Complexity of Shadows & Traversing Flows in Terms of the Simplicial Volume
aa r X i v : . [ m a t h . G T ] N ov COMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMSOF THE SIMPLICIAL VOLUME
GABRIEL KATZ
Abstract.
We combine Gromov’s amenable localization technique with the Poincar´eduality to study the traversally generic vector flows on smooth compact manifolds X withboundary. Such flows generate well-understood stratifications of X by the trajectoriesthat are tangent to the boundary in a particular canonical fashion. Specifically, we getlower estimates of the numbers of connected components of these flow-generated strataof any given codimension. These universal bounds are basically expressed in terms of thenormed homology of the fundamental groups π ( X ) and π ( DX ), where DX denotes thedouble of X . The norm here is the Gromov simplicial semi-norm in homology. It turnsout that some close relatives of the normed homology spaces H k +1 ( DX ; R ), H k ( X ; R )form obstructions to the existence of k -convex traversally generic vector flows on X . Introduction
This paper is a direct extension of [AK]. As the latter article, it draws its inspirationfrom the paper of Gromov [Gr1] where, among other things, the machinery of amenablelocalization has been developed. Here we combine the amenable localization with thePoincar´e duality to study the traversally generic vector flows (see [K2] or Section 2 for thedefinition) on smooth compact manifolds X with boundary.An example of a traversally generic field v on a surface X is shown in Fig. 1 (the field v is vertical). The trajectory space T ( v ) of the v -flow is a graph whose verticies are trivalentor univalent. So the local topology of T ( v ) is quite rigid; moreover, it is universal for alltraversally generic fields on surfaces. The fibers of the obvious map Γ : X → T ( v ) are closedsegments or singletons, hence Γ is a homotopy equivalence. Note that the trivalent verticiesof T ( v ) correspond the the v -trajectories γ that first pierce the boundary ∂X transversally,then tangentially touch it, then pierce it again traversally. We use the pattern (121) toencode this behavior of γ . The majority of γ ’s intersect ∂X at two points where γ istransversal to ∂X ; we use the pattern (11) to mark such γ ’s. Finally, there are points-trajectories that correspond to the univalent verticies of T ( v ); they are marked with thecombinatorial pattern (2). The traversally generic fields on surfaces do not admit othercombinatorial patterns of tangency (say, like (31) or (1221)).Similarly, for a smooth traversally generic vector field v on a compact ( n + 1)-manifold X with boundary, the trajectory space T ( v ) acquires a stratification {T ( v, ω ) } ω by thecombinatorial patterns of tangency ω that belong to a universal poset Ω • ′ h n ] (see [K2] andSection 2). It depends only on dim( X ). T (v)X Γ v Figure 1.
Traversally generic “vertical” field v on the surface X , its strat-ified trajectory space T ( v ), and the obvious map Γ : X → T ( v )The more numerous the connected components of these stratifications are, the more complex the v -flow is (thus the word “complexity” in the title of the paper). So our goalhere is to find some lower bounds of the numbers such components.The Ω • ′ h n ] -stratification of T ( v ) generates the stratification { X ( v, ω ) = def Γ − ( T ( v, ω )) } ω of X and the stratification { ∂X ( v, ω ) = def X ( v, ω ) ∩ ∂X } ω of ∂X . The X -stratification canbe refined by considering the connected components of the sets { ∂X ( v, ω ) , X ◦ ( v, ω ) } ω } ω , where X ◦ ( v, ω ) = def X ( v, ω ) \ ∂X ( v, ω ). In Fig. 1, the 0-dimensional strata of this strati-fication are the bold (light and dark) dots, the 1-dimensional strata are the segments thatconnect the dots (some of them are portions of trajectories, some are arcs that belong tothe boundary ∂X ), and the 2-dimensional strata are open cells in which the 1-dimensionalstrata divide the surface X .We consider an auxiliary closed manifold, the double DX = def X ∪ ∂X X of X . Thedouble comes equipped with an involution τ so that ( DX ) τ = ∂X and DX/ { τ } = X .We stratify DX by the connected components of the sets (see Fig. 3): { ∂X ( v, ω ) , X ◦ ( v, ω ) , τ ( X ◦ ( v, ω )) } ω All these v -induced stratifications of T ( v ), X , and DX are the foci of our investigation.Let Z be a topological space. Recall that the homology H j ( Z ; R ) comes equipped withthe Gromov simplicial semi-norm k ∼ k ∆ (see [Gr] and Definition 3.3). We denote by H ∆ j ( Z ; R ) the quotient of H j ( Z ; R ) by the subspace of elements whose simplicial semi-norm is zero. Thus, H ∆ j ( Z ; R ) is a normed space with respect to the quotient semi-norm. OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 3
For technical reasons, related to the application of the Gromov Localization Lemma 3.1,we substitute the relative homology H ∗ ( X, ∂X ) with the absolute homology H ∗ ( DX ).By Theorem 4.5, dim (cid:0) H ∆ k +1 ( X ; R ) (cid:1) ≤ C n − k ( v ), where C n − k ( v ) denotes the number ofconnected components of dimensions n − k of the strata { X ◦ ( v, ω ) } ω .Similarly, dim (cid:0) H ∆ k +1 ( DX ; R ) (cid:1) ≤ Σ C n − k ( v ), the number of connected components ofdimensions n − k of the strata { ∂X ( v, ω ) , X ◦ ( v, ω ) , τ ( X ◦ ( v, ω )) } ω . Thus, for any traversally generic v -flow on X , dim (cid:0) H ∆ k +1 ( X ; R ) (cid:1) and dim (cid:0) H ∆ k +1 ( DX ; R ) (cid:1) —two homotopy-theoretical invariants of ( X, ∂X )—deliver lower estimates for C n − k ( v )and Σ C n − k ( v ), the ( n − k )-dimensional traversing complexities of v ! By their very na-ture, C n − k ( X ) and Σ C n − k ( X )—the minima of such complexities, taken over all traversallygeneric vector fields v on X ,—are new invariants of the smooth structure on X . So, thehomotopy theory of the pair ( X, ∂X ) puts nontrivial restrictions on its traversing complex-ity.We stress that all our results are vacuous when the fundamental groups π ( DX ) or π ( X ) are amenable (see Definition 3.5).Let us describe another geometrical manifestation of basically the same phenomenon(see Theorem 4.2 and Corollary 4.1). Let f : M → X be a map from a closed ( k + 1)-dimensional pseudo-manifold M to X . It realizes the homology class f ∗ [ M ] ∈ H k +1 ( X ).We may assume that f is transversal to each pure ( n − k )-dimensional stratum from thestratification { X ◦ ( v, ω ) } ω of int( X ). Then there exists an universal constant c > n and k ) such that, for any X , any traversally generic v on it, and any f as above, the intersection number of the cycle f ( M ) with the union of ( n − k )-dimensionalstrata from the stratification { X ◦ ( v, ω ) } ω is greater then or equal to c · k f ∗ [ M ] k ∆ .Here is a couple of examples that show how these general conclusions apply to thetraversally generic flows on smooth compact 4-folds with boundary. When dim( X ) = 4,the only nontrivial lower bounds of traversing complexities can be provided by the groups H ∆ ( X ) , H ∆ ( X ) , H ∆ ( X ), and H ∆ ( DX ) , H ∆ ( DX ) , H ∆ ( DX ) , H ∆ ( DX ).The basic arguments that validate the following example are similar to the ones we usein Example 4.2.Let Z i be a fibration over a closed oriented surface M i of a genus g ( M i ) ≥ F i ( i = 1 , . . . , N ). Assume that Z i → M i admits a section s i . Formthe connected sum Z = Z Z . . . Z N and consider a smooth closed 4-fold W whichis homotopy equivalent to Z .By deleting the standard 4-ball from W we get the 4-fold X = W \ D whose boundary isthe sphere S . When at least one the fibers { F i } has the genus g ( F i ) ≥
2, then k [ DX ] k ∆ > (cid:0) H ∆ ( DX ) (cid:1) = 1. Under these hypotheses, our results imply that any transversallygeneric v on X must have at least one trajectory γ from the following list: (a) either γ pierces he boundary sphere transversally, then simply touches it 3 times, then transversallypierces it again (the combinatorial pattern of such γ is (12221)), or (b) γ is cubically tangentto the boundary, then simply tangent, then pierces S transversally (the combinatorialpattern of such γ is either (321) or (123)), or (c) γ is transversal to the boundary sphere,then quartically tangent to it, then meets it transversally again (the combinatorial pattern GABRIEL KATZ of such γ is (141)). Moreover, we prove that there exists a universal constant θ >
