Waist of maps measured via Urysohn width
WWAIST OF MAPS MEASURED VIA URYSOHN WIDTH
ALEXEY BALITSKIY ♣ AND ALEKSANDR BERDNIKOV ♠ Abstract.
We discuss various questions of the following kind: for a continuous map X → Y from a compact metric space to a simplicial complex, can one guarantee theexistence of a fiber large in the sense of Urysohn width? The d -width measures howwell a space can be approximated by a d -dimensional complex. The results of this paperinclude the following.(1) Any piecewise linear map f : [0 , m +2 → Y m from the unit euclidean ( m + 2)-cubeto an m -polyhedron must have a fiber of 1-width at least βm + m + m +1 , where β = sup y ∈ Y rk H ( f − ( y )) measures the topological complexity of the map.(2) There exists a piecewise smooth map X m +1 → R m , with X a riemannian (3 m +1)-manifold of large 3 m -width, and with all fibers being topological (2 m + 1)-balls ofarbitrarily small ( m + 1)-width. Introduction
The notion of the
Urysohn width of a compact metric space was suggested by PavelUrysohn in 1920s (and published much later by Pavel Alexandroff [3]). The d -widthmeasures how well a space can be approximated by a d -dimensional simplicial complex.A compact metric space X is said to have d -width at most w , if there is a continuous map X → Z d to a d -dimensional simplicial complex with all fibers having diameter at most w .The original definition of Urysohn was given in terms of closed coverings, and we give anoverview of different equivalent ways of defining width in Section 1.The Urysohn width of a riemannian manifold is related to other metric invariants. Forexample, the codimension 1 width does not exceed the n th root of the volume (see [13]),and bounds from above the filling radius of a manifold (see [9, Appendix 1]) and itshypersphericity (see [7, Proposition F ] or [10, Section 5]). Among the applications of theUrysohn width we mention a recent transparent proof [19] of Gromov’s systolic inequality,building on the ideas from [20, 11].The question raised in this paper is inspired by another famous Gromov’s inequality,namely the waist of the sphere theorem [8]. It says that any generic smooth map f : S n → R m , m < n , has a fiber of ( n − m )-volume at least the one of the ( n − m )-dimensional“equatorial” subsphere. The target space can be replaced by any m -manifold [14], while itis not clear if one can replace it by an m -polyhedron Y m . The only result in this directionwe are aware of is [2, Theorem 7.3], saying that any generic smooth map S n → Y n − hasa fiber of length ≥ π . A non-sharp version of the waist theorem, however, can be provedfor any m -dimensional target space by induction using the Federer–Fleming isoperimetricestimate. This type of argument is apparently goes back to Almgren, and it was used byGromov in [9] (see the exposition in [12, Section 7], which applies to any target space,or in [1, Section 7]). A discrete version of this non-sharp estimate is proven in [18]along the same lines. For riemannian metrics other than round, only the case n = 2 isunderstood [17, 4].The Urysohn width itself is a waist-type invariant, in which the size of a fiber is mea-sured via its diameter, instead of the volume. In this paper, we investigate (non-sharp)waist theorems, where the size of a fiber is measured via the Urysohn width. a r X i v : . [ m a t h . M G ] S e p AIST OF MAPS MEASURED VIA URYSOHN WIDTH 2
Prototype question.
Fix integers n, m, d . Let f : X n → Y m be a continuous map froma compact riemannian n -manifold to an m -dimensional simplicial complex. Let w be thesupremal Urysohn d -width of fibers f − ( y ) , y ∈ Y , viewed as compact metric spaces withthe extrinsic metric of X . Can one bound w from below in terms of the ( n − -width of X ? If not, can one bound w if the “topological complexity” of the fibers is restricted? It is natural to expect that the answer should be affirmative in some sense when n >m + d . When d = 1, and the first Betty number of the fibers is bounded, this is indeed thecase, as we will show in Section 3. However, in general this is far from true. In Section 4it will be shown that even for n = ( m + 1)( d − m ) + 2 m and topologically trivial fibersthe answer is negative. In a sense, this shows the failure of the notion of the d -width tomeasure the “defect of d -dimensionality”.Let us describe the answers for the first four non-trivial cases of Prototype question.These four claims are the simplest special cases of the theorems explained in this paper.(A) There is a map f : [0 , → [0 , with all fibers having arbitrarily small -width. We describe this example ([7, Example H (cid:48)(cid:48) ]) briefly. Consider an ε -fine cubicalgrid in R , and let Z be its 1-skeleton. Let Z be the 1-skeleton of the dual grid.Define f by setting f ( x ) = dist( x,Z )dist( x,Z )+dist( x,Z ) . It can be checked that every fiberΣ y = f − ( y ), y ∈ [0 , / Z with every point moving by distance (cid:46) ε ;hence it has small 1-width. Similarly, the fibers over y ∈ [1 / ,
1] are approximatedby Z .We explain how this example is generalized to higher dimensions, see Theo-rem 2.2. This might be known to experts, but we were not able to locate areference.(B) Notice that all regular fibers in the previous example have high genus. Whathappens if we bound their topological complexity? Suppose that a piecewise linear map f : [0 , → [0 , is such that all fibers f − ( y ) , y ∈ [0 , , are homeomorphic to [0 , . Then there is a fiber f − ( y ) ofUrysohn -width at least . This is the baby case of one of our main results, Theorem 3.14. Here is theidea of the proof that will be developed in Section 3. Suppose that every fiber X y = f − ( y ) has width UW d ( X y ) < c . So there are maps X y → Z y to graphs Z y whose fibers are of diameter less than c . A na¨ıve idea might be to assemble themtogether to get a map [0 , → (cid:83) Z y . If there was a nice way to interpret (cid:83) Z y as a two-dimensional space, then we would be done as long as c < UW n − ( X ). Acareful argument might try to assemble the maps X y → Z y by induction on theskeletal structure of Y , subdivided finely. The newly built intermediate maps willhave fibers with the size bounded in terms of c and the “topological complexity”of the fibers themselves.(C) The following is a special case of [7, Corollary H (cid:48) ], which we discuss in Section 2(see Theorem 2.1). Every continuous map f : X → Y from a compact metric space to a graphhas a fiber whose -width is at least the -width of X . (D) Another major result of this paper is Theorem 4.1, a family of examples of mapswith small and topologically trivial fibers; here is the simplest case. There is a map f : [0 , → [0 , with all fibers being topological -balls andhaving arbitrarily small -width. We sketch roughly the idea of the construction. The map f is just a coordinateprojection, and inside the fiber f − ( y ) (cid:39) [0 , the standard metric is modifiedas follows. Inside f − ( y ) (cid:39) [0 , consider the high-genus surface Σ y , as in the AIST OF MAPS MEASURED VIA URYSOHN WIDTH 3 example (A). In its small tubular neighborhood, blow up the metric in the normaldirection; then, squeeze the metric everywhere outside the tubular neighborhood.The result can be mapped to the suspension of Z or Z with small fibers. However,the entire space [0 , can be shown to have substantial 3-width. Acknowledgements.
