Local-to-global Urysohn width estimates
LLOCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES
ALEXEY BALITSKIY ♣ AND ALEKSANDR BERDNIKOV ♠ Abstract.
The notion of the Urysohn d -width measures to what extent a metric spacecan be approximated by a d -dimensional simplicial complex. We investigate how localUrysohn width bounds on a riemannian manifold affect its global width. We boundthe 1-width of a riemannian surface in terms of its genus and the supremal width ofits unit balls. Answering a question of Larry Guth, we give examples of n -manifolds ofconsiderable p n ´ q -width in which all unit balls have arbitrarily small 1-width. We alsogive examples of topologically simple manifolds that are locally nearly low-dimensional. Introduction
In the paper [7] Larry Guth proves that, on a riemannian manifold, local volume esti-mates translate into global information about the Urysohn width. This resolved a con-jecture of Gromov [4], and provided an alternative way to prove the celebrated systolicinequality of Gromov. Guth also conjectured a generalization of his theorem, dealing withthe Hausdorff content on compact metric spaces in place of volume, and his conjecturewas established by Liokumovich, Lishak, Nabutovsky, and Rotman [9]. Shortly after that,a simple and clever proof was given by Panos Papasoglu [11], and the method employedthere gives the simplest and cleanest proof [10] of Gromov’s systolic inequality, with thebest dimensional constants known so far.The notion of the Urysohn d -width, popularized by Gromov ([4, 5], etc.), is a metricinvariant measuring to what extent a metric space can be approximated by a d -dimensionalsimplicial complex. After several successful applications, as in the systolic inequality, itbecame an invariant of independent interest. The original definition (equivalent to theone we give below) by Pavel Urysohn was given in 1920s (and published much later byPavel Alexandroff [2]) in terms of closed coverings of bounded multiplicity. It was usedby Urysohn to state a result about the continuity of the dimension of a sequence of setsin R n . Definition 1.1.
The
Urysohn d -width of a closed subset S of a compact metric space X is UW d p S q “ inf π : S Ñ Y sup y P Y diam p π ´ p y qq , where the infimum is taken over all continuous maps π from S to any simplicial complex Y of dimension at most d . (The diameter of a set is measured using the distance functionin X : diam A “ sup a,a P A dist X p a, a q .)In the same paper [7], Guth gives an example of a metric on S with locally small butglobally large 2-width [7, Section 4]. Further, he asks if there is a setting in which localUrysohn width bounds translate into global ones. The goal of this paper is to answer thisquestion in different settings. Question . Suppose that M n is a riemannian manifold such thateach unit ball B Ă M has UW q p B q ă ε . If ε is sufficiently small, does this inequalityimply anything about UW q p M q for some q ě q ? a r X i v : . [ m a t h . M G ] A ug OCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES 2
Our first result is an estimate in dimension n “
2, depending on the topological com-plexity of M . Theorem 1.3.
Let M be a closed riemannian surface with the first Z { -Betti number β “ rk H p M ; Z { q . If every unit ball has -width less than { , then UW p M q ă β ` . The dependence on β does not seem optimal. Our second result is the following theorem,which in particular gives an oriented closed surface with UW p M q „ β { (see Figure 1). Theorem 1.4.
For any ε ą , there exists a riemannian manifold M n with all unit ballsof -width less than ε , and such that UW n ´ p M q ě . Figure 1.
A piece of the surface from Theorem 1.4 for n “
2. The wholesurface is made by replicating this piece periodically many times and clos-ing up the ends. Roughly speaking, the left half of this surface has smallUrysohn 1-width, as well as the right half, while the whole surface has largeUrysohn 1-widthBasically, this result answers Question 1.2 in the negative. The example establishingthis theorem has large Betti numbers. If one is looking for a topologically simple example,our third result gives it with M n being a ball (but with a worse dimension in the localwidth bound). Theorem 1.5.
For any ε ą , there is a metric on the n -ball (or n -sphere, or n -torus)such that its p n ´ q -width is at least but UW r log p n ` q s p B q ă ε for every unit ball B . Acknowledgements.
