n 2)-sphere. Finay, note that N (n, k) for k n 1 is the whoe (n 1)-simpex, which is a contractibe space. 1.1 Reduction to Eveny Spaced Arcs This section is not crucia for the determination of the Vietoris-Rips compexes for the circe, but it shows that the study of nerve compexes of arbitrary finite coections of arcs of S 1 can be reduced to the study of coections of eveny spaced arcs without osing any information about the homotopy types of the compexes. Hence, as a byproduct of the resuts of the foowing sections, the nerve compex of a finite coection of arcs in S 1 is aso determined up to homotopy. We start with the foowing observation. Lemma 5. Let U be an arbitrary finite coection of S 1 formed by arcs that are either open, cosed or semi-open. Then there exists another finite coection U formed soey by cosed arcs [a i, b i ] S 1 such that no two of the a i s or the b i s are equa, and N (U) = N (U ).

11 Chapter 1. Vietoris-Rips Compex of a Circe 6 Proof. Let (a, b) U. If there are no intervas of the form (c, a], [c, a] in U, changing (a, b) to [a, b) wi have no consequence on N (U); otherwise, et ε > 0 sma so that a + ε is stricty ess than the next endpoint of an arc going cockwise from a (this can be done since we work with finite coverings), and change (a, b) to [a + ε, b). By symmetry, we can do the same for right endpoints of arcs in U. In order to avoid repetitions of the endpoints, we can do the foowing. If the eft endpoints of m arcs are equa, say a, take ε sma so that a ε is greater than the next endpoint of an arc going countercockwise from a, and pace the m eft endpoints on a 1 m ε, a 2 mε,..., a ε. The same can be done with right endpoints going cockwise. Definition 6. Let K be a simpicia compex. The vertex v K is dominated by the vertex w K if any simpex σ K that contains v satisfies that σ {w} K. Some notation: if we have two cosed arcs U 1 = [a 1, b 1 ] and U 2 = [a 2, b 2 ], then U 1 U 2 wi mean that a 1 a 2 b 1 b 2 < a 1. The foowing Lemma wi aow us to reduce the nerve compexes N (U) to one of the form N (n, k) in the Theorem afterwards. We ca the a i s eft endpoints and the b i s right endpoints. Lemma 7. Let U = {U i = [a i, b i ] S 1 : i = 0,..., n 1} be a coection of n cosed arcs, and et U i, U j in U such that 1. U i U j, 2. U i U j and b k [a i, a j ) for a k, or 3. U j U i and a k (b j, b i ] for a k. Then the vertex i is dominated by the vertex j in N (U). Proof. Let σ N (U) and U σ = k σ U k with U σ U i (so that i σ). We must show that j σ, that is, U σ U i U j. Assuming 1., U σ U i U j = U σ U i. Assume 2. Suppose that U σ U i U j =. Hence U σ U i [a i, a j ). Since b k [a i, a j ) for a k (in particuar for k σ {i}), we must have a j U σ U i because the intersection of the U k s ies in [a i, a j ). Therefore, a j [a i, a j ). See Figure 1.2. Assuming 3., we concude anaogousy to the second case. Proposition 8. Let U be a finite coection of arcs in S 1. If N (U) has no dominated vertices, then there is an isomorphism N (U) = N (n, k) for some 0 k < n. Proof. By Lemma 5, we can assume that U is a coection of cosed arcs in S 1 with different endpoints, so et U = {U i = [a i, b i ] S 1 : i = 0,..., n 1}. Since the eft endpoints are a different, we can order the eements in U in such a way that a 0 < < a n 1 < a 0. We can further assume that no arc contains another because by Lemma 7(1.), there woud be a dominated vertex that we

12 Chapter 1. Vietoris-Rips Compex of a Circe 7 FIGURE 1.2: Proof Lemma 7 may take away without changing the homotopy type of the compex, so the right endpoints are aso ordered by b 0 < < b n < b 0. Let us show that, in this setting, the set of endpoints aternate between right and eft endpoints. If there were no right endpoint between a i and a i+1, the condition in Lemma 7(2.) woud be satisfied, and the vertex i + 1 of the compex woud dominate vertex i. Since the numbers of eft and right endpoints are the same, the endpoints must aternate, so there exists a constant 0 k < n such that a 0 < b k < a 1 < b k+1 < a 2 < < b k+n 1 < a 0. Since a 0 < b k impies that a k < b 0, two arcs ony intersect when their eft endpoints are separated by at most k eft endpoints of distance. This says that the maxima simpices of the compex N (U) are the nonempty intersections k j=0 U i+j for i = 0,..., n 1. Therefore, N (U) N (n, k). 1.2 Homotopy Types of Nerves of Eveny Spaced Arcs In the previous section we saw that in order to determine the homotopy types of the nerves N (U) we may simpy restrict to the case where U is of the form U n,k. Now we show how to actuay compute N (n, k) = N (U n,k ) up to homotopy. The next emma wi pay a crucia roe in the determination of the homotopy types of the N (n, k) in Proposition 12. Lemma 9. [Brown, 2006, Cor ] Let the space X be the union of cosed subspaces X 1 and X 2 with intersection X 0 and such that the incusions X 0 X 1, X 0 X 2 are cofibrations. Suppose that X 1 and X 2 are contractibe. Then X is of the homotopy type of the suspension ΣX 0. We can use a theorem for guing homotopy equivaences to prove Lemma 9. Theorem 10. [Brown, 2006, Thm ] Let X, Y be topoogica spaces, and f : X Y a continuous map. Suppose that:

13 Chapter 1. Vietoris-Rips Compex of a Circe 8 1. X = X 1 X 2, Y = Y 1 Y 2, where X i, Y i are cosed subspaces, and X 0 := X 1 X 2, Y 0 := Y 1 Y f(x n ) Y n for n = 0, 1, The restrictions f 0 : X 0 Y 0, f 1 : X 1 Y 1, f 2 : X 2 Y 2 are homotopy equivaences. 4. Each incusion X 0 X 1, X 0 X 2, Y 0 Y 1, Y 0 Y 2 is a cofibration. Then f is a homotopy equivaence. Proof. (of Lemma 9) Let i n : X 0 X n be the incusion for n = 1, 2. The maps i n are nu-homotopic because the spaces X n are contractibe. Let H n : X 0 [0, 1] X n for n = 1, 2 be the homotopies so that H n (x, 0) =, H n (x, 1) = i n (x), where is a point. Then i n can be extended to a map f i from a cone CX 0 = {(x, t) : x X 0, t [0, 1]}/{(x, 0) : x X 0 } over X 0 to X n for n = 1, 2, and these two maps f 1 and f 2 can be gued together to give a map f : ΣX 0 X 1 X 2 = X. The functions f i are maps between contractibe spaces (the cones and X 1, X 2 ), so they are homotopy equivaences. By hypothesis, the incusions are cofibrations. Hence, by Theorem 10, the map f is a homotopy equivaence, i.e. ΣX 0 X. The way we wi use Lemma 9 takes the foowing form. Lemma 11. Let K be a finite simpicia compex and et K 1, K 2 be contractibe simpicia subcompexes of K such that K 1 K 2 = K. Then K Σ(K 1 K 2 ). Proof. This is simpy a reformuation of Lemma 9 for simpicia compexes. The ony thing that may require specia attention is the hypothesis that the incusions must be cofibrations, but this is indeed the case for finite simpicia compexes (see [Hatcher, 2001, Prop. 0.17]). The next proposition wi give us a recurrence formua (moduo homotopy equivaence) to determine the homotopy type of N (n, k). Proposition 12. For n 2 k < n, there is a homotopy equivaence N (n, k) Σ 2 N (k, 2k n).

14 Chapter 1. Vietoris-Rips Compex of a Circe 9 FIGURE 1.3: Possibe intersections. Proof. Let us denote by σ i the maxima simpices {i, i + 1,..., i + k} of N (n, k). Note that we can write N (n, k) as the union of two subcompexes that are actuay cones, namey, N (n, k) = ( n k 2 i=0 } {{ } K 1 ) σ i n 1 j=n k 1 σ j } {{ } K 2 where every maxima simpex carries a its subsimpices. = K 1 K 2, They are both cones because if n k 1 j n 1, then n 1 j+k n+k 1, so every σ j has n 1 as a vertex, and since n 2 k < k+1, n < 2k+2, so n k 2 < k, which impies that every σ i contains k as a vertex. Since we have K 1 and K 2 are contractibe, by Lemma 11, N (n, k) Σ(K 1 K 2 ), and K 1 K 2 = {σ i σ j : i = 0, 1,..., n k 2; j = n k 1,..., n 1}. Since 0 i n k 2 impies k i + k n 2, we get that n 1 is not a vertex of K 1, so the compex K 1 K 2 has as vertices {0, 1,..., n 2}. It is cear from Figure 1.3 that there are three types of intersections among the maxima simpices σ i and σ j : 1. If 0 i j + k n i + k < j n 1, then σ i σ j = {i,..., j + k n} σ 0 σ n 1 = {0, 1,..., k 1}, where the incusion hods because the intersection ies on the right side of 0, the simpex σ j starts before 0 and every simpex has k + 1 vertices. 2. If 0 j + k n < i j i + k n 2, then σ i σ j = {j,..., i + k} σ n k 2 σ n k 1 = {n k 1,..., n 2}, because 0 j + k n impies j n k > n k 1 and i + k n If i j + k n and j i + k, then σ i σ j = {i,..., j + k n} {j,..., i + k}.

15 Chapter 1. Vietoris-Rips Compex of a Circe 10 Hence the maxima simpices of K 1 K 2 are the foowing: τ = {0,..., k 1}, τ = {n k 1,..., n 2}, τ i,j = {i,..., j + k n} {j,..., i + k} for 0 i j + k n, j i + k n 2. We now want to express K 1 K 2 as the union of two contractibe subcompexes to use Lemma 11 again. Let K 1 K 2 = }{{} τ K 1 τ (i,j) Λ τ i,j, } {{ } K 2 where every simpex carries a its subsimpices as before and Λ = {(i, j) : 0 i j + k n, j i + k n 2}. The subcompex K 1 is contractibe since it is the simpex of dimension k. Let us now show that K 2 τ (showing K 2 is contractibe too) by a sequence of reductions. Define K 2 () as the subcompex of K 2 on the vertices {,..., n 2} for = 0,..., n k 1. Note that if n k 1, then τ cannot be a simpex of K 2 () (hence does not contain ) and any maxima simpex τ i,j in K 2 that has as vertex in K 2 () must satisfy i j + k n and τ i,j {,..., n 2} τ,j = {,..., j + k n} {j,..., + k}, that is, it is of the form τ,j. This means that any simpex in K 2 () that contains aso contains +k, so +k dominates in K 2 () and we can take away preserving the homotopy type: K 2 () K 2 () \ {} = K 2 ( + 1). This gives K 2 = K 2(0) K 2(1) K 2(n k 1) = τ. Hence the maxima simpices of K 1 K 2 are τ τ = {n k 1,..., k 1} τ τ 0,j = {j,..., k 1} {0,..., j + k n} for n k j k 1. τ τ i,i+k = {i,..., i + 2k n} for 0 i n k 2, with vertex set {0,..., k 1}, but note that these sets are of the form [i, i+(2k n)] mod k, so K 1 K 2 = N (k, 2k n), and the resut foows. Reca the foowing fact about suspensions of wedge sums of spheres. Proposition 13. The suspension of a wedge sum of spheres is a wedge sum of spheres. More precisey, for nonnegative integers d i, Σ ( ) i I Sd i i I Sdi+1. We are ready to determine up to homotopy N (n, k).