We thank Larry Guth for helpful discussions.1.
Urysohn width
Everywhere in this section, X denotes a compact metric space. The diameter of a setis measured using the distance function in X : diam A = sup a,a (cid:48) ∈ A dist X ( a, a (cid:48) ). Definition 1.1.
The
Urysohn d -width of a closed subset S of a compact metric space X can be defined in either of the following ways.(UO) UW d ( S ) = inf (cid:83) U i ⊃ S sup i diam( U i ) , where the infimum is taken over all open covers of S of multiplicity at most d + 1.(UC) UW d ( S ) = inf (cid:83) C i = S sup i diam( C i ) , where the infimum is taken over all finite closed covers of S of multiplicity at most d + 1.(UM) UW d ( S ) = inf p : S → Z sup z ∈ Z diam( p − ( z )) , where the infimum is taken over all continuous maps p from S to any metrizable topologicalspace Z of covering dimension at most d .The quantity W( p ) = sup z ∈ Z diam( p − ( z )) will be called the width of the map p .The class of test spaces Z in (UM) can be narrowed down to d -dimensional simplicialcomplexes, without changing the width, as it will implicitly follow from the proof below. Proof of the equivalence of different definitions of the Urysohn width.
Denote by w c , w o , w m the width of a set S ⊂ X measured as in (UC), (UO), (UM),respectively.(UO ≤ UC) Given a finite closed covering S = (cid:83) C i , we can use compactness to argue that δ = min C i ∩ C j = ∅ dist( C i , C j ) > . Take 0 < ε < δ , and consider the open covering { U i } , where U i is the ε -neighborhoodof C i . It has the same multiplicity as the covering { C i } , and max diam U i ≤ max diam C i + 2 ε . Taking ε →
0, we get w o ≤ sup diam C i . Therefore, w o ≤ w c .(UC ≤ UM) Suppose we are given a map p : S → Z d to a metrizable space; fix a metric on Z .Recall that the width of p is defined as W( p ) = sup z ∈ Z diam( p − ( z )). Fix a smallnumber ε >
0. For each point z ∈ p ( S ) one can find radius r ( z ) > V r ( z ) ( z ), the r ( z )-neighborhood of z , has diameter smaller thanW( p ) + ε . Here we usedlim r → diam( p − ( V r ( z ))) = diam( p − ( z )) . By definition of dimension (and compactness), there is a finite open covering { V i } of p ( S ), refining { V r ( z ) ( z ) } , and with multiplicity at most d + 1. It follows fromLebesgue’s number lemma that there is a closed covering { D i } with D i ⊂ V i . Thenthe closed sets C i = p − ( D i ) have diameter less than W( p ) + ε , and cover S withmultiplicity at most d + 1. Repeating this with arbitrarily ε , one gets w c ≤ W( p ).Since this is true for all p , we conclude w c ≤ w m . AIST OF MAPS MEASURED VIA URYSOHN WIDTH 4 (UM ≤ UO) Given an open covering S ⊂ (cid:83) U i (which we can assume finite by compactness)with multiplicity d + 1, consider the mapping to its nerve ϕ : S → N d , associated to any subordinate partition of unity. The preimage of every point isentirely contained in some U i , hence W( ϕ ) ≤ sup diam U i . Therefore, w m ≤ w o . (cid:3) Definition 1.1 was given for a closed set S . We adopt the following convention: thewidth of a (not necessarily closed) set S ⊂ X is defined in terms of open coverings, (UO). Lemma 1.2.
Let f : X → Y be a continuous map from a compact metric space X to ametrizable topological space Y . The function y (cid:55)→ UW d ( f − ( y )) is upper semi-continuous for any d . Namely, UW d ( f − ( y )) ≥ lim sup y (cid:48) → y UW d ( f − ( y (cid:48) )) . Proof.
If a fiber f − ( y ) is covered by open sets U i ⊂ X , with diameters < UW d ( f − ( y ))+ ε and multiplicity at most d +1, then these open sets in fact cover neighboring fibers f − ( y (cid:48) )as well. (cid:3) Waist of maps with arbitrary fibers
Theorem 2.1 ([7, Corollary H (cid:48) ]) . Let X be a compact metric space, and let Y be ametrizable topological space of covering dimension m . Every continuous map f : X → Y has a fiber f − ( y ) of d -width UW d ( f − ( y )) ≥ UW n − ( X ) , where n = ( m + 1)( d + 1) .Proof. The assumptions on Y m imply that UW d ( f − ( y )) = inf open V (cid:51) y UW d ( f − ( V )). Sup-posing the contrary to the statement of the theorem, and pulling back a fine open coverof Y , we obtain an open cover { U i } of X of multiplicity at most m + 1, such thatUW d ( U i ) < u := UW n − ( X ) for all i . It follows from the definition of the d -width thatevery U i admits an open cover U i = (cid:83) j U ij of multiplicity at most d + 1, with diam U ij < u .The cover { U ij } of X has multiplicity at most ( m + 1)( d + 1), and it can be assumed finite(by compactness), so we get UW n − ( X ) < u , which is absurd. (cid:3) The relation between dimensions n, m, d in Theorem 2.1 is optimal, as the followingresult (generalizing example (A) from the introduction) shows.
Theorem 2.2.