0, suchthat the number of such trajectories grows as θ · k [ DX ] k ∆ at least.The images of the classes { [ s i ( M i )] } i are independent in H ∆ ( X ). Thus, dim (cid:0) H ∆ ( X ) (cid:1) ≥ N . Applying Theorem 4.5 to the traversally generic v on X , we get3 · π ( T ( v, π ( T ( v, π ( T ( v, ≥ N, where π ( T ( v, ω )) denotes the number of connected 1-dimensional components of thecombinatorial type ω in T ( v ).The Morse theory helps to exclude a priory the tangencies of local multiplicities 3 andhigher. Let W be as before and let f : W → R be a Morse function whose gradientfield v satisfies the Morse-Smale transversality property. We form a compact smooth 4-fold X by deleting from W sufficiently small standard 4-balls, centered on the f -criticalpoints. For such a choice of X and v , the combinatorial tangency patterns of the v -trajectories in X belong to the list: (11) , (121) , (1221) , (12221). Then by similar arguments,3 · π ( T ( v, ≥ N . ♦ Now let us describe the structure of the paper. In Section 2, for the reader’s conve-nience, we reintroduce the notion of a traversally generic vector field and describe its basicproperties, needed for Section 4. (they have been studied in a series of papers [K], [K1]-[K4], and [AK]).In Section 3, we study maps from a given compact PL -manifold X with boundary ontospecial compact CW -complexes K , dim( K ) = dim( X ) −
1. The local topology (the typesof singularities) of K is prescribed a priori; it is X -independent. We require the fibers of F : X → K to be PL -homeomorphic to closed segments or to singletons. We call suchmaps F the shadows of X . This setting is mimicking the maps Γ : X → T ( v ), generatedby traversally generic fields v on smooth manifolds X with boundary.The target space K of a shadow F comes equipped with a natural stratification, definedby the local topology of the singular loci in K . With the help of F , that stratificationinduces stratifications in X and in its double DX .We introduce the j -th complexities of a shadow F as the number of connected compo-nents of the strata of the fixed dimension j in F ( X ), X or in DX .The main results of Section 3 are Theorem 3.1-3.3. The first two deal with “the amenablelocalization of the Poincar´e duality operators”, in particular, with estimations of theirnorms in terms of the normed “homology” groups H ∆ ∗ ( DX ; R ) or H ∆ ∗ ( X ; R ). In Theorem3.3, the j -th complexities of any shadow F are estimated from below by the ranks of thegroups H ∆ n +1 − j ( DX ; R ) or H ∆ n +1 − j ( X ; R ), where n + 1 = dim( X ).In Section 4, we apply the results from Section 3 to the special shadows, produced bythe traversally generic vector fields. The applications deliver two main results, Theorem4.2 and Theorem 4.5. Recall that the v -flow canonically generates some well-understoodstratifications of the spaces T ( v ), X , and DX (see [K2]). As in the category of shadows,these stratifications lead to few competing notions of complexity for traversally genericflows. In Theorem 4.5, we get lower estimates of the numbers of connected componentsof these flow-generated strata of any given dimension. The estimates are universal for OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 5 the given homotopy type of the pair (
X, ∂X ) and any traversing field on X . Again, theseuniversal bounds are expressed in terms of the normed homology of DX or of X . Moreover,we prove that that the normed spaces H ∆ n +1 − j ( DX ; R ) and H ∆ n − j ( X ; R ) form obstructions to the existence of the globally j -convex (see Definition 2.2) traversally generic vector flowson a given X .In the process of studying various complexities of traversally generic flows, we introducefew graded differential complexes C ℧ ∗ ( T ( v )), C ℧ ∗ ( X, v ), C ℧ ∗ ( DX, v ) (see formulae (4.3),(4.7)). They are naturally produced by the filtrations of the spaces T ( v ), X , and DX bythe v -induced strata of a fixed codimension. The differential complexes C ℧ ∗ ( ∼ , v ) are morerefined invariants of the v -flow than the flow complexities: the complexities { tc j ( ∼ , v ) } j arejust the ranks of the corresponding j -graded terms of these complexes. Although C ℧ ∗ ( ∼ , v )seem to be the right instruments for studying traversing vector fields on X and ultimately X itself, their homological investigation belongs to a different paper.2. Basics of Traversally Generic Vector Fields
We start with presenting few basic definitions and facts related to the traversally genericvector fields.Let X be a compact connected smooth ( n + 1)-dimensional manifold with boundary.A vector field v is called traversing if each v -trajectory is ether a closed interval withboth ends residing in ∂X , or a singleton also residing in ∂X (see [K1] for the details). Inparticular, a traversing field does not vanish in X . In fact, v is traversing if and only if v = 0 and v is of the gradient type (see [K1]).For traversing fields v , the trajectory space T ( v ) is homology equivalent to X (Theorem5.1, [K3]).We denote by V trav ( X ) the space of traversing fields on X .In this paper, we consider an important subclass of traversing fields which we call traver-sally generic (see formula (2.4) and Definition 3.2 from [K2]).For a traversally generic field v , the trajectory space T ( v ) is stratified by closed sub-spaces, labeled by the elements ω of an universal poset Ω • ′ h n ] which depends only ondim( X ) = n +1 (see [K3], Section 2, for the definition and properties of Ω • ′ h n ] ). The elements ω ∈ Ω • ′ h n ] correspond to combinatorial patterns that describe the way in which v -trajectories γ ⊂ X intersect the boundary ∂ X = def ∂X . Each intersection point a ∈ γ ∩ ∂ X acquiresa well-defined multiplicity m ( a ), a natural number that reflects the order of tangency of γ to ∂ X at a (see [K1] and Definition 2.1 for the expanded definition of m ( a )). So γ ∩ ∂ X can be viewed as a divisor D γ on γ , an ordered set of points in γ with their multiplici-ties. Then ω is just the ordered sequence of multiplicities { m ( a ) } a ∈ γ ∩ ∂ X , the order beingprescribed by v .The support of the divisor D γ is either a singleton a , in which case m ( a ) ≡ D γ have odd multiplicities, and the rest ofthe points have even multiplicities. GABRIEL KATZ
Let m ( γ ) = def X a ∈ γ ∩ ∂ X m ( a ) and m ′ ( γ ) = def X a ∈ γ ∩ ∂ X ( m ( a ) − . (2.1)Similarly, for ω = def ( ω , ω , . . . , ω i , . . . ) we introduce the norm and the reduced norm of ω by the formulas: | ω | = def X i ω i and | ω | ′ = def X i ( ω i − . (2.2)Let ∂ j X = def ∂ j X ( v ) denote the locus of points a ∈ ∂ X such that the multiplicity ofthe v -trajectory γ a through a at a is greater than or equal to j . This locus has a descriptionin terms of an auxiliary function z : ˆ X → R which satisfies the following three properties:(2.3) • z , • z − (0) = ∂ X , and • z − (( −∞ , X .In terms of z , the locus ∂ j X = def ∂ j X ( v ) is defined by the equations: { z = 0 , L v z = 0 , . . . , L ( j − v z = 0 } , where L ( k ) v stands for the k -th iteration of the Lie derivative operator L v in the directionof v (see [K2]).The pure stratum ∂ j X ◦ ⊂ ∂ j X is defined by the additional constraint L ( j ) v z = 0. Definition 2.1
The multiplicity m ( a ), where a ∈ ∂X , is the index j such that a ∈ ∂ j X ◦ . ♦ The characteristic property of traversally generic fields is that they admit special flow-adjusted coordinate systems, in which the boundary is given by quite special polynomialequations (see formula (2.4)) and the trajectories are parallel to one of the preferred co-ordinate axis (see [K2], Lemma 3.4). For a traversally generic v on a ( n + 1)-dimensional X , the vicinity U ⊂ ˆ X of each v -trajectory γ of the combinatorial type ω has a specialcoordinate system ( u, ~x, ~y ) : U → R × R | ω | ′ × R n −| ω | ′ . By Lemma 3.4 and formula (3 .
17) from [K2], in these coordinates, the boundary ∂ X isgiven by the polynomial equation: ℘ ( u, ~x ) = def Y i (cid:2) ( u − i ) ω i + ω i − X l =0 x i,l ( u − i ) l (cid:3) = 0(2.4)of an even degree | ω | in u . Here i ∈ Z runs over the distinct roots of ℘ ( u,~
0) and ~x = def { x i,l } i,l . At the same time, X is given by the polynomial inequality { ℘ ( u, ~x ) ≤ } . Each v -trajectory in U is produced by freezing all the coordinates ~x, ~y , while letting u to be free. OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 7
We denote by X ( v, ω ) the union of v -trajectories whose divisors are of a given combina-torial type ω ∈ Ω • ′ h n ] . Its closure ∪ ω ′ (cid:22) • ω X ( v, ω ′ ) is denoted by X ( v, ω (cid:23) • ).Each pure stratum T ( v, ω ) ⊂ T ( v ) is an open smooth manifold and, as such, has a“conventional” tangent bundle.We denote by V ‡ ( X ) the space of traversally generic fields on X . It turns out that V ‡ ( X )is an open and dense (in the C ∞ -topology) subspace of V trav ( X ) (see [K2], Theorem 3.5). Definition 2.2.