We are grateful to Larry Guth for numerous helpful discussionsand his remarks on this paper. We also thank Hannah Alpert and Panos Papasoglu forthe discussions that led us to these questions.2.
Bounding width from below
Before we get to the main results, let us discuss the main tools one can use to show aspace has substantial Urysohn width.
OCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES 3
The first tool is the Lebesgue covering theorem (discovered by Lebesgue [8] and firstproved by Brouwer [3]), which can be used to show that p n ´ q -width of the unit euclidean n -cube equals 1. Lemma 2.1.
Every continuous map f : r , s n Ñ Y d from the unit n -cube to an d -dimensional simplicial complex, m ă n , has a fiber f ´ p y q meeting some two oppositefacets of the cube. The second tool amounts to the “fiber contraction” argument, which goes back toGromov [5, Proposition F ]. A detailed exposition can be found in [6, Section 5]. Wequote here a version of this argument due to Guth [6, Lemma 5.2]. Lemma 2.2.
Let W be a riemannian manifold of convexity radius at least ρ ; that is, anytwo points in a ball of radius ă ρ are connected by a unique minimal geodesic within thisball. Let π : X Ñ Y be a map from a metric space X to a simplicial complex Y , suchthat all fibers of π have diameter less than ρ . Then any -Lipschitz map f : X Ñ W ishomotopic to a map factoring as g ˝ π , for some g : Y Ñ W . Corollary 2.3.
Then f : X Ñ W be a -Lipschitz map from a metric space X to ariemannian manifold W of convexity radius at least ρ . Assume that the induced map f ˚ : H n p X q Ñ H n p W q is non-trivial for some n . Then UW n ´ p X q ě ρ . Surface width estimates
Proof of Theorem 1.3.
This proof follows closely the ideas from [6, Section 1]. The maintheorem therein says, basically, that in the case M » S , there is a universal way to mea-sure the Urysohn 1-width: it is given by the map to the set of the connected componentsof distance spheres around any point. The largest diameter of such a component givesthe value UW p M q within a factor of 7.We apply the same strategy. Pick any point p P M . Consider the distance spheres S r p p q .We show that UW p S r p p qq ă β ` r . For r ă { r ě { p S r p p qq ě β `
1, so there are points x and y distance β ` S r p p q . Denote γ a curve connecting x and y inside S r p p q (we can assume it exists by perturbing slightly the distance function dist p¨ , p q ). Denote x “ x , x β ` “ y , and pick points x k P γ , 1 ď k ě β , so that dist p x, x k q “ k . Noticethat dist p x i , x j q ě | i ´ j | . Denote g k a minimal geodesic from p to x k , for 0 ď k ď β ` (cid:96) k , 0 ď k ď β , the loop formed by the curves g k , g k ` and the part of γ between x k and x k ` . The loops (cid:96) , . . . , (cid:96) β cannot be independent in H p M ; Z { q ; hence, thereexist indices 0 ď i ă . . . ă i r ď β such that r (cid:96) i s ` . . . ` r (cid:96) i r s “ H p M ; Z { q .Assume that 1 is a regular value of dist p¨ , x i q (otherwise perturb this function slightly).The concatenation of (cid:96) i , . . . , (cid:96) i r bounds a 2-chain D in M . Define the domain D “ D X B p x i q . Now consider the map f : M Ñ R given by f p¨q “ p dist p¨ , p q , dist p¨ , x i qq (see Figure 3). Note it is ? f p D q covers a disk O of radius ? ´ in R ; formally speaking, the map f : p D , B D q Ñ p R , R z int O q is of degree 1. Inorder to apply Corollary 2.3, compose f with the ? -Lipschitz map wrapping the disk O around a standard sphere of radius ? ´ ? π (and convexity radius ? ´ ? ). Then our secondtool gives UW p D q ě ? ´ ? . On the other hand, UW p D q ď UW p B p x qq ă {
15, whichgives a contradiction.
Observation 1.