Let n = ( m + 1)( d + 1) − , and let ε > be any small number. Thereexists a continuous map f : B n → (cid:52) m from the unit euclidean n -ball to the m -simplex,whose fibers all have Urysohn d -width less than ε .Remark . It is easy to show that UW n − ( B n ) >
0. This can be deduced from theLebesgue covering theorem [16, 6], or from the Knaster–Kuratowski–Mazurkiewicz the-orem [15]. In fact, the exact value UW n − ( B n ) = (cid:113) n +2 n is known (see [22, pp. 84–85, 268] or [2, Remark 6.10]).The crucial tool used in the proof of Theorem 2.2 is the local join representation of R n ,which will be also used in Section 4. Lemma 2.4 (cf. [5, Lemma 4.1]) . Fix ε > . There is a locally finite triangulation of R n by simplices of diameter < ε , admitting a nice coloring: the vertices receive colors , , . . . , n so that each simplex receives all distinct colors. AIST OF MAPS MEASURED VIA URYSOHN WIDTH 5
Proof.
In fact, there is such a triangulation with simplices congruent to one another,via the reflection in the facets. Such a triangulation can be obtained from the type A root system and the corresponding affine Coxeter hyperplane arrangement (see [21,Chapter 6]). (Of course, simpler constructions are also possible.) (cid:3) Definition 2.5 (cf. [5, Definition 4.2]) . Let n = ( m + 1)( d + 1) −
1, and triangulate R n by ε -small simplices, as in Lemma 2.4. Define Z i , 0 ≤ i ≤ m , to be the union of all simplicesof the triangulation colored by colors ( d + 1) i through ( d + 1) i + d . We say that R n is the ε -local join of d -dimensional complexes Z , . . . , Z m .The name is justified by the following observation: every (top-dimensional) simplex σ of the triangulation can be written as the join ( σ ∩ Z ) ∗ . . . ∗ ( σ ∩ Z m ); that is, any point x ∈ σ can be written as x = m (cid:88) i =0 t i z i , where z i ∈ σ ∩ Z i , t i ≥ , m (cid:88) i =0 t i = 1 . The coefficients t i are determined uniquely, giving a well-defined join map τ : R n → (cid:52) m = (cid:40) ( t , . . . , t m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t i ≥ , m (cid:88) i =0 t i = 1 (cid:41) . Note that Z i = τ − ( v i ), where v , . . . , v m are the vertices of (cid:52) m . For each vertex v i , denotethe opposite facet of (cid:52) m by v ∨ i . For each complex Z i , its dual ( md + m − Z ∨ i = τ − ( v ∨ i ). There are natural retractions π i : R n \ Z ∨ i → Z i , defined by sending x = m (cid:80) i =0 t i z i ∈ σ to z i ∈ σ ∩ Z i ; they are well-defined since t i (cid:54) = 0whenever x / ∈ Z ∨ i . Note that π i moves each point by distance < ε . Proof of Theorem 2.2.
Represent R n as the ε/ -local join of d -dimensional complexes Z , . . . , Z m ; let τ : R n → (cid:52) m be its join map. Take f to be the restriction of τ onthe unit ball B n . Let us check that the d -width of any fiber F = f − ( t , . . . , t m ) is small.Fix any i for which t i (cid:54) = 0. The (restricted) retraction map π i | F : F → Z i has fibers ofdiameter < ε , so we are done. (cid:3) Waist of maps with fibers of bounded complexity
This section generalizes example (B) from the introduction. The main result, Theo-rem 3.14, which in particular implies the following waist inequality.
Any piecewise linear map f : X m +2 → Y m from a riemannian ( m + 2) -polyhedron to an m -polyhedron must have a fiber of -width at least UW m +1 ( X )2 βm + m + m +1 , where β = sup y ∈ Y rk H ( f − ( y )) measures the topological complexity of the map. PL maps of polyhedra.
We use the word polyhedron to refer to a topological spaceadmitting a structure of a finite simplicial complex (together with rectilinear structure oneach simplex), though we do not usually specify this structure. We say a continuous map X → Y of polyhedra is a piecewise linear map , or a PL map , if it is simplicial for somefine simplicial structures on X and Y .We use the words riemannian polyhedron for a polyhedron endowed with a smoothriemannian metric on each maximal simplex, so that the metrics on adjacent simplicesmatch in restriction to their common face. AIST OF MAPS MEASURED VIA URYSOHN WIDTH 6
For a map f : X → Y , we sometimes denote the preimage f − ( A ) of a subset A ⊂ Y by X A , if there is no confusion and f is understood from the context. If X and A ⊂ Y are polyhedra, and f is a PL map, then X A is naturally a polyhedron. If additionally X is riemannian, then X A is riemannian as well. Definition 3.1.
We measure the topological complexity using the first Betty number. Fora space X , we set tc( X ) = rk H ( X ). For a map f : X → Y , we set tc( f ) = sup y ∈ Y tc( X y ).For example, if X is a connected oriented surface then tc( X ) equals twice the genusplus the number of punctures/unbounded ends. Lemma 3.2.
Every PL map f : X → Y of polyhedra satisfies the following regularityassumption. Fix a simplicial structure on Y for which f is simplicial. Fix a simplex (cid:52) ⊂ Y (of any dimension), and let ˚ (cid:52) be its relative interior. Then one can pick a PLmap Ψ (cid:52) : (cid:52) × Σ (cid:52) → X (cid:52) , for some polyhedron Σ (cid:52) , such that • Ψ (cid:52) is fibered over (cid:52) : (cid:52) × Σ (cid:52) Ψ (cid:52) (cid:47) (cid:47) projection (cid:37) (cid:37) X (cid:52) f (cid:15) (cid:15) (cid:52) ⊂ Y • the restriction Ψ (cid:52) | ˚ (cid:52)× Σ (cid:52) : ˚ (cid:52) × Σ (cid:52) → X ˚ (cid:52) is a homeomorphism making f a fiber bundle over ˚ (cid:52) .Proof. For Σ (cid:52) , take the fiber over the center of (cid:52) , and the rest can be verified easily. (cid:3)
Connected maps.Definition 3.3.