We say that a traversing field v on X is globally k - convex if m ′ ( γ ) < k for any v -trajectory γ . ♦ Shadows of Manifolds with Boundary and their Complexity
The notion and properties of shadows (see Definition 3.1), the main subject of thissection, are inspired by the maps Γ : X → T ( v ), where the field v is traversally generic.We are going to pick a fixed and carefully chosen class of compact n -dimensional CW -complexes K whose local topological structure is prescribed.Let X be a compact connected PL -manifold of dimension n + 1 with boundary.We will consider a variety of surjective maps { F : X → K } with the F -fibers being aparticular type of contractible CW -complexes K = F ( X ) as “shadows” of the given manifold X . We consider singularities in K of particular types { K ( ω ) } ω and intend to count thecardinalities { π ( K ( ω )) } ω . We view this count of connected components of the strata K ( ω ) as measuring the complexity of the surjection F . Then we minimize these numbersover all F ’s to get various notions of complexity for the given manifold X .Let us start with a quite general setting. Let S be a poset equipped with two maps: amap µ : S → Z + and an order-preserving map µ ′ : S → Z + . By definition, for each ω ∈ S , µ ′ ( ω ) < µ ( ω ). For each n ∈ Z + , we assume that the poset S n = def ( µ ′ ) − ([0 , n ]) is finite.With each element ω ∈ S n we associate a model compact CW -complex T ω of dimension µ ′ ( ω ) and a model compact PL -manifold E ω of dimension µ ′ ( ω ) + 1. They are linked by a PL -map p ω : E ω → T ω whose fibers are closed intervals or singletons . In what follows, wewill list additional properties of the two collections, E = def { E ω } ω ∈S and T = def { T ω } ω ∈S (exhibiting topologically distinct T ω ’s). We will do it in a recursive fashion.Consider a set Cosp ( T , n ) of n -dimensional compact CW -complexes K such that eachpoint y ∈ K has a neighborhood which is PL -homeomorphic to the product T ω × D n − µ ′ ( ω ) for some ω ∈ S , where µ ′ ( ω ) ≤ n , and D n − µ ′ ( ω ) denotes the standard ball.We require that each model space T ω topologically will be a cone over a space S ω thatbelongs to the set Cosp ( T , n − Cosp ( T ,
1) consists of finite graphs whose verticies are of valencies 1 and3 only. “ Cosp ” is an abbreviation of “cospine”. “normal” to the ω -labeled stratum K ( ω ) in K GABRIEL KATZ
We denote by K ( ω ) the set of points in K whose neighborhoods are modeled after thespace T ω × D n − µ ( ω ) . It follows that each K ( ω ) ⊂ K is a locally closed PL -manifold.Let us also consider a filtration K = K − ⊃ K − ⊃ · · · ⊃ K − n of K by the closedsubcomplexes K − j = def [ { ω ∈S n | µ ′ ( ω ) ≥ j } K ( ω )(3.1)of dimensions n − j . Note that K − j = ∅ implies K − ( j +1) = ∅ . Definition 3.1.
Let X be a compact connected PL -manifold of dimension n + 1. Weassume that ∂X = ∅ . Consider the set Shad ( X, E ⇒ T ) of surjective PL -maps F : X → K such that: • K is a compact CW -complex of the type from Cosp ( T , n ), • each fiber of F is PL -homeomorphic to a closed interval I or to a singleton pt , where ∂I and pt reside in ∂X . • F ∂ = def F | : ∂X → K is a surjective map with finite fibers, • for each ω ∈ S , the map F | : F − ( K ( ω )) → K ( ω )is a trivial fibration with an orientable manifold base, and the map F ∂ | : ( F ∂ ) − ( K ( ω )) → K ( ω )is a trivial covering with the fiber of cardinality µ ( ω ) − µ ′ ( ω ), • each point y ∈ K ( ω ) has a regular neighborhood V y ⊂ K such that the map F : ( F − ( V y ) , F − ( ∂V y )) → ( V y , ∂V y )is conjugate to the model map p ω × id : ( E ω , δ E ω ) × D n − µ ( ω ) → ( T ω , S ω ) × D n − µ ( ω ) via a PL - homeomorphism which preserves the S n -stratifications in both spaces.Here δ E ω denotes the portion of the boundary ∂ E ω that is mapped to S ω .For F ∈ Shad ( X, E ⇒ T ), we call the CW -complex F ( X ) a S - shadow of X . ♦ Definition 3.2.
We say that a compact connected PL -manifold X with boundary is globally k -convex if it has a shadow F ∈ Shad ( X, E ⇒ T ) with the property F ( X ) − k = ∅ .Note that the global k -convexity implies global ( k + 1)-convexity. ♦ Part of this section is devoted to finding obstructions to the global k -convexity, selectedfrom the tool set of algebraic topology.Next, we will employ Gromov’s simplicial semi-norms [Gr] to give lower estimates ofthe complexities of shadows and of traversing flows on a given ( n + 1)-manifold X . Thenumber of v -trajectories of the (maximal) reduced multiplicity dim( X ) − Definition 3.3.
Let X ⊃ Y be topological spaces. Given a relative homology class h ∈ H k ( X, Y ; R ), consider all relative singular cycles c = P i r i σ i that represent h . Here OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 9 r i ∈ R and σ i : ∆ k → X are singular simplicies; each ( k − ∂c being mapped to Y . We assume that, for any compact K ⊂ X , only finitelymany images { σ i (∆) } intersect K . Put k c k l = def X i | r i | . We define the simplicial semi-norm of a homology class h by the formula: k h k ∆ = def inf c (cid:8) k c k l (cid:9) . ♦ This construction defines a semi-norm k ∼ k ∆ on the vector space H k ( X, Y ; R ). Thesemi-norm is monotone decreasing under continuous maps of pairs of spaces: k h k ∆ ≥ k f ∗ ( h ) k ∆ for any h ∈ H k ( X , Y Z ) and a continuous map f : ( X , Y ) → ( X , Y ). Moreover, if f : X → X is a continuous map such that f ∗ : π ( X ) → π ( X ) is an isomorphism ofthe fundamental groups, then k f ∗ ( h ) k ∆ = k h k ∆ for any h ∈ H k ( X ; R ) [Gr].If M is any closed, oriented hyperbolic manifold, thenVol( M ) = c ( n ) · k [ M ] k ∆ , (3.2)where [ M ] denotes the fundamental class of M and c ( n ) is an universal positive constant(this is the Proportionality Theorem, page 11 of [Gr]). For this reason, the simplicial normof the fundamental class [ X ] is often called the simplicial volume .Gromov’s Localization Lemma 3.1 below relies on the notion of the stratified simplicialsemi-norm , available for stratified spaces X and pairs X ⊃ Y of stratified spaces.We consider stratified spaces such that if a stratum S intersects the closure S ′ of anotherstratum S ′ , then S ⊆ S ′ . In this case we write S (cid:22) S ′ . If neither S (cid:22) S ′ nor S ′ (cid:22) S , thenwe say the two strata are incomparable .Recall that the corank of a stratum Y ω in a S -stratification of a given space Y is themaximal length k of a filtration Y ω ⊂ ¯ Y ω ⊂ · · · ⊂ ¯ Y ω k by the distinct strata whose closurescontain Y ω . Definition 3.4.
Let X be a S -stratified topological space. Given a real homology class h ∈ H k ( X ; R ), consider all singular cycles c = P i r i σ i that represent h , where r i ∈ R and σ i : ∆ k → X are singular simplicies that are consistent with the stratification S , in thefollowing sense : • We require that for each simplex σ i of c , the image of the interior of each face(of any dimension) must be contained in one stratum. We call this the cellular condition. Gromov gives two conditions: ord (er) and int (ernality) ([Gr1], p. 27). We use these two conditions plustwo more! a b cd e f
Figure 2.
Examples of a singular 2-simplex in relation to a stratification ofthe plane by a single stratum of codimension 2, three strata of codimension1, and two strata of codimension 0. Diagrams (a), (b), (c) are consistentwith the four bullet list from Definition 3.4; diagram (d) violates the secondbullet, diagram (e) violates the third bullet, diagram (f) violates the firstbullet. • The ( ord ) condition states that the image of each simplex of c must be containedin a totally ordered chain of strata; that is, the simplex does not intersect any twoincomparable strata. • The ( int ) condition states that for each simplex of c , if the boundary of a face (ofany dimension) maps into a stratum S , then the whole face maps into S . • For technical reasons (involving the Amenable Reduction Lemma in [AK]), werequire that if two vertices of a simplex σ i map to the same point v ∈ X , then theedge between them must be constant at v . We call this the loop condition.We define the S - stratified simplicial semi-norm of a homology class h by the formula: k h k S ∆ = def inf c (cid:8) k c k l (cid:9) , where c runs over all the cycles c = P i r i σ i that represent h , subject to the four propertiesabove.Similar definition of k h k S ∆ is available for relative homology classes h ∈ H k ( X, Y ; R ),where the inclusion Y ⊂ X is a S -stratified map. ♦ Definition 3.5.
A discrete group G is called amenable if for every finite subset S ⊂ G and every ǫ >
0, there exists a finite non-empty set A ⊂ G such that the proportion of OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 11 cardinalities | (( g · A ) ∪ A ) \ (( g · A ) ∩ A ) || A | < ǫ for all g ∈ S . ♦ Finally, we are in position to state the pivotal Gromov’s Localization Lemma from [Gr1],page 772. Its proof there is a bit rough; a detailed proof can be found in [AK].
Lemma 3.1 ( Gromov’s Localization Lemma).