16 Chapter 1. Vietoris-Rips Compex of a Circe 11 Theorem 14. Let 0 k n 2. Then { n k 1 S 2 N (n, k) if k for some = 0, 1, 2,... Proof. The proof is done by induction on. n = +1, S 2+1 if +1 < k n < +1 +2, For = 0, if k n =, then k = 0 and N (n, 0) is a discrete set of n points which is the same as n 1 S 0. Now suppose that k n = for some > 0. We have that thus 2k n k 1 = 2 n k 1 +1 = N (k, 2k n) k (2k n) 1 S 2( 1) = n k 1 S 2 2. Therefore, by Proposition 12 and Proposition 13, we have N (n, k) Σ 2 N (k, 2k n) n k 1 S 2. = 0, Simiary, reca that if 1 k < n 2 (this is the case = 0), then N (n, k) S1. Assume +1 < k n < for some > 0. Then and hence k n = k n k > 2k n k 1 = 2 n k = 0 > = 0, < 2k n k < +1. By induction hypothesis and Proposition 12, N (n, k) Σ 2 N (k, 2k n) Σ 2 S 2( 1) S 2, as we wanted to show. 1.3 Homotopy Equivaence Nerves and Ciques Thus far we have ony taked about Čech compexes when we are actuay interested in Vietoris-Rips compexes. Just as we defined N (n, k) and this can be seen as Čech compexes of eveny spaced arcs, we now define N (n, k), which can be seen as Vietoris-Rips compexes of some eveny spaced subsets of S 1. Definition 15. The simpicia compex N (n, k) has vertex set the coection U n,k in 1.1, and there is a face [i 0,..., i n ] if [ ir n, ir+k n ] [ i sn, ir+k ] n for a ir, i s {i 0,..., i n }.

17 Chapter 1. Vietoris-Rips Compex of a Circe 12 FIGURE 1.4: Visua representation VR ( {0, 1 6,..., 5 6 }; 1 3) S 2. Aternativey, N (n, k) is the maxima ( simpicia compex with 1-skeeton N (n, k) (1). We note that N (n, k) = VR {0, 1 n, 2 n,..., n 1 n }; n) k. In this section the two kinds of compexes wi be connected via a homotopy equivaence (for appropritate n and k) and we wi be abe to determine the homotopy type of the N (n, k) by using Theorem 14. We wi need the foowing resut that is the version of Quien s A theorem on conditions for a functor to induce a homotopy equivaence on cassifying spaces appied to compexes. Theorem 16. [Barmak, 2011] Let ϕ : K L be a simpicia map between two finite compexes. Suppose that the preimage of each cosed simpex of L is contractibe. Then ϕ is a simpe homotopy equivaence. Reca that a simpe homotopy equivaence is a homotopy equivaence that is induced by a sequence of eementary expansions and coapses. In particuar ϕ preserves the homotopy type of the spaces. We have the foowing resut. Theorem 17. Let n 1 and k 0. Then N (n + k, k) N (n, k). Proof. The caim is true if n 2 k because in both simpicia compexes any two vertices form an edge, so we et k n 2. Let us define ϕ : {0, 1,..., n + k 1} {0, 1,..., n 1} by ϕ(i) = i (mod n). This induces a surjective simpicia map ϕ : N (n+k, k) N (n, k). It is surjective because [i, i + k] N (n + k, k) [i, i + k] N (n, k) for 0 i < n, and in genera, for an arbitrary simpex σ N (n + k, k), we have two cases for ϕ(σ). Let t := min σ (in the order 0 1 n + k 1), so σ has no vertices in {0, 1,..., t 1}, and note that σ [t k, t + k]. 0 t k : Since σ [0, 1,..., t 1] =, we have σ [t, t + k], so ϕ(σ) [t (mod n), t + k (mod n)]. Hence ϕ(σ) is a face in N (n, k). t k < 0 : We have that σ [t k (mod n + k), n + k 1] [t, t + k] = [t + n, n + k 1] [t, t + k],

18 Chapter 1. Vietoris-Rips Compex of a Circe 13 FIGURE 1.5: Cases i. and iii. for the choice of w given v. hence, appying ϕ, we have and again ϕ(σ) is a face in N (n, k). ϕ(σ) [t, t + k] [t, k 1] [t, t + k], Our goa now is to show that ϕ satisfies the hypothesis of Theorem 16 to prove that it is a homotopy equivaence. Let σ be a simpex in N (n, k) with vertex set {i 1,..., i s } {j 1,..., j t } where 0 i 1 < < i s < k j 1 < j t n 1. We then have that σ := ϕ 1 (σ ) has vertex set {i 1,..., i s } {i 1 + n,..., i s + n} {j 1,..., j t }. Let us show that there exists w σ such that σ [w k, w + k], so that σ is a cone with apex w and hence contractibe. Reca that the maxima simpices of N (n, k) are of the form [i, i + k]. Therefore, there is v σ so that σ [v, v + k]. We have the foowing cases for w according to the choice of v. i. For v = i q (for some 1 q s) et w = i q. ii. For v = j 1 and s = 0 et w = j 1. iii. For v = j 1 and s > 0 et w = i s + n. iv. We cannot have v = j q for q 2 since j q 1 is not in the interva [j q, j q + k]. See Figure 1.5 for a representation of the first and third cases. This shows that ϕ 1 (σ ) is contractibe for any simpex σ in N (n, k). We concude that N (n + k, k) N (n, k). We aready knew the two spaces were homotopy equivaent for an especific choice of n and k. Letting n = 4 and k = 2, we have N (4, 2) S 2 N (4 + 2, 2) (cf. Figure 1.1 and Figure 1.4). We can now determine N (n, k) up to homotopy. Theorem 18. Let 0 k n 2. Then { n 2k 1 S 2 N (n, k) if k for some = 0, 1, 2,... n = 2+1, S 2+1 if 2+1 < k n < ,