Let f : X → Y be a continuous map of topological spaces. It is called connected if the fibers f − ( z ), z ∈ Z , are (nonempty and) path-connected. Every map f ,connected or not, cannot be factored as X (cid:101) f → (cid:101) Y → Y, with (cid:101) f connected, and with (cid:101) Y being the space of path-connected components of the fibersof f (topologized by the finest topology making (cid:101) f continuous). The map (cid:101) f is called the connected map associated to f .If f is a PL map of polyhedra, then (cid:101) f is also PL, and (cid:101) Y is a polyhedron having thesame dimension as f ( X ). Lemma 3.4.
Let f : X → Y be a connected PL map of polyhedra.(1) If Y is connected then X is connected.(2) The induced map f ∗ : H ( X ) → H ( Y ) is onto.Proof. Let γ : [0 , → Y be a path in the base. Fix a simplicial structure of Y forwhich f is simplicial. Let us build a path (cid:101) γ : [0 , → X covering γ in the followingweak sense: there is a monotone reparametrization map r : [0 , → [0 ,
1] such that f ( (cid:101) γ ( t )) = γ ( r ( t )). First, split γ into arcs each of which belongs to a single cell of Y .Without loss of generality, there are finitely many of these arcs (this can be achieved byhomotoping γ slightly, while fixing endpoints). For each such arc [ t (cid:48) , t (cid:48)(cid:48) ] → Y , one can lift γ by Lemma 3.2. If γ is lifted independently over [ t (cid:48) , t ] and [ t, t (cid:48)(cid:48) ], the two lifted patches AIST OF MAPS MEASURED VIA URYSOHN WIDTH 7 can be connected inside the fiber f − ( γ ( t )). This is how (cid:101) γ can be built. For the firstassertion of the lemma, having two points x, x (cid:48) ∈ X , one can connect f ( x ) to f ( x (cid:48) ) in thebase, and lift the path as above. The endpoints of the lifted path can be connected to x and x (cid:48) in the corresponding fibers. This proves that X is connected. For the secondassertion, one can notice that if γ were a closed loop in the base, the lifted (cid:101) γ could bemade closed as well. (cid:3) Foliations.Definition 3.5.
Let Σ be a topological space. We use the word foliation to denote acontinuous map p : Σ → Z to a graph (finite 1-dimensional simplicial complex), in thesense that Σ is foliated by the fibers p − ( z ), z ∈ Z (the leaves ). Definition 3.6.
Let Σ be a polyhedron. We say a foliation p : Σ → Z is simple if it is aconnected PL map.Lemma 3.4 shows that a simple foliation induces an epimorphism in the first homology;in this case, tc( Z ) is bounded by tc(Σ).For a foliation p of a compact metric space Σ, recall the notation W( p ) = sup z ∈ Z diam p − ( z )for its width. Lemma 3.7. If Σ is a riemannian polyhedron, any its foliation of width < can be“simplified” while keeping its width < .Proof. Let p : Σ → Z be a foliation of width <
1. Subdivide Z finely so that thepreimage of the open star S v of every vertex v ∈ Z has diameter <
1. Use the simplicialapproximation theorem to approximate p by a simplicial (for some subdivision of Σ) map p (cid:48) such that for each x ∈ Σ, p (cid:48) ( x ) belongs to the minimal closed cell of Z containing p ( x ).It implies that for each vertex v ∈ Z , ( p (cid:48) ) − ( v ) ⊂ p − ( S v ), so p (cid:48) has width < p (cid:48) by the associated connected map (cid:101) p (cid:48) (which is also PL), we arrive atthe situation where the leaves ( (cid:101) p (cid:48) ) − ( z ) are (nonempty and) connected for all z ∈ Z , andhave diameter < (cid:3) Interpolation lemma.Definition 3.8.
Let Σ be a topological space, and let p : Σ → Z , p : Σ → Z be itsfoliations. An interpolation between these is a family of foliations p t : Σ → Z t , t ∈ [0 , • There are 2-dimensional cell complex Z [0 , together with a parametrization map π : Z [0 , → [0 , π − ( t ) = Z t ⊂ Z [0 , . • There is a continuous map P : [0 , × Σ → Z [0 , fibered over [0 , p t when restricted over { t } :[0 , × Σ P (cid:47) (cid:47) projection (cid:37) (cid:37) Z [0 , π (cid:15) (cid:15) [0 , { t } × Σ p t (cid:47) (cid:47) projection (cid:38) (cid:38) Z t ⊂ Z [0 , π (cid:15) (cid:15) { t } Lemma 3.9.
Let Σ be a riemannian polyhedron of topological complexity β = tc(Σ) , andlet p : Σ → Z , p : Σ → Z be simple foliations. It is possible to interpolate betweenthem through simple foliations of width at most ( β + 2) W( p ) + ( β + 1) W( p ) . The open star of a vertex of a simplicial complex is the union of the relative interiors of all facescontaining the given vertex. In a graph, the open star of a vertex is the vertex itself together with allincident open edges.
AIST OF MAPS MEASURED VIA URYSOHN WIDTH 8
We only outline the proof, since a more general statement will be proved in the nextsubsection. However, this outline illustrates the main method of this section.We can assume Σ connected (by dealing with each connected component separately).
Lemma 3.10.