Let X be a closed ( n + 1) -manifoldwith stratification S consisting of finitely many connected locally closed submanifolds. Pickan integer j from the interval [0 , n + 1] . Let Z be a space with the contractible universalcover , and let α : X → Z be a continuous map such that the α -image of the fundamentalgroup of each stratum of codimension less than j is an amenable subgroup of π ( Z ) .Let X − j ⊆ X denote the union of strata with codimension at least j , and let U bea neighborhood of X − j in X . Then the α -image of every j -dimensional homology class h ∈ H j ( X ) satisfies the upper bound k α ∗ ( h ) k ∆ ≤ k h U k S ∆ , where h U ∈ H j ( U, ∂U ) denotes the restriction of h to U , obtained via the composite homo-morphism H j ( X ) → H j ( X, X \ U ) → H j ( U, ∂U ) , where the last map is the excision isomorphism. ♦ Let X be a compact oriented manifold with boundary. For technical reasons relatedto the application of Lemma 3.1, many arguments to follow will deal with the double DX = X ∪ ∂X X of X instead of the pair ( X, ∂X ). The orientation on X extends to anorientation on its double. Then there is an orientation-reversing involution τ : DX → DX whose orbit space is X and whose fixed point set DX τ = ∂X .Note that any absolute homology class ˜ h ∈ H ∗ ( DX ), via the restriction to X , gives riseto a relative class h ∈ H ∗ ( X, ∂X ). Conversely, every relative class h , with the help of theinvolution τ gives rise to an absolute class ˜ h ∈ H ∗ ( DX ) whose restriction to X ⊂ DX produces h . Therefore, H ∗ ( X, ∂X ) can be viewed as a direct summand of H ∗ ( DX ).However, the relation between the simplicial semi-norms of ˜ h ∈ H ∗ ( DX ) and of h ∈ H ∗ ( X, ∂X ) is not so transparent. For example, if X is a cylindrical surface and h ∈ H ( X, ∂X ) is a generator, then k h k ∆ = 1, while k ˜ h k ∆ = 0 since homologically the longitude˜ h of the torus DX can be represented by the singular rational cycle N · { f : [0 , → DX } ,where f wraps the segment [0 , N times around the longitude; so k ˜ h k ∆ ≤ N for all N .Note that X/∂X is a torus with one of its meridians being collapsed to a point. Thesimplicial semi-norm of the longitude in
X/∂X is zero. So, although H ( X, ∂X ) is canoni-cally isomorphic to the reduced homology H ( X/∂X, ∂X/∂X ), the isomorphism is not anisometry in the two simplicial semi-norms!At the same time, similar considerations imply that, for the cylinder X , the simplicialsemi-norms on both H ( X, ∂X ) and H ( DX ) are trivial. that is, a K ( π, Lemma 3.2.
Let X be a connected compact and oriented ( n + 1) -manifold with boundary.Let β : DX → X/∂X be the degree 1 map which collapses the second copy τ ( X ) ⊂ DX of X to the point ⋆ = ∂X/∂X ∈ X/∂X , and let α : ( X, ∂X ) → ( X/∂X, ⋆ ) be the obvious mapof pairs. Then, for any h ∈ H j ( X, ∂X ) , we get k h k ∆ ≥ k h − τ ∗ ( h ) || ∆ ≥ k α ∗ ( h ) k ∆ , where the middle term is the simplicial semi-norm of the class h − τ ∗ ( h ) ∈ H j ( DX ) .In particular, for the fundamental class h = [ X, ∂X ] ∈ H n +1 ( X, ∂X ; R ) , the class h − τ ∗ ( h ) = [ DX ] is fundamental; so we get k [ X, ∂X ] k ∆ ≥ k [ DX ] k ∆ ≥ k [ X/∂X ] k ∆ . Proof. If c = P i r i σ i is a relative cycle with the l -norm k c k being ǫ -close to k h k ∆ , thenthe chain P i r i σ i − P i r i τ ( σ i ) is an absolute cycle in DX whose l -norm is 2 k c k at most.Thus 2 k h k ∆ ≥ k h − τ ∗ ( h ) k ∆ .On the other hand, consider the degree 1 map β : DX → X/∂X . By [Gr], the simplicialvolume does not increase under continuous maps. Therefore, observing that α ∗ ( h ) = β ∗ ( h − τ ∗ ( h )), we get k h − τ ∗ ( h ) k ∆ ≥ k α ∗ ( h )] k ∆ . (cid:3) Let S be a stratification of X by strata S such that the sets { S ∩ ∂X } S ∈S form astratification S ∂ of the boundary ∂X . Then S gives rise to a stratification DS of thedouble DX so that any stratum S ∈ DS either belongs to ∂X (in which case S ∈ S ∂ )or to DX \ ∂X . Conversely, any τ -invariant stratification DS of the double induces astratification S of X .Next, we are going to associate few useful differential chain complexes and their homologygroups with shadows of manifolds X with boundary. Eventually they will become usefulinstruments in our investigations of various notions of complexity of traversing flows.Let F ∈ Shad ( X, E ⇒ T ) be a shadow (see Definition 3.1) of a compact orientable PL -manifold X of dimension n + 1. Put K = def F ( X ). We consider the finite S n -stratification { K ( ω ) } ω ∈S n of K by the model (normal to the strata) spaces { T ω } of dimensions { µ ′ ( ω ) } .We denote by X ( F, ω ) the F -preimage of the pure stratum K ( ω ). These sets form astratification S F ( X ) of X .The CW -complex K ∈ Cosp ( T , n ) comes equipped with a filtration K = def K − ⊃ K − ⊃ · · · ⊃ K − n which has been introduced in (2.1) and employs the more refined S n -stratification.Let A be an abelian coefficient system on K . As a default, A = R , the trivial coefficientsystem with the real numbers for a stalk.For each j ∈ [0 , n ], consider the relative homology groups { C ℧ j ( K ) := H j ( K − n + j , K − n + j − ; A ) } j (3.3)associated with the filtration. Note that dim( K − n + j ) = j , so C ℧ j ( K ) is the top reducedhomology of the quotient K − n + j /K − n + j − . in our notations, we suppress the dependance of these homology groups on the coefficients A . OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 13
These homology groups can be organized into a differential complex C ℧ ∗ ( K ) = def (cid:8) → C ℧ n ( K ) ∂ n → C ℧ n − ( K ) ∂ n − → · · · ∂ → C ℧ ( K ) → (cid:9) , (3.4)where the differentials { ∂ j } are the boundary homomorphisms from the long exact homol-ogy sequences of the triples { K − n + j ⊃ K − n + j − ⊃ K − n + j − } j . Since the pure strata K ◦− n + j = def K − n + j \ K − n + j − are open orientable manifolds ofdimension j , the A -modules { C ℧ j ( K ) } j are free and finitely generated, the number of gen-erators being the number of connected components of the locus K ◦− n + j .We call the homology groups H ℧ ∗ ( K ), associated with the differential complex (3.4), the ℧ - homology of K .In fact, the ℧ - homology is an ingredient of the homology spectral sequence associatedwith the filtration { K − j } j and converging to the regular homology H ∗ ( K ) ≈ H ∗ ( X ) for anyshadow K = F ( X ). Therefore the is a canonical homomorphism A ℧ ∗ : H ℧ ∗ ( K ) → H ∗ ( X ). Definition 3.6.
With any filtered CW -complex K ∈ Shad ( T , n ) and its filtration as in(2.1), we associate the ordered collection of ranks (cid:8) c j ( K ) = def rk A ( C ℧ j ( K )) (cid:9) ≤ j ≤ n , where the groups { C ℧ j ( K )) } j were introduced in (3.3).We call c j ( K ) the j -th ℧ -complexity of K . ♦ Definition 3.7.
Let X be a compact connected PL -manifold with boundary, dim( X ) = n + 1. Consider the variety of maps F : X → K from the set Shad ( X, E ⇒ T ) as inDefinition 3.1. Each F produces the sequence of ℧ -complexities: c ( F ) = def (cid:8) c ( F ( X )) , c ( F ( X )) , . . . , c n ( F ( X )) (cid:9) Consider the lexicographical minima c shad ( X, E ⇒ T ) = def lex.min { F ∈ Shad ( X, E ⇒ T ) } c ( F ( X ))(3.5)We call them the lexicographic shadow complexity of X .We denote by c shad j ( X, E ⇒ T ) the ( j + 1)-component of the vectors c shad ( X, E ⇒ T ). ♦ Remark 3.1.
Of course, the definitions above rely on the set
Shad ( X, E ⇒ T ) beingnonempty, a nontrivial fact which requires a carefully designed poset S and a system ofmodels { p ω : E ω → T ω } ω ∈S (as the ones introduced in [K2]). ♦ Remark 3.2.