19 Chapter 1. Vietoris-Rips Compex of a Circe 14 Proof. By Theorem 17, we have N (n, k) N (n k, k). Now k/(n k) = /( + 1) impies k/n = /(2 + 1), and + 1 < k n k < impies The resut foows now from Theorem < k n < See [Adamaszek, 2013] for an aternative way to arrive at this resut. 1.4 Cycic Graphs In this section we define cycic graphs (for which the 1-skeeton of Vietoris-Rips compexes VR(X; r) with X S 1 are exampes considering directed edges) and ook at some of their properties. A particuar invariant wi be attached to these graphs, and this wi aow us to contro the behaviour of the compexes in terms of r. A cycic graph is a directed graph with a fixed cycic order in its vertices, denoted by v 0 v 1... v n 1, in such a way that if there is an edge v i v j for i j, then either j = i + 1 (mod n) or there are edges v i v j 1 and v i+1 v j (where the indices are taken mod n). This in particuar impies that if there is an edge between v i and v j, there are edges between v i and v k, and between v k and v j for a k between i and j. As we mentioned earier, the exampes of cycic graphs to have in mind are the ones associated to Vietoris-Rips compexes. Definition 19. If X S 1 is a finite set and 0 < r < 1 2, then we define the directed Vietoris-Rips graph VR < (X; r) as the graph with vertex set X and a directed edge x i x j if the cockwise distance 1 d(xi, x j ) between x i and x j is positive and stricty ess than r. The graph VR (X; r) is defined anaogousy. Definition 20. Let G be a graph. The cique compex C(G) has vertex set G and a simpex for each compete subgraph of G. Note that VR < (X; r) = C( VR < (X; r)) and VR (X; r) = C( VR (X; r)) when we forget the direction of the edges. A specific famiy of cycic graphs (to which every other cycic graph wi be reduced) is the foowing. Given integers n and k such that 0 k 1 2n, then the graph Cn k has vertex set {0,..., n 1} and i i + s for a i = 0,..., n 1 and s = 1,..., k. Note that Cn k = ({ VR 0, 1 n,..., n 1 } ; k ). n n 1 For exampe, the cockwise distance in S 1 = R/Z between 1 4 and 1 2 is 1 4, and the cockwise distance between 1 2 and 1 4 is 3 4.

20 Chapter 1. Vietoris-Rips Compex of a Circe 15 The associated morphisms to cycic graphs are caed cycic homomorphisms. Given cycic graphs G and H, a cycic homomorphism f : G H is a homomorphism of directed graphs such that if v 0 < < v n 1 v 0 in G, then f(v 0 ) f(v n 1 ) f(v 0 ) in H, and is not constant if G has a directed cyce. We note that he composition of two cycic homomorphisms is a cycic homomorphism, and that when we have a homomorphism of directed graphs f : G VR < (X; r) (or VR (X; r)), being a non-constant cycic homomorphism is equivaent to having where G = {v 0, v 1,..., v n }. n 1 i=0 d(f(v i ), f(v i+1 )) = 1, (1.2) In order to determine the homotopy type of the Vietoris-Rips compexes VR(X; r), we now define an invariant associated to cycic graphs that wi aow us to transform r into information about the 1-skeeton of the simpicia compex. Definition 21. Let G be a cycic graph. The winding fraction wf(g) of G is defined as { k wf(g) = sup n : there is a cycic homomorphism } Cn k G. Remark 22. Note that having a cycic homomorphism f : G H immediatey impies that wf(g) wf(h) because any cycic homomorphism C k n G extends to one C k n H. We said that we wanted to contro r in VR(X; r) in terms of the winding fraction of its 1-skeeton, so et us see now how the two are reated and how to compute the winding fraction of C k n. Proposition 23. Let X be a finite subset of S 1 and et 0 < r < 1. Then 1. wf( VR < (X; r)) < r, 2. wf( VR (X; r)) r, 3. wf( VR < (X; r)) wf( VR (X; r)). Proof. 1. Let f : C k n VR < (X; r) be a cycic homomorphism with k 1 (since if there are ony maps with k = 0, the caim foows). Since i i + k in C k n

21 Chapter 1. Vietoris-Rips Compex of a Circe 16 for i = 0,..., n 1, we have d(f(i), f(i + k)) < r, hence nr > = n 1 i=0 n 1 n 1 d(f(i), f(i + k)) = j j=0 i=j+1 k i=0 i+k 1 j=i d(f(j), f(j + 1)) = d(f(j), f(j + 1)) n 1 j=0 d(f(j), f(j + 1)) } {{ } =1 j i=j+1 k 1 = k, by Equation 1.2. Therefore, k n < r and wf( VR < (X; r)) < r. 2. The same as 1. changing d(f(i), f(i + k)) < r by d(f(i), f(i + k)) r. 3. It is cear since we have an incusion map VR < (X; r) VR (X; r). Incusions are cycic homomorphisms so we have the desired inequaity by Remark 22. Proposition 24. The winding fraction wf( C k n) of C k n is k n. Proof. The identity map Cn k Cn k is a cycic homomorphism, so, by definition of winding fraction, wf( Cn) k k n. On the other hand, since Cn k is a Vietoris-Rips graph with r = k n, Proposition gives wf( Cn) k k n. We aready know the winding fractions of the graphs Cn, k but what about other cycic graphs? It turns out that these k n are the ony possibe vaues for wf(g) for a cycic graph G. To see this, we remove some vertices of the graphs and show that this remova does not affect the winding fraction. The vertices to be removed wi be caed dominated. 2 A vertex v i in a cycic graph G with cycic ordering v 0 < < v n 1 < v n is dominated by the vertex v i+1 if, for any vertex w in G, there is an edge w v i if and ony if there is one w v i+1. Whenever we have a cycic graph with a dominated vertex, we can take it away and obtain a new cycic graph. Suppose G is a cycic graph and v i is dominated by v i+1. Define the map f : G G \ v i by f(v j ) = v j if j i, and f(v i ) = v i+1. This map f is a cycic homomorphism and ceary G \ v i G ψ G \ v i is the identity map, but we actuay have more. Proposition 25. Let G be a cycic graph, v i, v i+1 G, with v i dominated by v i+1. Then G \ v i G and ψ : G G \ v i induce homotopy equivaences of cique compexes. Proof. We have C(G\v i ) G ψ C(G\v i ) is the identity map, so we ony need to check C(G) ψ C(G \ v i ) C(G) is homotopic to the identity on C(G). 2 There is aready a notion of dominated vertex in simpicia compexes, but it wi be cear from the context to which one we refer.