The image of f lies above the straight line λ connecting points p r, q and p , r q . Strictly speaking, Corollary 2.3 is applied to the evident map from D {B D to the 2-sphere. OCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES 4
Figure 2.
The map f covers a substantial triangular region Observation 2.
Consider the triangle ∆ with the vertices p r, q , p r, q , and p r ´ { , { q , and observe that f pB D q X int ∆ “ ∅ . Indeed, B D Ă B D Y S p x i q , so the image f pB D q is contained in the union of the following curves: ‚ the line λ , where f p g i q lies; ‚ the vertical straight line through p r, q , where f p γ q lies; ‚ the horizontal line through p r, q , where f p S p x i qq lies; ‚ the curves f p g k q , k ą i , each of which can be viewed as the graph of a 1-Lipschitzfunction of argument dist p¨ , p q ; they all lie above the straight line connecting p r, q and p r ´ { , { q . Observation 3.
Let q “ g i X S { p x i q and observe that the geodesic segment r q, x i s Ă g i is present in the 1-chain B D . The image f pr q, x i sq is the straight line segment between p r, q and p r ´ { , { q (traversed once). Other parts of f pB D q are all contained in theunion f p γ q Y f p S p x i qq Y Ť k ą i f p g k q , avoiding this straight line segment. In view of theprevious two observations, f pB D q winds around ∆ nontrivially. Observation 4.
The disk O inscribed in ∆ is of radius ? ´ .This concludes the proof. (cid:3) Remark . Under the same assumptions (every unit ball in the surface M has 1-widthless than 1 { homological systole (the length of the shortest loopthat is not null-homologous) is less than 2, regardless of genus. One way to show it is toadjust the proof of [6, Theorem 4.1].4. Manifolds of small local but large global width
The constructions of this section are inspired by the mother of examples [5, Exam-ple H ].4.1. Local join representation.
A crucial ingredient for the constructions below is adecomposition of R n ´ as the “local join” of several 1-dimensional complexes. Lemma 4.1.
Fix ă ε ă . It is possible to triangulate R n ´ by simplices with thefollowing properties: (1) Each simplex is c n -bi-Lipschitz to a regular simplex with edge length ε . OCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES 5 (2)
The vertices of triangulation can be colored by colors through n so that eachsimplex receives all distinct colors. (3) The colored triangulation can be taken periodic with respect to n almost orthogonaltranslation vectors of length « c n ; hence the colored triangulation descends tothe product T n ˆ R n ´ , where T n is a flat torus with convexity radius at least .Proof. In fact, one can take all simplices congruent to one another. For instance, one cantake the (scaled) set of alcoves for the affine Coxeter group r A n (see [12, Chapter 6]); thiswill give an example with a good value of c n (perhaps, the best). If we are not chasingafter good constants, much simpler constructions are possible. One way is to considerthe cubical subdivision of R n ´ with the set of vertices ε Z n ´ , split each ε -size cubeinto p n ´ q ! simplices, and then take the barycentric subdivision, which can be colorednaturally. Either of these constructions can be made periodic easily. (cid:3) Definition 4.2.
Let X n ´ be R n ´ or T n ˆ R n ´ . Triangulate it as in Lemma 4.1, anddefine Z i , 1 ď i ď n , to be the union of all edges of the triangulation between the verticesof colors 2 i ´ i . We say that X is the ε -local join of Z , . . . , Z n .The motivation behind this definition is that every (top-dimensional) simplex σ of thetriangulation can be written as the join p σ X Z q ˚ . . . ˚ p σ X Z n q ; that is, any point x P σ can be written as x “ n ÿ i “ t i z i , where z i P σ X Z i , t i ě , n ÿ i “ t i “ . The coefficients t i are determined uniquely; if t i ‰
0, the corresponding z i is determineduniquely too. This defines a map x ÞÑ p t , . . . , t n q from σ to the standard p n ´ q -dimensional simplex (cid:52) n ´ ; for adjacent simplices of the triangulation, those maps agreeon their intersection; hence, we have a well-defined map τ : X Ñ (cid:52) n ´ , which we call the join map . Note that Z i “ τ ´ p v i q , where v , . . . , v n are the vertices of (cid:52) n ´ . For each vertex v i , denote the opposite facet of (cid:52) n ´ by v _ i . For each complex Z i , introduce its dual complex Z _ i “ τ ´ p v _ i q . In other words, Z _ i is the union of all p n ´ q -dimensional cell of our triangulation that do not intersect Z i . There are naturalretractions π i : X z Z _ i Ñ Z i , defined by sending x “ n ř i “ t i z i P σ to z i P σ X Z i ; they are well-defined since t i ‰ x R Z _ i . Note that π i moves each point by distance at most sup diam σ „ ε .4.2. Manifolds that are locally nearly one-dimensional.