If a compact connected PL -manifold X of dimension n + 1 is globally k -convex (see Definition 3.2), then for some shadow F and all j ≥ k , we get F ( X ) − j = ∅ .Thus C ℧ n − j ( F ( X )) = 0 for all j ≥ k , implying c shad n − j ( X, E ⇒ T ) = 0 for all j ≥ k . In otherwords, c shad l ( X, E ⇒ T ) = 0 for all l ≤ n − k . ♦ Our next goal is to find some lower estimates of the complexities { c shad j ( X, E ⇒ T ) } j interms of the algebraic topology of X .Let us consider a refinement S • F ( X ) of the stratification S F ( X ) of X , formed by the connected components of the sets: (cid:8) X ◦ ( F, ω ) = def X ( F, ω ) \ ( ∂X ∩ X ( F, ω )) , X ∂ ( F, ω ) = def ∂X ∩ X ( F, ω ) (cid:9) ω ∈S . (3.6) Fig. 1 shows an example of such a stratification S • F ( X ) on a surface X .The double DX acquires the τ -equivariant stratification S F ( DX ) whose pure strata are: { X ◦ ( F, ω ) , τ ( X ◦ ( F, ω )) , X ∂ ( ω ) } ω . and its refinement S • F ( DX ), formed by the connected components of the strata from S F ( DX ).The stratifications S • F ( X ) and S • F ( DX ) give rise to the filtrations: X = def X F − ⊃ X F − ⊃ · · · ⊃ X F − ( n +1) ,DX = def DX F − ⊃ DX F − ⊃ · · · ⊃ DX F − ( n +1) by the union of strata of a fixed codimension, each pure stratum being an open manifold.Analogously to (3.3) we consider the relative homology and cohomology groups withcoefficients in A : { C ℧ j (cid:0) DX, F (cid:1) = def H j (cid:0) DX Fj − n − , DX Fj − n ; A (cid:1) } j , (3.7) { C j ℧ (cid:0) DX, F (cid:1) = def H j (cid:0) DX Fj − n − , DX Fj − n ; A (cid:1) } j . They can be organized into a differential complex: C ℧ ∗ ( DX, F ) = def (3.8) (cid:8) → C ℧ n +1 ( DX, F ) ∂ n +1 → C ℧ n ( DX, F ) ∂ n → · · · ∂ → C ℧ ( DX, F ) → (cid:9) , where the differentials { ∂ j } are the boundary homomorphisms from the long exact homol-ogy sequences of triples { DX F − j +1 ⊃ DX F − j ⊃ DX F − j − } j . Similarly, we introduce the dual differential complex C ∗ ℧ ( DX, F ) = def (3.9) (cid:8) ← C n +1 ℧ ( DX, F ) ∂ ∗ n +1 ← C n ℧ ( DX, F ) ∂ ∗ n ← · · · ∂ ∗ ← C ℧ ( DX, F ) ← (cid:9) When A = Z , since DX Fj − n − \ DX Fj − n is an open orientable j -manifold, by [Hat],Corollary 3.28, the torsion tor (cid:8) H j − (cid:0) DX Fj − n − , DX Fj − n ; Z (cid:1)(cid:9) = 0, which results in thenatural isomorphism C j ℧ ( DX, F ) ≈ Hom Z ( C ℧ j ( DX, F ) , Z ) . In the notations to follow, we drop the dependence of the constructions on the choice ofthe coefficient group A , but presume that A is Z or R .So the complexes C ℧ ∗ ( DX, F ) and C ∗ ℧ ( DX, F ) are comprised of free A -modules whosegenerators { [ σ ] } are in 1-to-1 correspondence with the strata σ ∈ S • F ( DX ) (see (3.7)).We denote by Z ℧ j ( DX, F ) the kernels of the differentials ∂ j and by B ℧ j ( DX, F ) the imagesof the differentials ∂ j +1 from (3.9). Next, we form the ℧ - homology H ℧ j ( DX, F ) = def Z ℧ j ( DX, F ) / B ℧ j ( DX, F )of the double DX , associated with its stratification S • F ( DX ). OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 15
Since each cycle ζ ∈ Z ℧ j ( DX, F ) is a common singular cycle in DX (indeed, by (3.8)and (3.9), the boundary of the j -chain ζ is mapped in the subcomplex of DX of dimension j − A ℧ ∗ : H ℧ ∗ ( DX, F ) → H ∗ ( DX ) . Since the differential complex (3.9) is the dual of the differential complex (3.8), thenatural paring C n − k ℧ ( DX, F ) ⊗ C ℧ n − k ( DX, F ) → R produces an isomorphismΦ : C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) ≈ ( Z ℧ n − k ( DX, F )) ∗ , where ( Z ℧ n − k ( DX, F )) ∗ denotes the dual space of Z ℧ n − k ( DX, F )).The next localization construction is central to the rest of this section. Using thePoincar´e duality D on the closed oriented manifold DX , for each k ∈ [ − , n ], we introducethe localization transfer map L Fk +1 : H k +1 (cid:0) DX (cid:1) ≈D −→ H n − k (cid:0) DX (cid:1) i ∗ → H n − k (cid:0) DX F − ( k +1) (cid:1) , where i : DX F − ( k +1) ⊂ DX is the embedding.Because the locus DX F − ( k +1) is a ( n − k )-dimensional CW -complex, the top cohomologygroup H n − k ( DX F − ( k +1) ) can be identified with the quotient C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F )from the complex in (3.9). This fact follows from the diagram chase in the two long exactcohomology sequences of the pairs (cid:0) DX F − ( k +1) , DX F − ( k +2) (cid:1) , (cid:0) DX F − ( k +2) , DX F − ( k +3) (cid:1) . Thefirst pair gives rise to the fragment of the exact sequence → H n − k − ( DX F − ( k +2) ) δ → C n − k ℧ ( DX, F ) j ∗ → H n − k ( DX F − ( k +1) ) → , while the second pair to the fragment → C n − k − ℧ ( DX, F ) J ∗ → H n − k − ( DX F − ( k +2) ) → , so that the composition J ∗ ◦ δ coincides with the boundary map δ n − k − : C n − k − ℧ ( DX, F ) → C n − k ℧ ( DX, F )from the differential complex C ∗ ℧ ( DX, F ). Therefore H n − k ( DX F − ( k +1) ) ≈ C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) . Similar diagram chase in homology leads to an isomorphism H n − k ( DX F − ( k +1) ) ≈ Z ℧ n − k ( DX, F ) . As a result, for each shadow F , we get the transfer homomorphism L F, ℧ k +1 : H k +1 ( DX ) ≈D −→ H n − k ( DX ) i ∗ → C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) . (3.10)Our next goal is to introduce a norm | [ ∼ ] | ℧ in the target space C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F )of the homomorphism L F, ℧ k +1 and to show that k h k ∆ ≤ const ( n ) · | [ L F, ℧ k +1 ( h )] | ℧ for any h ∈ H k +1 ( DX ) and some universal constant const ( n ) > { E ω → T ω } ω as in the last bullet of Definition 3.1. Consider the l -norm k ∼ k l on the space C n − k ℧ ( DX, F ) in the unitary basis { [ σ ∗ ] } ,labeled by the strata σ ∈ S • F ( DX ) of dimension n − k , and dual to the preferred basis { [ σ ] } of C ℧ n − k ( DX, F ). Then the norm | [ ∼ ] | ℧ on C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) is, by definition,the quotient norm induced by the norm k ∼ k l on the space C n − k ℧ ( DX, F ).We may regard the maximal proportionsup { h ∈ H k +1 ( DX ) , k h k ∆ =0 } (cid:8) | [ L F, ℧ k +1 ( h )] | ℧ / k h k ∆ (cid:9) as the norm of the operator L F, ℧ k +1 . So the inequality above (see also (3.11)) testifies thatthese norms admit a positive lower bound, universal for all X , dim( X ) = n + 1, and F .As the reader examines the hypotheses of the next theorem, it is helpful to keep in mindthe following simple example. Let X be a 2-dimensional ball with two holes. Then DX isa closed surface of genus 2. Note that the fundamental group of each component of ∂X isabelian. So its image in π ( DX ) is an amenable group. Theorem 3.1 ( The 1 st Amenable Localization of the Poincar´e Duality).