22 Chapter 1. Vietoris-Rips Compex of a Circe 17 By definition of cycic graph, if we have an edge v i v k for k > i + 1, then there is an edge v i+1 v k. This, together with the hypothesis that v i is dominated by v i+1, impies that every vertex w adjacent to v i is adjacent to v i+1. Hence, for every σ = [v j0,..., v jk ] in k C(G) (v i ) = {σ C(G) : v i σ, σ {v i } C(G)} we have edges v js v i in G for 1 s k, so there are edges v js v i+1 by the previous remark, which means that [v i+1, v j1,..., v js ] k C(G) (v i ). This impies that k C(G) (v i ) is a cone with apex v i+1 and so C(G) can be ontained from C(G\ v i ) by attaching a cone (over the cone k C(G) (v i )). Therefore C(G \ v i ) C(G) is a homotopy equivaence. Since we know that when we have a dominated vertex we may reduce the graph by deeting it, the natura question to ask is what happens when we deete a the dominated vertices. The next proposition gives an answer to this. Proposition 26. A cycic graph without dominated vertices is isomorphic to C k n for some 0 k < 1 2 n. Proof. Let G be a cycic graph without dominated vertices and with cyce order v 0 < < v n 1 < v 0. For every j there is a k j > 0 such that v j v j+1,..., v j v j+kj and v j v j+kj +1. Hence ({v i } {w : w v i }) \ {w : w v i+1 } = {v j : j + k j = i} =: N(v i ). Note that N(v i ) = if and ony if v i is a dominated vertex. Writing G = n 1 i=0 N(v i), since the N(v i ) are a disjoint, we have n 1 i=0 N(v i) = n, so N(v i ) = 1 for a i. This impies that {w : w v i } = {w : w v i+1 } = = {w : w v i 1 } =: k. Therefore G = C k n as we wanted to show. Definition 27. We say that G dismantes to C k n if there is a sequence G G 1 G = C k n (1.3) where every map induces a homotopy equivaence at the eve of cique compexes, and the composition C k n G C k n is the identity. As a consequence of Propositions 25 and 26, we have the foowing coroary. Coroary 28. Let G be a cycic graph G. Then either G = C k n if G does not have any dominated vertices, or G dismantes to some C k n. Therefore, the ony homotopy types of C(G) for a cycic graph G are the ones appearing in Theorem 18.

23 Chapter 1. Vietoris-Rips Compex of a Circe 18 Remark 29. Note that the reduction in 28 shows that wf(g) = k n by Remark 22 and the definition of winding fraction. Hence, the homotopy type of C(G) is determined by wf(g) subject to the same conditions for k n in Theorem Vietoris-Rips Circe As can be noticed from a gimpse of Theorems 2 and 3, there is a difference in the behaviour of the Vietoris-Rips compexes when we take r from intervas of the form ( 2+1, ), and when we take r from the extremes of those intervas. Hence, this section is divided in two according to the vaues of r we take. The former vaues wi be caed generic and the atter ones wi be caed singuar Generic Vaues In this section the first thing we prove is a theorem of stabiity of the homotopy type of the cique compexes of cycic graphs in the sense that whenever there is a cycic homomorphism f : G H and the winding fractions of G and H both ie within certain vaues, then f is a homotopy equivaence at the eve of simpicia compexes (after taking cique compexes of the graphs). This is done by first considering the cases when G is a subgraph of H by taking away vertices and edges (this is Proposition and 30.2.), and, after that, the genera case (Proposition 30.3.). Proposition 30. Let G and H be cycic graphs, v and e a vertex and an edge of H, respectivey. Let f : G H be a cycic homomorphism and suppose aso that + 1 < wf(g) wf(h) < (1.4) for some 0. Then 1. If G = H \ v and f is the incusion H \ v H, then f induces a homotopy equivaence of cique compexes. 2. Suppose H \ e is a cycic graph. If G = H \ e and f is the incusion H \ v H, then f induces a homotopy equivaence of cique compexes. 3. The map f induces a homotopy equivaence of cique compexes. Proof. 1. Let H v be the subgraph of H induced by the set of vertices adjacent to v and write C(H) = C(G) (C(H v ) v) so that C(G) (C(H v ) v) = C(H v ). With this cover of C(H), we have the Mayer-Vietoris sequence (in reduced homoogy) given by H n (C(H v )) H n (C(G)) H n (C(H v ) v) H n (C(H)) H n 1 (C(H)) By Remark 29 and 1.4, both compexes C(G) and C(H) are homotopy equivaent to S 2+1, and C(H v ) v is contractibe, so the ony non-zero groups in the