Given a (finite) connected graph Z (viewed as a topological space), thereis a filtration by closed subspaces Z ( t ) ⊂ Z , t ∈ [0 , , such that • Z ( t ) = α − ([0 , t ]) , for some continuous function α : Z → [1 / , ; • Z (1 / = α − (1 / consists of a single point; • every preimage α − ( t ) , t ∈ [1 / , , consists of finitely many points (informally,this condition says that Z ( t ) depends continuously on t ).One can also consider a satellite filtration by open subspaces ˚ Z ( t ) = (cid:83) t (cid:48) ∈ [0 ,t ) Z ( t (cid:48) ) = α − ([0 , t )) .Proof. Such a filtration can be constructed using α ( z ) = dist Z ( z , z )2 sup z (cid:48) ∈ Z dist Z ( z , z (cid:48) ) + 1 / z ∈ Z and any metrization of Z . (cid:3) The graph Z is connected, since Σ is connected, and p is simple (hence surjective).Filter Z as in Lemma 3.10: Z (0)1 ⊂ . . . ⊂ Z ( t )1 ⊂ . . . ⊂ Z (1)1 , t ∈ [0 , p and p through foliations p t : Σ → Z t , which can be roughly described asfollows. To get a picture of p t , first you draw the fibers of p over Z ( t )1 . Then in theremaining room we draw the fibers of p (their parts that fit). The resulting picture isinterpreted as a foliation by connected leaves, and we call it p t (see Figure 1).Let us rigorously describe the space of leaves Z t and the foliation map p t . • Define Z ( t )0 , t ∈ [0 , Z such that p − ( Z ( t )0 ) ∪ p − ( ˚ Z ( t )1 ) = Σ; in other words, Z ( t )0 = p (cid:16) Σ \ p ( ˚ Z ( t )1 ) (cid:17) . We write Σ ( t ) = Σ \ p ( ˚ Z ( t )1 ) for short. • The map p | Σ ( t ) : Σ ( t ) → Z ( t )0 might not have all fibers connected, so we factor itthrough its associated connected map:Σ ( t ) (cid:101) p ( t )0 → (cid:101) Z ( t )0 → Z ( t )0 . • The graph Z t is defined as (cid:16) (cid:101) Z ( t )0 (cid:116) Z ( t )1 (cid:17) / t ∼ , where t ∼ is the following equivalence relation. Let us write z t ≈ z (cid:48) if z ∈ (cid:101) Z ( t )0 , z (cid:48) ∈ Z ( t )1 , and ( (cid:101) p ( t )0 ) − ( z ) intersects p − ( z (cid:48) ). Define t ∼ to be the transitive closureof t ≈ . There are natural maps ι ( t )0 : (cid:101) Z ( t )0 → Z t and ι ( t )1 : Z ( t )1 → Z t . • The map p t : Σ → Z t is defined as p t ( x ) = (cid:40) ι ( t )1 ( p ( x )) , if p ( x ) ∈ Z ( t )1 ι ( t )0 (cid:16)(cid:101) p ( t )0 ( x ) (cid:17) , otherwise . Observe that for t = 0 , p and p . AIST OF MAPS MEASURED VIA URYSOHN WIDTH 9
Figure 1.
Interpolation between foliations. Each rectangle represents afoliation of Σ, given by a map to a graph. The foliations p and p arepictured in green and red, respectivelyThis describes the intermediate foliations p t , but in order to describe the interpolationcompletely we also need to explain how the graphs Z t assemble into a 2-complex Z [0 , , AIST OF MAPS MEASURED VIA URYSOHN WIDTH 10 and how the maps p t assemble into a continuous map P : [0 , × Σ → Z [0 , . We do notgive these details here, because a more general construction will be explained in the nextsubsection.To finish the proof, we need to bound the size of the fibers of p t . Why could it bepossibly large? Because in the process of interpolating some vertices of the target graphmerged under the t ∼ -identification, so multiple fibers of p and p might have been united.Consider a fiber of p t . For this fiber, consider the longest chain of identifications z ≈ z (cid:48) ≈ z ≈ z (cid:48) ≈ . . . with z j ∈ (cid:101) Z ( t )0 , and with z (cid:48) j ∈ Z ( t )1 all distinct. Suppose it has more than 1+tc(Σ) elementsof Z ( t )1 . To every subchain z (cid:48) j ≈ z j ≈ z (cid:48) j +1 assign a loop γ j ⊂ Σ in the following way. By thedefinition of t ≈ , there is an arc inside ( (cid:101) p ( t )0 ) − ( z j ) connecting some two points x ∈ p − ( z (cid:48) j )and y ∈ p − ( z (cid:48) j +1 ), such that only the endpoints x and y are not in the interior of Σ ( t ) . Onthe other hand x and y belong to the set p − ( Z ( t )1 ), which is connected by Lemma 3.4, sothere is another arc between x and y completely avoiding the interior of Σ ( t ) . Those twoarcs form a loop γ j , which represents a non-trivial element of H (Σ), since it projects toa non-trivial loop in Z ( t ) . If we are given more than tc(Σ) cycles in Z ( t ) , there must be arelation between them in H ( Z ( t ) ) (recall that tc( Z ( t ) ) ≤ β by Lemma 3.4). It follows thatsome z (cid:48) j repeats in the chain, which proves such a chain has at most 1 + tc(Σ) elements of Z ( t )1 , hence at most 2 + tc(Σ) elements of (cid:101) Z ( t )0 . We conclude that the diameter of a fiberof p t is at most ( β + 2) W( p ) + ( β + 1) W( p ). The proof outline is finished.3.5. Parametric interpolation lemma.Definition 3.11.
Let Σ be a topological space, and let π : Z K → K be a map ofpolyhedra such that every fiber is a (nonempty and) connected graph. A continuousmap P : K × Σ → Z K is called a parametric foliation over K , or a family of foliationsparametrized by K , if the composition π ◦ P : K × Σ → K is the projection onto the firstfactor: K × Σ P (cid:47) (cid:47) projection (cid:36) (cid:36) Z Kπ (cid:15) (cid:15) K We call Z K the space of leaves , and π the parametrization map . For s ∈ K , the restriction P | { s }× Σ can be viewed as a foliation p s : Σ → π − ( s ), and we think of P as the family offoliations p s parametrized by s ∈ K . We say that P is simple it is PL and connected.For a parametric foliation P : K × Σ → Z K of a metric space Σ, we keep using thenotation W( P ) = sup z ∈ Z K diam P − ( z ) for the width. Definition 3.12.
Let Σ be a topological space.(1) Let P : K × Σ → Z K and P : K × Σ → Z K be parametric foliations overthe same complex K . An interpolation between them is a parametric foliation P : ([0 , × K ) × Σ → Z [0 , × K over the prism [0 , × K , restricting to P j on( { j } × K ) × Σ, j = 0 , P : K × Σ → Z K be a family of foliations, and let p : Σ → Z be anotherfoliation. An interpolation between them is a parametric foliation P : ( CK ) × Σ → Z CK over the cone CK = ([0 , × K ) / ( { } × K ), restricting to P over the base { } × K of CK , and to p over the apex of CK . AIST OF MAPS MEASURED VIA URYSOHN WIDTH 11
We are in position to prove the principal lemma of this section.