Let X be a compact connected ( n + 1) -dimensional PL -manifold with a nonempty boundary.Assume that for each connected component of the boundary ∂X , the image of its funda-mental group in π ( DX ) is an amenable group .Then, there exists an universal constant Θ > such that, for any shadow F ∈ Shad ( X, E ⇒ T ) , the space C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) admits a norm | [ ] | ℧ so that k h k ∆ ≤ Θ · | [ L F, ℧ k +1 ( h )] | ℧ (3.11) for all h ∈ H k +1 ( DX ) . Here the Poincar´e duality localizing operator L F, ℧ k +1 is introducedin (3.10), and the constant Θ depends only on n , the poset S , and the list of model maps { E ω → T ω } ω ∈S in the way that is described in formula (3.16).Proof. Let U be a regular neighborhood of the set DX F − ( k +1) in DX . So the PL -manifold U has a homotopy type of a ( n − k )-dimensional CW -complex DX F − ( k +1) .Let Π = def π ( DX ). In order to apply Localization Lemma 3.1 to the neighborhood U , the homology class h ∈ H k +1 ( DX ; Z ), and the classifying map β : DX → K (Π , σ ∈ S • F ( DX ), the subgroup β ∗ π ( σ ) of Π is anamenable group. This is true if σ ⊆ ∂X since for each connected component of ∂X , theimage of its fundamental group in Π is amenable, and every subgroup of an amenable groupis amenable. Otherwise, σ is contained in some locus X ◦ ( ω ) or in some locus τ ( X ◦ ( ω )).By the fourth bullet of Definition 3.1, this preimage X ◦ ( F, ω ) is a trivial oriented bundle F : X ◦ ( F, ω ) → K ( ω ) whose fiber is a disjointed union of open intervals. Therefore F : σ → F ( σ ) ⊂ K ( ω ) is a trivial fibration whose fiber is an open interval. As a result, F ∗ : π ( σ ) → π ( F ( σ )) is an isomorphism.On the other hand, the covering F : X ∂ ( F, ω ) → K ( ω ) is trivial and therefore admitsa section ρ : K ( ω ) → X ∂ ( F, ω ) such that its image intersects with the closure cl( σ ) of Evidently, if for each connected component of the boundary ∂X , the image of its fundamental groupin π ( X ) is amenable, then it is automatically amenable in π ( DX ). OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 17 σ in X . Put ¯ σ = def σ ∪ ρ ( F ( σ )). The map F : ¯ σ → F ( σ ) is a trivial fibration witha semi-open interval for the fiber. Therefore the imbedding j : σ ⊂ ¯ σ is a homotopyequivalence, and so is the obvious imbedding q : ρ ( F ( σ )) ⊂ ¯ σ . Hence j ∗ : π ( σ ) → π (¯ σ )and q ∗ : π ( ρ ( F ( σ ))) → π (¯ σ ) are isomorphisms. Since ρ ( F ( σ )) ⊂ ∂X and, for everychoice of the base point x ⋆ ∈ ∂X , the β ∗ -image of each fundamental group π ( ∂X, x ⋆ ) inΠ is amenable, so is the β ∗ -image of π ( ρ ( F ( σ )))—a subgroup of an amenable group isamenable. By the previous arguments, the β ∗ -image of π ( σ ) is amenable.Now, applying localization Lemma 3.1 to the classifying map β : DX → K (Π , k h k ∆ ≤ k h U k S • F ( U )∆ , (3.12)where h U denotes the restriction of the absolute homology class h to ( U, ∂U ). Indeedby [Gr], k β ∗ ( h ) k ∆ = k h k ∆ since, by its construction, β induces tan isomorphism of thefundamental groups of DX and K (Π , F(X)X σ F( σ ) hD σ σ ’F( σ ’ ) σ ’ D D(X)
Figure 3.
The cycle h ∈ H k +1 ( DX ), its transversal intersections with thestrata { σ ⊂ DX F − ( k +1) } , and the normal disks { D k +1 σ } which represent thelocalization h U .For each ( n − k )-dimensional stratum σ ∈ S • F ( DX ), consider an oriented disk( D k +1 σ , ∂D k +1 σ ) ⊂ ( U, ∂U ) , normal to the open manifold σ ⊂ U at its typical point. Taking a smaller regular neighbor-hood U of the subcomplex DX F − ( k +1) ⊂ DX if necessarily, we can arrange for { D k +1 σ } σ to bedisjointed, so that each disk D k +1 σ hits its stratum σ transversally at a singleton and misses the rest of the strata. Note that the relative integral homology classes { [ D k +1 σ , ∂D k +1 σ ] } σ may be dependent in H k +1 ( U, ∂U ).We claim that any element h U ∈ H k +1 ( U, ∂U ) can be written as a linear combination ofrelative cycles { [ D k +1 σ , ∂D k +1 σ ] ∈ H k +1 ( U, ∂U ) } σ : h U = X σ ∈S • F ( DX ) , dim σ = n − k r σ · [ D k +1 σ , ∂D k +1 σ ] . (3.13)In fact, H k +1 ( U, ∂U ) can be recovered from the free Z -module, generated by the elements { [ D k +1 σ , ∂D k +1 σ ] } σ , by factoring the module by the appropriate relations.To justify the presentation in (3.13), we notice that, since DX F − ( k +1) is a deformationretract of U , any ( n − k )-dimensional homology class in U is represented by a cycle z in DX F − ( k +1) , a combination P σ n σ · σ of the top strata of the ( n − k )-dimensional CW -complex DX F − ( k +1) . The algebraic intersection of z ◦ D k +1 σ = n σ , since σ ◦ D k +1 σ = 1 and σ ′ ◦ D k +1 σ = 0for any σ ′ = σ by the very choice of the normal disks D k +1 σ ’s. For the R -coefficients, by thePoincar´e duality, any ( k + 1)-dimensional homology class ˜ h ∈ H k +1 ( U, ∂U ) is determinedby its algebraic intersections with the cycles z ∈ H n − k ( U ) ≈ H n − k ( DX F − ( k +1) ). Thereforeany ˜ h is a linear combination of { [ D k +1 σ , ∂D k +1 σ ] } σ .To simplify notations, put [ D k +1 σ ] = def [ D k +1 σ , ∂D k +1 σ ].We introduce the norm of h U by the formula | [ h U ] | = def inf { representations of h U } n X σ ∈S • F ( DX ) , dim σ = n − k | r σ | o , (3.14)the minimum being taken over all representations of h U as in (3.13).Applying (3.12) to any presentation of h U as in (3.13), we get k h k ∆ ≤ X σ ∈S • F ( DX ) , dim σ = n − k | r σ | · (cid:13)(cid:13) [ D k +1 σ ] (cid:13)(cid:13) S • F ( U )∆ . (3.15)Thus, k h k ∆ ≤ Θ · X σ ∈S • F ( DX ) , dim σ = n − k | r σ | , where Θ = def max σ ∈S • F ( DX ) , dim σ = n − k n(cid:13)(cid:13) [ D k +1 σ ] (cid:13)(cid:13) S • F ( U )∆ o . (3.16)Here we stratify the normal disk D k +1 σ by intersecting it with the S • F ( DX )-stratificationin the ambient space DX . Employing Definition 3.1, the normal to σ disk D k +1 σ can beviewed as a subspace of the model double space D E ω σ , stratified with the help of the modelmap p ω σ : E ω σ → T ω σ . In D E ω σ , the two copies of the space E ω σ , given by ℘ ( u, ~x ) ≤ ℘ ( u, ~x ) = 0. The normal disk acquires its stratificationfrom the ambient space D E ω σ . Hence, as a stratified topological space, D k +1 σ depends onlyon the position of the stratum σ in the canonical stratification of D E ω σ . That position is OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 19 determined by the appropriate combinatorial data, provided by formula (2.4). As a result,the stratified simplicial norms (cid:13)(cid:13) [ D k +1 σ ] (cid:13)(cid:13) S • F ( U )∆ take a finite set of values , which depend onlyon the list of model maps { p ω : E ω → T ω } ω . So the constant Θ > X and its shadow F ∈ Shad ( X, E ⇒ T ).Therefore, in line with formula (3.12) and the definition in (3.14), we get k h k ∆ ≤ Θ · | [ h U ] | (3.17)where Θ > X , F , and h .Next, we are going to reinterpret the norm | [ h U ] | in terms of the differential complexes C ∗ ℧ ( DX, F ) and C ℧ ∗ ( DX, F ) to make it “more computable”.Let { [ σ ] ∗ } σ be the basis of C n − k ℧ ( DX, F ), dual to the basis { [ σ ] } σ in C ℧ n − k ( DX, F ).The | [ ∼ ] | ℧ -norm of a class η ∗ ∈ C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) is the quotient norm, in-duced by the l -norm on C n − k ℧ ( DX, F ); it is defined by the formula (which resembles theformula in (3.14)) | [ η ∗ ] | ℧ = def inf { ζ ∗ ≡ η ∗ mod B n − k ℧ ( DX,F ) } n X σ ∈S • F ( DX ) , dim σ = j | r σ | o , where ζ ∗ = X σ ∈S • F ( DX ) , dim σ = j r σ · [ σ ] ∗ . Recall that in (3.13) we have considered an epimorphism A : C † k +1 → H k +1 ( U, ∂U ),where C † k +1 denotes the free module over R (or over Z ), generated by the normal relativedisks { D k +1 σ } σ . Put R † k +1 = def ker( A ), so that C † k +1 / R † k +1 ≈ H k +1 ( U, ∂U ) . On the other hand, the Poincar´e duality D U produces an isomorphism B : H k +1 ( U, ∂U ) ≈D U −→ H n − k ( U ) ≈ −→ C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) . The composition B ◦ A takes each generator D k +1 σ ∈ C † k +1 to the class of [ σ ] ∗ and identifiesthe space of relations R † k +1 with the space B n − k ℧ ( DX, F ).Let D U ( h U ) denotes the Poincar´e dual in U of the relative homology class h U . Exam-ining the definitions of the norm in (3.14) and of the norm | [ ∼ ] | ℧ and tracing the natureof the Poincar´e duality (in terms of the intersections of relative and absolute cycles ofcomplementary dimensions in U ), we see that the norm | [ h U ] | in (3.14) coincides with thenorm | [ D U ( h U )] | ℧ , where D U ( h U ) ∈ H n − k ( DX F − ( k +1) ) ≈ C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) . The unit balls in the l -norms on the spaces C j ℧ ( DX, F ) are convex closures of the vectors {± [ σ ] ∗ } σ ,where σ are strata of dimension j . Therefore, combining this observation with (3.17), we get the desired inequality: k h k ∆ ≤ Θ · | [ L F, ℧ k +1 ( h )] | ℧ . (3.18) (cid:3) Theorem 3.1 has a “more geometric” interpretation (see Fig. 3).
Corollary 3.1.