24 Chapter 1. Vietoris-Rips Compex of a Circe 19 sequence are 0 H 2+1 (C(H v )) Z Z H 2 (C(H v )) 0. (1.5) We know H v is a cycic graph, so, by Remark 30, the homoogy of C(H v ) is free and is not zero ony in at most one dimension. If, for instance, H 2+1 (C(H v )) = Z r, r > 0 and H 2 (C(H v )) = 0, the midde map in 1.5 is an isomorphism, but then there woud be an injective map Z r 0, hence r = 0. Simiary assuming non-zero homoogy appears at dimension 2. We have H (C(H v )) = 0 and the midde map in 1.5 is an isomorphism. Since f : C(H) C(G) induces isomorphisms in homoogy and C(G) C(H) S 2+1, the map f is a homotopy equivaence by Whitehead s theorem. 2. Let H e be the cycic subgraph induced by the vertices adjacent to both a and b that the edge a e b connects. Then we write C(H) = C(G) (C(H e ) e) with intersection C(H e ) {a, b} = Σ C(H e ). By Mayer-Vietoris, we get 0 H 2+1 (Σ C(H e )) H 2+1 (C(G)) H 2+1 (C(H)) H 2 (Σ C(H e )) 0, but since H k (C(H e )) = H k+1 (Σ C(H e )), we have 0 H 2 (C(H e )) Z Z H 2 1 (C(H e )) 0. The same anaysis as in Part 1. of this Proposition shows that f is a homotopy equivaence. 3. Let f : V (G) V (H) be injective so that G can be seen as a subgraph of H. Then f is the composition of eementary cycic homomorphisms G G 1 H where G i+1 is obtained from G i by adding a vertex or an edge. In this case Parts 1. and 2. appy, and hence f induces a homotopy equivaence on cique compexes. Now et f arbitrary, G Cn k and H Cn k as in 28, and consider the composition C k n G f H C k where means that it induces a homotopy equivaence on cique compexes. If we proved the composition induces a homotopy equivaence on cique compexes, the map f woud do too, so it is enough to prove the caim for an arbitrary cycic homomorphism f : Cn k Cn k for 2+1 < k k n n < This is what we do now. n,

25 Chapter 1. Vietoris-Rips Compex of a Circe 20 FIGURE 1.6: Definition of the function f d. If for some d > 0 we managed to create cycic homomorphisms f d, τ and γ so that we have a commutative diagram of the form C k n f d C dk dn γ C k n f τ C k n then f woud have the required form if τ does, and τ does because γ does. The maps γ and τ are actuay easy to define once we have d: set γ(i) = di and τ(j) = j d. The tricky one is f d. See Figure 1.6 for a guide for the definition of the function f d beow. Let f( Cn) k = {j 0 < < j s < j 0 } Cn k for some 1 s n 1. The preimage of each j q by f is an interva ( mod n) so we can reabe the vertices so that f 1 (j q ) = {i q, i q + 1,..., i q+1 1}. Let d := max{ f 1 (j q ) : 0 q s}. Define f d : C k n C dk dn by i dj q + d n (i q, i) for i f 1 (j q ). Let us see that f d is a graph homomorphism, that is, for 0 i n 1, we must have d dn (f d (i), f d (i + k)) dk. If i, i + k f 1 (j q ), then d dn (f d (i), f d (i + k)) d dk. If i f 1 (j q ) and i + k f 1 (j q ), then d n (j q, j q ) k because f is a graph homomorphism. If d n (j q, j q ) < k, then d dn (f d (i), f d (i + k)) d nd (dj q, dj q + d) dk.

26 Chapter 1. Vietoris-Rips Compex of a Circe 21 If d n (j q, j q ) = k so that j q = j q + k, then d dn (f d (i), f d (i + k)) = dk + d n (i q, i + k) d n (i q, i) = dk + d n (i + k) d n (i q, i q ) = dk + k d n (i q, i q ) dk, where the ast inequaity hods because otherwise d n (i q 1, i q ) k and hence d n (f(i q 1), f(i q )) = d n (j q 1, j q ) > k contradicting the fact that f is a homomorphism. Finay, note that within each f 1 (j q ) the map f d preserves the cycic ordering, so f d is a cycic homomorphism. We now have to make the transition from finite subsets to arbitrary subsets of the circe. To do this, we first assign to the geometric reaization of VR(X; r) the weak topoogy with respect to subcompexes induced by finite subsets of X. That is, if F (X) denotes the poset of a finite subsets of X ordered by incusion, then VR(X; r) = coim VR(X; r). Y in F (X) Given a finite subset Y 0 of X, the poset of a finite subsets of X containing Y 0 is cofina in F (X) and is denoted by F (X; Y 0). Hence we aso have VR(X; r) = coim VR(X; r). (1.6) Y in F (X;Y 0 ) We wi reduce 1.6 further (moduo homotopy equivaences) by taking homotopy coimits instead of simpy coimits. This can be done by the Projection Lemma (see [Weker, Zieger, and Živajević, 1999]) that says that this can be done when the maps in the diagram are cosed cofibrations, which is the case here. We extend the definition of winding fraction to arbitrary sets by setting wf( VR < (X; r)) = sup{wf( VR < (Y ; r)) : Y X, Y finite}, and simiary for wf( VR (X; r)). Proposition 31. Suppose that X is a non-empty subset of S 1 and 0 < r < 1 2. If, for some = 0, 1,..., either < wf( VR < (X; r)) < , or 2. wf( VR < (X; r)) = and wf( VR < (Y ; r)) wf( VR < (X; r)) for a finite subsets Y of X, is true, then VR < (X; r) S 2+1. If r r and conditions 1. or 2. hod with the same for r, the incusion VR < (X; r) VR < (X; r ) is a homotopy equivaence. Proof. Conditions 1. and 2. are imposed to justify the existence of a finite subset Y 0 of X such that if Y F (X; Y 0 ), then < wf(vr(y ; r)) < 2+3. Hence the