Proof of Theorem 1.4.
Pick a torus T n with convexity radius ě
1, as in Lemma 4.1. Onscale ε , represent X “ T n ˆ R n ´ as the local join of one-dimensional complexes Z , . . . , Z n ,as in Definition 4.2. The goal is to build a manifold M n Ă X so that on the large scale( „
1) it resembles T n homologically, but on the small ( „ ε ) scale it will become porousin a way that makes its local 1-width small.Recall the join map τ : X Ñ (cid:52) n ´ arising from the local join structure of X . Inthis proof, it will be convenient to think of the target simplex as a regular simplex ofinradius 3, placed in R n ´ and centered at the origin. We make use of the join map τ : T n ˆ R n ´ Ñ (cid:52) n ´ Ă R n ´ to “perturb” the projection p : T n ˆ R n ´ Ñ R n ´ ontothe second factor: OCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES 6 r p : “ p ´ τ { . The choice of the factor 1 { p -term dominates the τ -term in the sense that r p does not vanish outsideof T n ˆ int (cid:52) n ´ .Finally, define the “perturbation of T n “ p ´ p q by the Z i “ τ ´ p v i q ”: M n : “ r p ´ p q . Note: as defined, M is a PL-manifold; but we can perturb τ slightly to make it smoothand to make 0 a regular value of r p ; then M becomes a smooth manifold. Observe that M is contained in T n ˆ int (cid:52) n ´ , so M is closed; it is also orientable by construction. SeeFigure 1 for an illustration of the case n “ B p x q Ă M , we want to find a projection on one of the Z i with ε -small fibers. Recall the notation introduced after Definition 4.2: the dual complexes Z _ i “ τ ´ p v _ i q , and the retractions π i : X z Z _ i Ñ Z i . One of these retraction would doif we find i , depending on x , so that B p x q X Z _ i “ ∅ . Pick i maximizing the distancebetween τ p x q and v _ i in (cid:52) n ´ ; this distance is at least 3 by our choice of metric on (cid:52) n ´ . When we move x to x P B p x q , its p -projection changes by at most 1, whereasits τ -projection changes by at most 2 to compensate, so τ p x q never reaches v _ i . We cannow use the retraction π i : B p x q Ñ Z i , showing UW p B p x qq À ε . (Notation À meansinequality that holds up to a factor depending on dimension only.)To show that UW n ´ p M q ě
1, we use our second tool for estimating widths. ApplyCorollary 2.3 to the 1-Lipschitz projection map M Ñ T n (the composition M ã Ñ T n ˆ R n ´ Ñ T n ), sending the fundamental class r M s P H n p M q to r T n s ‰
0. Indeed, (thePoincar´e duals of) the classes of M and T n are the same in H n ´ p T n ˆ (cid:52) n ´ , T n ˆ B (cid:52) n ´ q as zero level sets for homotopic mappings r p and p , respectively; the homotopy p ´ tτ , t P r , { s , does not vanish on T n ˆ B (cid:52) n ´ since the p -term dominates the τ -term. (cid:3) Remark . In this argument, we used a torus as the “base space to be perturbed”. Infact, this construction can be repeated for any reasonable base space, provided that ithas sufficient convexity radius. One can easily adapt Lemma 4.1 for this case, and therest of the proof goes unchanged. Morally, the outcome is that any manifold can be“homologically perturbed” to make its local (on the scale comparable with its convexityradius) 1-width arbitrarily small.4.3.