Let a homology class h ∈ H k +1 ( DX ; Z ) be realized my a singular pseudo-manifold f : M → DX , dim( M ) = k + 1 .Under the hypotheses of Theorem 3.2, the number of intersections of f ( M ) with the locus DX F − ( k +1) is greater than or equal to Θ − · k h k ∆ , provided that f is in general position withthe subcomplex DX F − ( k +1) ⊂ DX .Proof. Recall that any integral homology class h ∈ H k +1 ( DX ; Z ) can be realized by apseudo-manifold f : M → DX ([Hat], pages 108-109).By a small perturbation of f , we can assume that f ( M ) intersects transversally onlywith the pure ( n − k )-dimensional strata in DX F − ( k +1) from the poset S • F ( DX ).Consider the transversal intersections from the set f ( M ) ∩ DX F − ( k +1) . Then, for asufficiently narrow regular neighborhood U of DX F − ( k +1) , the localized class h U has arepresentation h U = X x σ ∈ f ( M ) ∩ DX F − ( k +1) r σ · [ D k +1 x σ ] , where the relative disk ( D k +1 x σ , ∂D k +1 x σ ) ⊂ ( U, ∂U ) ∩ f ( M ) and r σ = ±
1. By (3.18), thenorm | [ h U ] | ≤ X x σ ∈ f ( M ) ∩ DX F − ( k +1) (cid:16) f ( M ) ∩ DX F − ( k +1) (cid:17) . Again, by (3.18), we get k h k ∆ ≤ Θ · (cid:16) f ( M ) ∩ DX F − ( k +1) (cid:17) . Thus (cid:16) f ( M ) ∩ DX F − ( k +1) (cid:17) ≥ Θ − · k h k ∆ , (3.19)where the positive constant Θ − depends only on n , the poset S , and the list of modelmaps E ⇒ T in the way that is described in formula (3.16). (cid:3) In order to get similar results about absolute homology classes h ∈ H k +1 ( X ), we considerthe F -induced filtration X F ◦− ⊃ X F ◦− ⊃ · · · ⊃ X F ◦− n of int ( X ) by the connected components of the strata { X ◦ ( F, ω ) } as in (3.6) and the relativehomology/cohomology groups { C ℧ j (cid:0) X, F ◦ (cid:1) = def H j (cid:0) X F • j − n − , X F • j − n ∪ ( X F • j − n − ∩ ∂X ); A (cid:1) } j , { C j ℧ (cid:0) X, F ◦ (cid:1) = def H j (cid:0) X F • j − n − , X F • j − n ∪ ( X F • j − n − ∩ ∂X ); A (cid:9) j . (3.20) a compact simplicial complex whose singular set has codimension 2 at least OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 21
They are designed to emphasize the role of the connected components of { X ◦ ( F, ω ) } fromthe stratification S ◦ F ( X ) of int ( X ).Using the excision property, the long exact sequences of the triples { X F •− j +1 ∪ ∂X ⊃ X F •− j ∪ ∂X ⊃ X F •− j − ∪ ∂X } j help to organize the groups in (3.20) into differential complexes C ℧ ∗ ( X, F ◦ ) and C ∗ ℧ ( X, F ◦ ),similar to the ones in (3.8) and (3.9).Employing the Poincar´e duality D again, we introduce the localization homomorphism M Fk +1 : H k +1 ( X ) ≈D −→ H n − k ( X, ∂X ) i ∗ → H n − k (cid:0) X F •− ( k +1) , X F •− ( k +1) ∩ ∂X (cid:1) whose target can be identified with the quotient C n − k ℧ ( X, F ◦ ) / B n − k ℧ ( X, F ◦ ) . This isomor-phism can be validated in a way that is similar to the one that led to (3.10). Indeed, put A = def X F •− ( k +1) , B = def ( X F •− ( k +1) ∩ ∂X ) ∪ X F •− ( k +2) , and C = def ( X F •− ( k +1) ∩ ∂X ). Considerthe fragment J : H n − k ( A, B ) → H n − k ( A, C ) of the long exact sequence of the triple A ⊃ B ⊃ C . By definition, H n − k ( A, C ) is the target of M Fk +1 , and H n − k ( A, B ) = C n − k ℧ ( X, F ◦ ).Since dim( B ) < n − k , H n − k ( B, C ) = 0; so J is an epimorphism.Similar arguments, which involve a different triple of spaces, imply that ker J can beidentified with im (cid:0) ∂ ∗ : C n − k − ℧ ( X, F ◦ ) → C n − k ℧ ( X, F ◦ ) (cid:1) .As a result, for each shadow F , we get a localization of the Poincar´e duality: M F, ℧ k +1 : H k +1 ( X ) ≈D −→ H n − k ( X, ∂X ) I ∗ → C n − k ℧ ( X, F ◦ ) / B n − k ℧ ( X, F ◦ ) . (3.21)As before, the target space of M F, ℧ k +1 admits the | [ ∼ ] | ℧ quotient norm. It is inducedby the l -norm on the based space C n − k ℧ ( X, F ◦ ), the base being indexed by the connectedcomponents of { X ◦ ( F, ω ) } ω .These considerations lead to a twin of Theorem 3.1. Theorem 3.2 ( The 2 nd Amenable Localization of the Poincar´e Duality).
Let X be a compact connected ( n + 1) -dimensional PL -manifold with a nonempty boundary.Assume that for each connected component of the boundary ∂X , the image of its funda-mental group in π ( X ) is an amenable group.Then, there exists an universal constant Θ ◦ > such that, for every shadow F ∈ Shad ( X, E ⇒ T ) , the space C n − k ℧ ( X, F ◦ ) / B n − k ℧ ( X, F ◦ ) admits a norm | [ ] | ℧ so that k h k ∆ ≤ Θ ◦ · | [ M F, ℧ k +1 ( h )] | ℧ (3.22) for all h ∈ H k +1 ( X ) . Here the Poincar´e duality localizing operator M F, ℧ k +1 is introduced in(3.21), and the constant Θ ◦ > depends only on n , the poset S , and the list of model maps { E ω → T ω } ω ∈S in the way that is described in formula (3.23).Proof. When h is represented by a singular cycle whose image is contained in int ( X ), itinteracts only with the strata from S ◦ F ( X ) (and misses the strata from S • F ( ∂X )). Therefore,we can pick a regular neighborhood U of X F ◦− ( k +1) in X so small that the localized class h U ∈ H k +1 ( U, ∂U ) is a linear combination of a disjoint union of disks { ( D k +1 x σ , ∂D k +1 x σ ) } ,normal to the strata σ in X F ◦− ( k +1) .We apply the Localization Lemma 3.1 to the image of the class h under the classifyingmap α : X → K ( π ( X ) , S ◦ F ( X ) are amenable in π ( X ) .Since only the disks { D k +1 x σ } σ ∈S ◦ F ( X ) participate in the localization of h to U , the constantΘ − in the estimate (3.19) can be improved by replacing Θ from formula (3.16) by a smallerpositive constant Θ ◦ = def max σ ∈S ◦ F ( X ) , dim σ = n − k n(cid:13)(cid:13) [ D k +1 σ ] (cid:13)(cid:13) S • F ( U )∆ o (3.23)which has a smaller universal upper bound, also depending on S , n , and k only.The rest of the arguments are identical with the ones from the proof of Theorem 3.1. (cid:3) Retracing the proof of Theorem 3.2, we get the following
Corollary 3.2.
Under the amenability hypotheses of Theorem 3.2, if h ∈ H k +1 ( X ; Z ) isrepresented by a singular pseudo-manifold f : M → X , then (Θ ◦ ) − · k h k ∆ gives a lowerbound of the number of transversal intersections of the absolute cycle f ( M ) with the locus X F ◦− ( k +1) = def int ( X ) ∩ X F − ( k +1) . ♦ For a pair of topological spaces Z ⊃ W and an integer j ≥
0, consider the sub-space/subgroup K ∆=0 j ( Z, W ) ⊂ H j ( Z, W ) , formed by the homology classes h ∈ H j ( Z, W )whose simplicial semi-norm k h k ∆ = 0. Then k ∼ k ∆ becomes a norm on the quotient space H ∆ j ( Z, W ) = def H j ( Z, W ) / K ∆=0 j ( Z, W ) . Note that if Z admits a continuous self map φ : ( Z, W ) → ( Z, W ) whose action φ ∗ : H j ( Z, W ) → H j ( Z, W ) on homology has an eigen-element h such that φ ∗ ( h ) = λ · h for ascalar λ , subject to | λ | >
1, then k h k ∆ ≥ k φ ∗ ( h ) k ∆ = | λ | · k h k ∆ . Thus k h k ∆ = 0; so such an element h ∈ H j ( Z, W ) dies in the quotient H ∆ j ( Z, W ).In fact, the construction of H ∆ ( ∼ ) always produces a trivial result: for any Z , H ∆ ( Z ) =0. However, if Z is a closed surface of genus g >
1, then H ∆ ( Z ) = 0. Moreover, when Z is a product of many closed surfaces of genera g ( M i ) ≥
2, then H ∆ ( Z ) = 0 is rich.If f : Z → Y is a continuous map of topological spaces, then k h k ∆ ≥ k f ∗ ( h ) k ∆ . There-fore, f induces a continuous linear map of normed spaces f ∗ : H ∆ j ( Z ) → H ∆ j ( Y ) whoseoperator norm is ≤
1. It is not difficult to verify that when f is a homotopy equiva-lence, then this map f ∗ is an isometry (in the simplicial norm) between the normed spaces H ∆ j ( Z ) and H ∆ j ( Y ). In particular, the shape of the unit ball in H ∆ j ( Z ) is an invariant ofthe homotopy type of Z .We can take this observation one step further. The Mapping Theorem from [Gr],section 3.1, implies that the classifying map f : Z → K ( π ( Z ) ,
1) induces an isometry f ∗ : H ∆ j ( Z ) → H ∆ j (cid:0) K ( π ( Z ) , (cid:1) (so that f ∗ is a monomorphism). OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 23
Question 3.1.