27 Chapter 1. Vietoris-Rips Compex of a Circe 22 hypothesis of Proposition 30 are satisfied, and we have VR(X; r) hocoim VR(Y ; r) Y in F (X;Y 0 ) S2+1. The ast caim aso foows from Proposition 30 because then we have that VR < (Y ; r) VR < (Y ; r ) is a homotopy equivaence for a Y F (X; Y 0 ). Therefore, the same is true for VR < (X; r) VR < (X; r ) by the Homotopy Lemma in [Weker, Zieger, and Živajević, 1999]. Lemma 32. Let X be a finite subset of S 1 and 0 < r < 1 2. If for any two consecutive points v i and v i+1 of X, we have d(v i, v i+1 ) < 2ε for some ε > 0, then wf( VR < (X; r)) > r 2ε. Proof. Assume r 2ε > 0 (otherwise it is trivia). Choose another ε < ε such that d(v i, v i+1 ) < 2ε. We are going to find a cycic homomorphism ϕ : Cn k VR < (X; r) (which impies wf( VR < (X; r)) k n ) for any k n < r 2ε so that wf( VR < (X; r)) must be greater than or equa to r 2ε, and hence greater than r 2ε. For every 0 i n 1 define ϕ(i) = x i X as the nearest point to i n. Hence ϕ preserves the cycic ordering and is a graph homomorphism because d(x i, x i+k ) d(x i, i n ) + d( i n, i+k i+k n ) + d( n, x i+k) < 2ε + k n < r, so we have the homomorphism ϕ required and the resut foows. We are finay ready to prove Theorem 2. Theorem 1. If X S 1 is dense (in particuar when X = S 1 ) and 0 < r < 1 2, then VR < (X; r) S 2+1 for + 1 < r + 1, = 0, 1, Moreover, if 2+1 < r r , then the incusion VR < (X; r) VR < (X; r ) is a homotopy equivaence. Proof. We first note that wf( VR < (X; r)) = r. This foows from Lemma 32 as we can find, for any ε > 0, a finite subset of X such that the distance between consecutive vertices is ess than 2ε, so wf( VR < (X; r)) r. On the other hand, Proposition 23 gives the reverse inequaity and aso shows the suppremum is not attained by any finite subset Y of X. Condition < r 2+3 then transforms in either Condition 1. or 2. of Proposition 31 and the resut foows.

28 Chapter 1. Vietoris-Rips Compex of a Circe Singuar Vaues Now that we have aready deat with Vietoris-Rips compexes VR(X; r) for r within intervas of the form ( +1, ), we have to anayze what happens when r is equa to a extreme of those intervas and the winding fraction is attained by some finite subset Y of X. The idea of the proof is the foowing. First it is shown that if r = 2+1 for some = 1,..., then VR (X; r) is simpy-connected, H (VR (X; r)) is torsion-free and H n (VR (X; r)) = 0 if n 2. If we somehow knew that H 2 (VR (X; r)) is a free abeian group, then we woud have VR (X; r) M( κ Z, 2) k S 2 (1.7) by the uniqueness up to homotopy of (CW-compex) Moore spaces M(G, n), consequence of Whitehead s Theorem (see [Hatcher, 2001, Cor. 4.33]), for n > 1. Our goa then is to prove that H 2 (VR (X; r)) is free abeian by studying its generators (seen as images of certain homoogy casses from other spaces induced by cycic homomorphisms). Finay, a counting argument wi show that κ = c. We start by defining the casses we wi use to describe generators for the aforementioned homoogy group. Reca that the cross-poytope K n in R n is the convex hu of the set of points {±e 1, ±e 2,..., ±e n } R n, where e 1 = (1, 0,..., 0), e 2 = (0, 1,..., 0), and so on. Consider C 2 2(2+1) = VR ({0, , ,..., 1 2,..., } ; ) Let us note that any two vertices in C2+1 are adjacent except for antipoda vertices, that is, eements of the form { i 4 + 2, i } Identifying i e i and i e i, we see that the boundary of the cross-poytope K 2+1 is equa to the cique compex of C2(2+1) This, in particuar, gives another way to visuaize VR ( {0, 1 6,..., 5 6 }; 1 3 ) S 2 as the octahedron in R 3 (cf. Figure 1.4).

29 Chapter 1. Vietoris-Rips Compex of a Circe 24 In H 2 (C( C 2 )) we fix the homoogy cass 2(2+1) ι 2 = ( 1) (+3) 2 ([0] [2 + 1]) ([1] [2 + 2]) ([2] [4 + 1]) = [0, 2,..., 4] [1, 3,..., 4 + 1] ±. (1.8) Let G be a cycic graph. We say a homoogy cass 0 α H 2 (C(G)) is crosspoytopa if it is the image f (ι 2 ) of a map induced by a cycic homomorphism f : C2(2+1) 2 G. The next proposition describes a the cycic homomorphisms from injective homomorphisms C2(2+1) 2 Cn. k It is enough to consider injective homomorphisms because a homoogy cass of degree 2 in a cique compex must have as underying vertex set at east vertices, as can be seen in [Kahe, 2009, Lem. 5.3]. Proposition 33. Let d Every cycic homomorphism θ : C2+1 Cd(2+1) d is of the form θ a(i) = a + di (mod d(2 + 1)) for some a = 0,..., d(2 + 1) Every injective cycic homomorphism α : C2(2+1) 2 Cd(2+1) d is of the form α a,b defined by { a + d i α a,b (i) = 2 (mod d(2 + 1)) i even b + d i 1 2 (mod d(2 + 1)) i odd, Proof. for some a = 0,..., d(2 + 1) 1 and b = a + 1,..., a + d 1 1. For = 0, we simpy have a point on the domain and the resut foows, so et > 0. Since θ is a cycic homomorphism, we have which impies, using an anaogue to Equation 1.2, (2 + 1)(d) 2 i=0 d d(2+1) (θ(i), θ(i + )) d, (1.9) 2 d d(2+1) (θ(i), θ(i + )) = d d(2+1) (θ(i), θ(i + 1)) = d(2 + 1), hence 1.9 is an equaity for a i. Note aso that d d(2+1) (θ(i), θ(i + 1)) equas n d d(2+1) (θ(i + 1), θ(i + + 1)) d d(2+1) (θ(i + + 1), θ(i)) = n 2k = d. Defining a := θ(0) and from the fact that θ(i + 1) = θ(i) + d, the resut foows. 2. Note that C2(2+1) 2 can be written as the union of two copies of C2+1 with the vertex sets {0, 2,..., 4} and {1, 3,..., 4 + 1}. Defining a := α(0) and b := α(1), we have by Part 1. that α(2i) = a+di and α(2i+1) = b+di. Since α is an injective i=0