Lemma 3.13 (Parametric interpolation) . Let Σ be a riemannian polyhedron of topologicalcomplexity β = tc(Σ) . Let P K : K × Σ → Z K be a family of simple foliations over a d -dimensional complex K , and let p : Σ → Z be a simple foliation. It is possible tointerpolate between P K and p via a simple family CK × Σ → Z CK of width at most ( β + 2) W( P ) + ( β + 1) W( p ) .Proof. We can assume Σ connected (by dealing with each connected component sepa-rately).The parametric foliation P K splits into simple foliations p s : Σ → Z s , where Z s = π − ( s ), s ∈ K , π : Z K → K is the parametrization of the foliation base.The proof idea is simple: for each s ∈ K , interpolate between p s and p as in Lemma 3.9,and make sure that the interpolation depends nicely on s , in order to assemble themaltogether to a parametric interpolation. The details are pretty technical, and now wewrite them out.The graph Z is finite and connected, since Σ is compact and connected, and p issimple (hence surjective). Filter Z as in Lemma 3.10: Z (0)1 ⊂ . . . ⊂ Z ( t )1 ⊂ . . . ⊂ Z (1)1 , t ∈ [0 , P K and p via a family P : CK × Σ → Z CK to bedescribed. With a little abuse of notation, we use coordinates ( t, s ) ∈ [0 , × K on CK ,with a convention that all points (1 , s ) are identified with the apex of CK . The restriction P | { ( t,s ) }× Σ is a foliation p ( t,s ) : Σ → Z ( t,s ) , which can be pictured as follows. First, drawthe fibers of p over Z ( t )1 ; then fill in the remaining room with the fibers of p s (with theirparts that fit). The resulting picture is interpreted as a foliation by connected leaves, andwe call it p ( t,s ) .We now describe P : CK × Σ → Z CK formally. • Define P : [0 , × K × Σ → [0 , × Z K ( t, s, x ) (cid:55)→ ( t, p s ( x )) P : CK × Σ → CK × Z ( c, x ) (cid:55)→ ( c, p ( x )) • Define Z = (cid:91) ( t,s ) ∈ CK Z ( t )1 ⊂ CK × Z where we think of Z ( t )1 as sitting in { ( t, s ) } × Z . The interior of Z is˚ Z = (cid:91) ( t,s ) ∈ CK ˚ Z ( t )1 ⊂ CK × Z . Define S = ([0 , × K × Σ) \ P − (˚ Z ) ⊂ [0 , × K × Σand Z = P ( S ) ⊂ [0 , × Z K . • The map P | S might not be connected, so we factor it through its associatedconnected map: S (cid:101) P → (cid:102) Z → Z . The space (cid:102) Z inherits t - and s -coordinates from Z . AIST OF MAPS MEASURED VIA URYSOHN WIDTH 12 • The space of leaves is Z CK = (cid:16) (cid:102) Z (cid:116) Z (cid:17) / ∼ , where ∼ is the following equivalence relation. Let us write z ≈ z (cid:48) if z ∈ (cid:102) Z , z (cid:48) ∈Z , and (cid:101) P − ( z ) intersects P − ( z (cid:48) ), as subsets of CK × Σ. (Recall our conventionfor coordinates in a cone, in which [0 , × K ⊂ CK .) Define ∼ to be the transitiveclosure of ≈ . There are natural maps ι : (cid:102) Z → Z CK and ι : Z → Z CK . • The parametric foliation P is defined as P : CK × Σ → Z CK ξ (cid:55)→ (cid:40) ι ( P ( ξ )) , if P ( ξ ) ∈ Z ι (cid:16) (cid:101) P ( ξ ) (cid:17) , otherwise . It is easy to see that P indeed interpolates between P K and p .Clearly, P is connected. It is rather technical but straightforward to make sure that P is PL.The analysis of the width was already done in Lemma 3.9. Any foliation from the family P belongs to an interpolation between certain p s , s ∈ K , and p , as in the constructionof Lemma 3.9. Therefore, W( P ) ≤ ( β + 2) W( P ) + ( β + 1) W( p ). (cid:3) Waist of a PL map.
Finally, we are ready to prove the main theorem of thissection.
Theorem 3.14.