Topologically simple n -manifolds that are locally nearly log n -dimensional. The next result is an “amplification” of Guth’s example [7, Section 4] of a 3-sphere withlarge 2-width but all unit balls UW -small.We start by taking a reasonable base space X n ´ endowed with an ε -fine triangulation, as in Lemma 4.1. A particular choice of X is not really important; the only assumptions we need are its substantial codimension 1width, and the existence of a colored triangulation (local join representation). For exam-ple, one can take X to be a unit cube in the euclidean space R n ´ ; then UW n ´ p X q ě ě {
2; then one can take X to be the ball of radius 2 in order to have UW n ´ p X q ě X a sphere, or a torus, etc.The triangulation of X comes equipped with one-dimensional complexes Z , . . . , Z n ,and the join map τ : X Ñ (cid:52) n ´ mapping Z i to the i th vertex of the simplex. Nowwe blow up the metric in X along the (cid:52) n ´ -direction and leave it unchanged along thefibers of τ . Formally speaking, consider a metric making (cid:52) n ´ a regular simplex with OCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES 7 inradius ą X .The resulting metric is piecewise riemannian, and after a slight smoothening, we get ariemannian manifold X . Proposition 4.4. UW n ´ p X q ě but UW n p B q À ε for every unit ball B Ă X .Proof. By assumption, UW n ´ p X q ě
1. The metric on X is even larger, so UW n ´ p X q ě
1. Now take a unit ball B Ă X , and observe that UW n p B q ď UW n p τ ´ p τ p B qqq . Byconstruction of the blown-up metric of X , τ p B q misses at least one facet of (cid:52) n ´ , say,the i th one. Then the retraction π i (in the notation introduced after Definition 4.2) givesa map τ ´ p τ p B qq Ñ Z i , whose fibers are small in the original metric of X . The map B Ñ Z i ˆ (cid:52) n ´ x ÞÑ p π i p x q , τ p x qq gives a desired bound on UW n p B q . (cid:3) For n “ (cid:52) n ´ to getbetter estimates. Namely, we can use the same procedure starting with (cid:52) n ´ (in placeof X ). Start from a metric making (cid:52) n ´ a regular simplex with inradius 2. Every unitball in (cid:52) n ´ misses a facet; this property is preserved after we will increase the metric atcertain places. The goal is to increase metric so that every unit ball B (cid:52) in (cid:52) n ´ is smallin the sense of some width. Let n “ m . Repeat the construction above: pick m skeleta,each of dimension 1, in a fine colored triangulation inside (cid:52) n ´ , and blow up the metricin certain directions in order to have maps B (cid:52) Ñ Y m with small fibers, for every unitball B (cid:52) Ă (cid:52) n ´ . Now, once we defined a (modified) metric on (cid:52) n ´ , we blow up themetric of X to match it in the (cid:52) n ´ -direction (and leave unchanged along the transversaldirection). Call the resulting metric space X . Proposition 4.5. UW m ´ p X q ě but UW m ` p B q À ε for every unit ball B Ă X .Proof. UW m ` p B q ď UW m ` p τ ´ p τ p B qqq , where τ p B q lies in a unit ball B (cid:52) Ă (cid:52) n ´ ,missing, say, the i th facet of (cid:52) n ´ . Then there is a map τ ´ p τ p B qq Ñ Z i ˆ Y m with smallfibers, defined as follows: a point x gets mapped to p π i p x q , y q P Z i ˆ Y m , where y is theimage of τ p x q under the map B (cid:52) Ñ Y m . (cid:3) Iterating this procedure (cid:96) times, we arrive at the following conclusion.
Proposition 4.6.