30 Chapter 1. Vietoris-Rips Compex of a Circe 25 cycic homomorphism, we have a < b < a + d (this is α(0) < α(1) < α(2)), so any option for a is possibe, but b must be in the set {a + 1,..., a + d 1}. The foowing Proposition wi show (among other things) that the homoogy of cique compexes of cycic graphs dismanting to a Cd(2+1) d, are generated by cross-poytopa casses. Proposition 34. Let G be a cycic graph dismanting to Cd(2+1) d. Then H 2 (C(G)) = Z d 1 has a basis {e 1,..., e d 1 } such that a the cross-poytopa eements in H 2 (C(G)) are ±e 1,..., ±e d 1, e i e j for 1 i, j d 1 and i j, for a tota of d(d 1) crosspoytopa eements in H 2 (C(G)). Proof. First, note that if n = d(2 + 1) and k = d, then n 2k 1 = d 1, so, by Theorem 18, we have that H 2 (C(G)) = Z d 1. Assume that the proposition is true for Cd(2+1) d. Then, if G is a cycic graph that dismantes to C d, we have cycic homomorphisms d(2+1) C d d(2+1) f G g Cd(2+1) d that compose to the identity and induce homotopy equivaences at the eve of cique compexes (see Coroary 28). Hence H 2 (C( Cd(2+1) d )) = H2 (C(G)) = H2 (C( C d d(2+1) )), and this shows both that f ({e 1,..., e d 1 }) is a basis (of cross-poytopa casses, by definition) for H 2 (C(G)), and that ±f (e i ), f (e i ) f (e j ) for a i j, are cross-poytopa casses. Furthermore, if α H 2 (C(G)) is cross-poytopa, so is β := g (α), but then g (α) = β = g f (β) shows that α = f (β), so α is a cross-poytopa cass of the required form. By the previous argument, it is then enough to show the proposition for G = Cd(2+1) d. If σ is an oriented simpex, et σ denote the cochain assigning 1 to σ, 1 to σ with the opposite direction, and 0 to any other simpex. For a = 0,..., n 1, et γ a := [a, a + d,..., a + 2d]. We have γ a is a maxima simpex in Cd(2+1) d, so [β a] := [γa ] is a cohomoogy cass in H 2 (C( C d )). Now et, from 1.8 and Proposition 33.2, d(2+1) [α a,b ] := (α a,b ) (ι 2 ) = [a, a + d,..., a + 2d] [b, b + d,..., b + 2d] ±.

31 Chapter 1. Vietoris-Rips Compex of a Circe 26 It foows that [β i ]([α j,d ]) = δ i,j for 1 i, j d 1, so {[α 1,d ],..., [α d 1,d ]} and {[β 1 ],..., [β d 1 ]} are (dua) basis for H 2 (C( Cd(2+1) d )) and H 2 (C( Cd(2+1) d )), respectivey. Note that [α a+d,b+d ] = [a + d, a + 2d,..., a + (2 + 1)d] [b + d, b + 2d,..., b + (2 + 1)d] ± = ( 1) 2 [a, a + d,..., a + 2d] + ( 1) 2 [b, b + d,..., b + 2d] ± = [α a,b ] hence every cross-poytopa cass is of the form [α a,b ] for 0 a d 1 and a + 1 b a + d 1 (reca Proposition 33.2). We now show the reationships between the α a,b. For every [v] H 2 (C( C d )) we have We consider the foowing cases: d(2+1) d 1 [v] = [β i ]([v]) [α i,d ] (1.10) i=1 a = 0, 1 b d 1 : Then [v] = [α a,b ] in 1.10 shows that [α 0,b ] = [α b,d ]. 1 a < b d 1 : Then [v] = [α a,b ] in 1.10 shows that [α a,b ] = [α a,d ] [α b,d ] since [β a ]([α a,b ]) = 1 and [β b ]([α a,b ]) = 1. 1 a d 1, b = d : Then [α a,b ] = [α a,d ], so [α a,b ] is a generator. 1 a d 1, d + 1 b a + d 1 : Then 1 b d < a d 1 and [v] = [α a,b ] in 1.10 shows that [α a,b ] = [α a,d ] [α b,d ] since [β a ]([α a,b ]) = 1 and [β b d ]([α a,b ]) = 1 (this by a simiar computation to the one that showed that [α a,b ] = [α a+d,b+d ]). Taking e i := α i,d for i = 1,..., d 1, the resut foows. We are now ready to prove Theorem 3. Theorem 2. For 0 r < 1 2, there is a homotopy equivaence ( VR S 1 ; r ) { c S 2 if r = +1 S 2+1 if 2+1 < r < , = 0, 1, 2,..., where c is the cardinaity of the continuum. Moreover, if 2+1 < r r , the incusion VR (X; r) VR (X; r ) is a homotopy equivaence. ( Proof. We note, as we did in the proof of Theorem 1., that wf(vr S 1 ; r ) ) = r, and in this case the supremum is attained. For 2+1 < r < , the resut foows from Theorem 1. If r = 2+1, then the supremum is attained by the a finite subset Y 0 of S 1 (given by the vertex set of a reguar (2 + 1)-gon). Hence, for any finite set Y 0 Y S 1 we have wf(vr (Y ; r)) = 2+1, so VR (Y ; r) is a finite wedge