Let f : X → Y m be a PL map from a riemannian polyhedron X toan m -dimensional polyhedron Y . Let β = tc( f ) be its topological complexity, that is, β = sup y ∈ Y tc( f − ( y )) . Then there is a fiber X y = f − ( y ) of Urysohn width UW ( X y ) ≥ c ( m, β ) UW m +1 ( X ) , for some positive constant c depending only on m and β .Proof. Replacing f with its associated connected map, we can assume that f is connected.Even if f is not a fiber bundle, still locally this is almost the case by Lemma 3.2. Foreach simplex (cid:52) ⊂ Y in a fine triangulation of Y (of any dimension), the map f can be“almost” trivialized over (cid:52) via a PL mapΨ (cid:52) : (cid:52) × Σ (cid:52) → X (cid:52) , for some polyhedron Σ (cid:52) ; this map is a genuine trivialization over the open simplex ˚ (cid:52) ,the relative interior of (cid:52) . For y ∈ (cid:52) , this map induces a metric on Σ (cid:52) , the pullback ofthe piecewise riemannian metric on X y ; we denote the corresponding distance functionby d (cid:52) y . Refining the triangulation of Y if needed, we can assume that all metrics d (cid:52) y over y ∈ (cid:52) are ε -close to one another in the following sense: the “layers” Ψ (cid:52) ( (cid:52) × { x } ) havediameter less than ε/ x ∈ Σ (cid:52) , hence for any x, x (cid:48) ∈ Σ (cid:52) and any y, y (cid:48) ∈ (cid:52) we have | d (cid:52) y ( x, x (cid:48) ) − d (cid:52) y (cid:48) ( x, x (cid:48) ) | ≤ ε .Suppose that UW ( X y ) < w , for all y ∈ Y , with w = c ( m, β ) UW d +1 ( X ) to bespecified later. We get a foliation of X y of width less than w , which can be assumedsimple without loss of generality. The idea of the proof is to pick a dense discrete set ofpoints in Y , and use those foliations to build a map F : X → Z m +1 of controlled width.This is done inductively on skeleta of Y .At the zeroth step, for each vertex v of Y , pick a simple foliation F v : X y → Z v of widthless than w .At the k th step, 1 ≤ k ≤ m , we assume that we already defined F k − : X Y ( k − → Z Y ( k − , over the ( k − Y , of width less than w k − , and we need to extend AIST OF MAPS MEASURED VIA URYSOHN WIDTH 13 it over Y ( k ) . Take a k -simplex (cid:52) ⊂ Y , and consider the corresponding “trivialization”Ψ (cid:52) : (cid:52) × Σ (cid:52) → X (cid:52) . Pick a point y in the relative interior of (cid:52) , and a simple foliation p y of Σ (cid:52) of d (cid:52) y -width < c . We would like to use Lemma 3.9 to build a parametric foliation P (cid:52) : (cid:52) × Σ (cid:52) → Z (cid:52) interpolating between p y : Σ (cid:52) and the family of foliations ∂ (cid:52) × Σ (cid:52) Ψ (cid:52) → X ∂ (cid:52) F k − → Z Y ( k − (here ∂ denotes the relative boundary). In order to apply that lemma, we need to fix ametric on Σ (cid:52) , so we use d (cid:52) y (recall that the are all ε -close). We get a map P (cid:52) : (cid:52) × Σ (cid:52) → Z (cid:52) width less than ( β + 2) w k − + ( β + 1) c . The desired map F (cid:52) : X (cid:52) → Z (cid:52) that we arelooking for is already defined over ∂ (cid:52) , so we specify it over ˚ (cid:52) : X ˚ (cid:52) Ψ − (cid:52) → ˚ (cid:52) × Σ P (cid:52) → Z (cid:52) . The resulting map F (cid:52) is continuous and has width less than w k = ( β + 2) w k − + ( β + 1) c + ε. As ε →
0, the solution of this recurrence tends to w k = (2( β + 2) k − w . Therefore, UW m +1 ( X ) ≤ (2( β + 2) m − c ( m, β ) UW m +1 ( X ). Hence, for each c < β +2) m − , there is a fiber X y ( c ) of width at least c UW m +1 ( X ). Finally, send c → β +2) m − ,pick a limit point ¯ y of { y ( c ) } , and note that UW ( X ¯ y ) ≥ UW m +1 ( X )2( β +2) m − by upper semi-continuity of width (Lemma 1.2). (cid:3) This proof gives the value c = β +2) m − . Let us give a more careful estimate, showingthat one can do much better, namely take c = βm + m + m +1 . Lemma 3.15.
Let Σ be a riemannian polyhedron of topological complexity β = tc(Σ) .Let p j : Σ → Z j , j = 0 , , . . . , m , be simple foliations of width at most . Suppose aparametric foliation P : (cid:52) × Σ → Z (cid:52) over an m -simplex (restricting to p j over the j th vertex of (cid:52) ) is obtained by inductively applying Lemma 3.13; that is, first interpolatebetween p and p , then between the result and p , and so on. Then the width of P is atmost βm + m + m + 1 .Proof. Recall the idea behind the construction in Lemma 3.13. A foliation of family P can be pictured as follows. First, draw the fibers of p m over Z ( t m ) m , a subgraph of Z m (connected or empty). In the remaining room, draw (the parts of) the fibers of p m − over Z ( t m − ) m − , a subgraph of Z m − . Continue in the same fashion. At the last step, fill in theremaining room with (the parts of) the fibers of p . The touching fibers of different p j getmerged to a single fiber of the resulting foliation, which we call p : Σ → Z . We assumethat none of the graphs Z ( t j ) j is empty (otherwise the result follows by induction on m ).Denote by Σ j the closed subset of Σ covered by the fibers of p j , . . . , p m (in particular,Σ = Σ). Notice that for 1 ≤ j ≤ m , Σ j consists of at most m − j + 1 connectedcomponents, since each set p − j ( Z ( t j ) j ) is connected by Lemma 3.4. From the long exactsequence . . . → H (Σ) → H (Σ , Σ j ) → ˜ H (Σ j ) → . . . one gets that rk H (Σ , Σ j ) ≤ rk H (Σ) + rk ˜ H (Σ j ) ≤ β + m − j .We need to bound the number of fibers in a merged chain. Fix two points x, y ∈ Σin a single fiber p − ( z ), and connect them by a path α : [0 , → Σ inside this fiber.For each t , notice which of the regions Σ j \ Σ j +1 the point α ( t ) belongs to, and write AIST OF MAPS MEASURED VIA URYSOHN WIDTH 14 down the corresponding index J ( t ) (here Σ m +1 is assumed empty). We have a piecewiseconstant function J : [0 , → { , , . . . , m } . Denote the number of its discontinuities by D ; without loss of generality, D is finite. Note that dist( x, y ) ≤ D + 1. We will transform α (while keeping it inside the same fiber of p , and fixing its endpoints x, y ) to achieve D ≤ (2 β + m + 1) m . Consider the following property, which α may or may not enjoy. Desired property.
For 1 ≤ j ≤ m , we say that a path α is j -nice if the superlevel set I ≥ j = { t ∈ [0 , | J ( t ) ≥ j } consists of at most β + m − j + 1 components. We say that α is nice if it j -nice for all 1 ≤ j ≤ m .Suppose first α is not nice, and take the smallest index j such that α is not j -nice.Mark a point in each component of I ≥ j , so that we have marked points t , . . . , t k , k >β + m − j + 1. Each arc α ([ t i , t i +1 ]) represents an element of H (Σ , Σ j ). Recall thatrk H (Σ , Σ j ) ≤ β + m − j . It follows that some two points α ( t i ), α ( t i (cid:48) ) can be connectedinside p − ( z ) ∩ Σ j . Replace α ([ t i , t i (cid:48) ]) with this new curve. We decreased the number ofcomponents of I ≥ j . Proceeding in the same fashion, we can make α j -nice. Repeatingthis procedure for larger j if needed, we make α nice.Now that α is nice, we bound its number D of discontinuities. Clearly, D is boundedby the total number of the endpoints of all I ≥ j . Since α is nice, D ≤ m (cid:88) j =1 β + m − j + 1) = (2 β + m + 1) m. (cid:3) This analysis shows that the constant c in Theorem 3.14 can be taken equal βm + m + m +1 .We remark that the improved bound still does not seem sharp. In Gromov’s example(example (A) of the introduction) the dependence on β is of order β − / while our boundonly guarantees β − ,4. Fibered manifolds with topologically trivial fibers of small width
The following result generalizes example (D) from the introduction.