For a unit euclidean cube X (or a ball, or a sphere, or a torus) ofdimension (cid:96) k ´ , there is a way to blow up the metric in order to get a space X p (cid:96) q suchthat UW (cid:96) k ´ p X p (cid:96) q q ě but UW k ` (cid:96) ´ p B q À ε for every unit ball B Ă X p (cid:96) q .Proof. The original metric on X satisfies UW (cid:96) k ´ p X q ě
1, and an ε -local join structureon X gives a join map τ : X Ñ (cid:52) (cid:96) ´ k ´ . We blow up the metric of X along the (cid:52) (cid:96) ´ k ´ -direction in a carefully chosen way. Inducting on (cid:96) , we may assume that there is a metricon (cid:52) (cid:96) ´ k ´ such that no unit ball meets all its facets, and every unit ball is small in thesense of UW k `p (cid:96) ´ q´ . We use this metric to blow up the metric of X and this way get X p (cid:96) q . Arguing as in the proof of Proposition 4.5, one makes sure that every unit ball of X p (cid:96) q is small in the sense of UW k ` (cid:96) ´ . (cid:3) Proof of Theorem 1.5.
Denote (cid:96) “ r log p n ` q s , and apply Proposition 4.6 with k “ X (or a sphere, or a torus) of dimension 2 (cid:96) ´ p (cid:96) ´ q -widthbut small local (cid:96) -width. Now we can build M n as a subspace of X . (cid:3) OCAL-TO-GLOBAL URYSOHN WIDTH ESTIMATES 8 Open problems
Question . Let M be a closed riemannian surface with the first Z { β ,and with every unit ball having 1-width less than ε , for some fixed small absolute constant ε . In the optimal bound UW p M q À f p β q , what is the order of magnitude of the righthand side? It must be between β { (by Theorem 1.4) and β (by Theorem 1.3). Question . Let M n be a closed riemannian n -sphere, with every unit ball having d -width less than ε n , for some fixed small dimensional constant ε n . What is the smallest d “ d p n q such that the assumption on local width would imply UW n ´ p M q À
1? Theorem 1.5implies d p n q ď r log p n ` q s . References [1] A. Akopyan, R. Karasev, and A. Volovikov. Borsuk-Ulam type theorems for metric spaces. arXivpreprint arXiv:1209.1249 , 2012.[2] P. Alexandroff.
P.S. Urysohn: Works on topology and other fields of mathematics , volume 1. Goste-hizdat, Moscow, 1951.[3] L. E. J. Brouwer. ¨Uber den nat¨urlichen Dimensionsbegriff.
Journal f¨ur die reine und angewandteMathematik , 142:146–152, 1913.[4] M. Gromov. Filling Riemannian manifolds.
Journal of Differential Geometry , 18(1):1–147, 1983.[5] M. Gromov. Width and related invariants of Riemannian manifolds.
Ast´erisque , 163–164:93–109,1988.[6] L. Guth. Lipshitz maps from surfaces.
Geometric & Functional Analysis GAFA , 15(5):1052–1090,2005.[7] L. Guth. Volumes of balls in Riemannian manifolds and Uryson width.
Journal of Topology andAnalysis , 9(02):195–219, 2017.[8] H. Lebesgue. Sur la non-applicabilit´e de deux domaines appartenant respectivement `a des espaces `a n et n ` p dimensions. Mathematische Annalen , 70(2):166–168, 1911.[9] Y. Liokumovich, B. Lishak, A. Nabutovsky, and R. Rotman. Filling metric spaces. arXiv preprintarXiv:1905.06522 , 2019.[10] A. Nabutovsky. Linear bounds for constants in gromov’s systolic inequality and related results. arXivpreprint arXiv:1909.12225 , 2019.[11] P. Papasoglu. Uryson width and volume.
Geometric and Functional Analysis , pages 1–14, 2020.[12] J.-Y. Shi.
The Kazhdan–Lusztig cells in certain affine Weyl groups , volume 1179 of
Lecture Notes inMathematics . Springer-Verlag Berlin Heidelberg, 1986.
E-mail address : ♣ [email protected] E-mail address : ♠ [email protected] ♣♠ Dept. of Mathematics, Massachusetts Institute of Technology, 182 Memorial Dr.,Cambridge, MA 02142, USA ♣♣