Theorem 4.1.
For any non-negative integers m , k , and any ε > , there exists a map X → Y such that • X = F × Y , and the map is the trivial fiber bundle F × Y → Y ; • Y and F are closed topological balls of dimensions m and mk + m + k , respectively; • X is endowed with a riemannian metric with UW n − ( X ) ≥ , where n = dim X = mk + 2 m + k ; • for each y ∈ Y , the fiber X y (cid:39) F has UW k + m ( X y ) < ε .Remark . Consider the trivial bundle X (cid:48) = F (cid:48) × Y (cid:48) → Y (cid:48) , where Y (cid:48) is the euclidean m -ball of radius ∼ ε , and F (cid:48) is the euclidean ( mk + m + k )-ball of radius ∼ ε . Thebundle X in the theorem will be constructed in a way so that near its boundary X willlook exactly like X (cid:48) . This allows to modify the construction to make X a closed manifold(e.g., a sphere or a torus), or to take the connected sum with other fibrations, etc. Proof.
To start with, take Y = R m , F = R mk + m + k , X = F × Y = R mk +2 m + k , and ignorefor the moment that they are not closed balls. Let p : X → Y and p F : X → F be theprojection maps. We start from the euclidean metric on X , modify it, and then cut X tomake it compact. Then the (restricted) map p will be the one we are looking for.On the first factor F = R mk + m + k , consider the structure of the ε -local join of k -dimensional complexes Z , . . . , Z m in the sense of 2.5. The construction is based on theidea of blowing up the metric in between the Z i (cf. [5, Subsection 4.2], where a similar AIST OF MAPS MEASURED VIA URYSOHN WIDTH 15 idea is used). Let τ : F → (cid:52) m be the join map. We think of (cid:52) m as sitting in R m with the center at the origin, scaled so that the inradius of (cid:52) m equals 3. Consider the“perturbation of the projection via the join map” p τ : X → Y, p τ = p − τ ◦ p F . One can observe that the fibers of p τ are PL homeomorphic to F , and it will be usefulto look at X in the coordinates Φ = ( p F , p τ ). Namely, Φ : X → X is the map given byΦ( x ) = ( p F ( x ) , p τ ( x )) ∈ F × Y = X .Let φ : [0 , + ∞ ) → R be a monotone cut-off function that equals 1 on [0 ,
1] and 0 on[1 . , ∞ ). Denote by φ kr : R k → R an r -sized bump function φ kr ( x ) := φ ( | x | /r ); here | · | is the euclidean norm in R k . Let g euc X , g euc Y be the standard metrics on the correspondingeuclidean spaces, viewed as symmetric 2-forms. To define a new metric on X we take g euc X ,blow it up transversely to (cid:101) p − ( x ) for x close to the origin of R m , and squeeze everywhereelse. Formally, g X = Φ ∗ g (cid:48) X , where g (cid:48) X = εg euc X + (1 − ε )( φ m g euc Y ) × ( φ mk + m + k g euc F ) . In order for this to be well-defined, one might want to approximate Φ by a smooth map.From now on, we assume that X is endowed with metric g X . To make X compact, onecan replace it by its subset B g euc F (0) × B g euc Y m (0). Radius 3 + m here is chosen so that the2 . (cid:52) m is covered by p ( X ). We write X (cid:48) for the space Φ( X ) with metric g (cid:48) X ; clearly, X and X (cid:48) are isometric.Figure 2 depicts the case m = 1 , k = 0: there, X = R is sliced by lines p − ( y ) (boldblack curves in the figure), each of which is the local join of a green point set Z and ablue point set Z . On the left, the geometry of g X is depicted by stretching X along thevertical direction, so that it corresponds to the value of p τ . On the right, one sees X inthe coordinates Φ = ( p f , p τ ), with the pinching in the region where | p τ ( x ) | > g X . To see that UW n − ( X ) ≥ B g (cid:48) X (0) is just the usual euclidean ball, and its width is > p have small width. Consider a fiber X y = p − ( y ), y ∈ Y , and the restriction of g X on it. It equals εg euc F plus a term supported in τ − ( B g euc Y . ( y )).The ball B g euc Y . ( y ) does not reach one of the faces v ∨ i of (cid:52) m . We would like to use theretraction π i (as in the discussion after Definition 2.5) to map p − ( y ) to Z i ; this is notpossible for the points in the dual complex Z ∨ i , which is entirely contained in the squeezedzone, so we will not lose much if we just send it to a single point. Here is the map witnessingUW k + m ( X y ) (cid:46) ε : X y (cid:39) F → ( Z i × (cid:52) m ) / ( Z i × v ∨ i ) x (cid:55)→ (cid:40) ( π i ( x ) , τ ( x )) , if x / ∈ Z ∨ i (cid:63), otherwise . where (cid:63) denotes the pinched copy of Z i × v ∨ i in the quotient. The fiber of this map over (cid:63) is ε -small since the metric is squeezed around Z ∨ i . Consider the fiber over any otherpoint ( z, t ) of the quotient; since it is contained in τ − ( t ), its g X -size does not exceed its g F -size; since it is contained in π − i ( z ), its g F -size is ε -small. (cid:3) References [1] A. Akopyan, A. Hubard, R. Karasev, et al. Lower and upper bounds for the waists of different spaces.
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E-mail address : ♣ [email protected] E-mail address : ♠ [email protected] ♣♠ Dept. of Mathematics, Massachusetts Institute of Technology, 182 Memorial Dr.,Cambridge, MA 02142, USA ♣♣