For a given group π , how to describe the shape of the unit ball B j in thenormed space H ∆ j (cid:0) K ( π, R (cid:1) in terms of π (say, in terms of its subgroups or in terms ofthe representation theory)? Is B j a polyhedron for a finitely-presented π ? ♦ It turns out that formula (3.18) gives a nontrivial lower boundary for the number ofstrata σ ∈ S • F ( DX ) of dimension n − k for any shadow F ∈ Shad ( X, E ⇒ T ). Thisuniversal boundary is constructed in terms of the group H ∆ k +1 ( DX ).The next theorem is also an expression of the amenable localization, coupled with thePoincar´e duality. It can be viewed as a version of Morse Inequalities , where a Morsefunction f : X → R is replaced by a shadow F , the f -critical points are replaced by the F -induced strata in the double DX , and the homology of X by the “homology” H ∆ ∗ ( DX ).It shows, in particular, that the groups H ∆ k +1 ( DX ) provide lower bounds of the ranks ofthe groups C n − k ℧ ( DX, F ) from the differential complex in (3.9).
Theorem 3.3.
Let X be a compact connected ( n + 1) -dimensional PL -manifold with anonempty boundary. • Let, for each connected component of the boundary ∂X , the image of its fundamentalgroup in π ( DX ) be amenable. Then, for each k ∈ [ − , n ] and every shadow F ∈ Shad ( X, E ⇒ T ) the Poincar´e duality localizing operator L F, ℧ k +1 from (3.10) hasthe property rk (cid:0) im( L F, ℧ k +1 ) (cid:1) ≥ rk (cid:0) H ∆ k +1 ( DX ) (cid:1) . (3.24) As a result, rk (cid:0) C n − k ℧ ( DX, F ) (cid:1) —the number of strata σ ∈ S • F ( DX ) of dimension ( n − k ) —is greater than or equal to rk (cid:0) H ∆ k +1 ( DX ) (cid:1) . • Let, for each connected component of the boundary ∂X , the image of its fundamentalgroup in π ( X ) be amenable. Then, for for each k ∈ [0 , n ] and every shadow F ∈ Shad ( X, E ⇒ T ) , the Poincar´e duality localizing operator M F, ℧ k +1 from (3.21)has the property rk (cid:0) im( M F, ℧ k +1 ) (cid:1) ≥ rk (cid:0) H ∆ k +1 ( X ) (cid:1) . (3.25) Thus, rk (cid:0) C n − k ℧ ( X, F ◦ ) (cid:1) —the number of strata σ ∈ S ◦ F ( X ) of dimension ( n − k ) —isgreater than or equal to rk (cid:0) H ∆ k +1 ( X ) (cid:1) .Proof. Put K ∆=0 k +1 = def K ∆=0 k +1 ( DX ). If h ∈ H k +1 ( DX ) belongs to the kernel of the ho-momorphism L F, ℧ k +1 , then evidently | [ L F, ℧ k +1 ( h )] | ℧ = | [0] | ℧ = 0. By the inequality (3.17), k h k ∆ = 0. Therefore, h ∈ K ∆=0 k +1 . In other words, ker( L F, ℧ k +1 ) ⊂ K ∆=0 k +1 . Hence,rk (cid:0) H k +1 ( DX ) / K ∆=0 k +1 (cid:1) ≤ rk (cid:0) H k +1 ( DX ) / ker( L F, ℧ k +1 ) (cid:1) = rk (cid:0) im( L F, ℧ k +1 ) (cid:1) . So we get rk (cid:0) H ∆ k +1 ( DX ) (cid:1) ≤ rk (cid:0) C n − k ℧ ( DX, F ) / B n − k ℧ ( DX, F ) (cid:1) ≤ rk (cid:0) C n − k ℧ ( DX, F ) (cid:1) , (3.26)the later rank is the number of strata σ of dimension n − k in the stratification S • F ( DX ). Similar arguments, based on Theorem 3.2, prove thatrk (cid:0) H ∆ k +1 ( X ) (cid:1) ≤ rk (cid:0) C n − k ℧ ( X, F ◦ ) / B n − k ℧ ( X, F ◦ ) (cid:1) ≤ rk (cid:0) C n − k ℧ ( X, F ◦ ) (cid:1) . (3.27) (cid:3) Remark 3.1.
Since H ∆ ( DX ) = 0 and H ∆ ( X ) = 0 for all X , the statements of Theorem3.3 are vacuous for k = 0. Also, for k = n , the statement in the second bullet is vacuoussince H ∆ n +1 ( X ) = 0.Since the classifying map β : DX → K (Π , π ( DX ), induces an iso-morphism of the fundamental groups, β ∗ : H ∗ ( DX ) → H ∗ ( K (Π , β ∗ : H ∆ ∗ ( DX ) → H ∆ ∗ ( K (Π , the best lower estimate that formula (3.26) could provide is equal tork (cid:0) H ∆ k +1 (cid:0) K (Π , (cid:1)(cid:1) ≥ rk (cid:0) H ∆ k +1 ( DX ) (cid:1) . Similarly, with π = π ( X ), the best lower estimate that formula (3.27) could provide isequal to rk (cid:0) H ∆ k +1 (cid:0) K ( π, (cid:1)(cid:1) ≥ rk (cid:0) H ∆ k +1 ( X ) (cid:1) . ♦ Question 3.2.
Can one probe faithfully the shape of the unit ball ˜ B k +1 in the quotientnorm k ∼ k ∆ on H ∆ k +1 ( DX ) by counting the cardinalities (cid:16) f ( M ) ∩ DX F − ( k +1) (cid:17) for variousshadows F and singular pseudo-manifolds f : M → DX ? Similar question can be posedabout the shape of the unit ball B k +1 in the quotient norm k ∼ k ∆ on H ∆ k +1 ( X ). ♦ In one interesting special case that deals with the strata of maximal codimension The-orems 3.1-3.3 can be made more explicit. The proposition below is a generalization ofTheorem 2 from [AK].
Theorem 3.4.
Let X be a compact connected and oriented ( n +1) -dimensional PL -manifoldwith a nonempty boundary.Assume that, for each connected component of the boundary ∂X , the image of its funda-mental group in π ( DX ) is amenable. Then there is a X -independent universal constant θ > such that, for any shadow F ∈ Shad ( X, E ⇒ T ) , the cardinality of the finite set F ( X ) − n satisfies the inequality (cid:0) F ( X ) − n (cid:1) ≥ θ · k [ DX ] k ∆ . (3.28) Proof.
The argument is based on the proof of Theorem 3.1. Note that the fundamentalclass [ DX ] ∈ H n +1 ( DX ), being restricted to the regular neighborhood U = def a { σ ∈S • F ( DX ) | µ ′ ( ω σ )= n } D n +1 σ OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 25 of the locus DX F − ( n +1) , equals the sum X { σ ∈S • F ( DX ) | µ ′ ( ω σ )= n } [ D n +1 σ , ∂D n +1 σ ] . Therefore, by (3.14) , [ | [ DX ] U | ] = (cid:16) DX F − ( n +1) (cid:17) . Employing inequality (3.18), we get k [ DX ] | ∆ ≤ Θ · (cid:16) DX F − ( n +1) (cid:17) ≤ Θ · κ · (cid:0) F ( X ) − n (cid:1) , where κ = def max { ω | µ ′ ( ω )= n } ( µ ( ω ) − µ ′ ( ω )) is the maximal cardinality of the fibers of themap F : DX F − ( n +1) = F − (cid:0) F ( X ) − n (cid:1) ∩ ∂X −→ F ( X ) − n . Choosing θ = (Θ · κ ) − completes the argument.Note that X has no absolute fundamental cycle and that X F ◦− ( n +1) = ∅ ; therefore applyingthe second bullet of Theorem 3.3 leads to a tautology. (cid:3) The next implication of Theorem 3.3 reveals the groups H ∆ k +1 ( DX ) and H ∆ k ( X ) as obstructions to the existence of globally k -convex shadows of X . Corollary 3.3.
Let X be a compact connected ( n + 1) -dimensional PL -manifold with anonempty boundary. Assume that, for each connected component of the boundary ∂X , theimage of its fundamental group in π ( X ) is an amenable group.If X is globally k -convex, then the simplicial semi-norm is trivial on H j +1 ( DX ) and on H j ( X ) for all j ≥ k .In particular, if an oriented X admits a shadow F ∈ Shad ( X, E ⇒ T ) such that F ( X ) − n = ∅ , then k [ DX ] k ∆ = 0 and H ∆ n ( X ) = 0 .Proof. By Remark 3.2, F ( X ) − k = ∅ implies F ( X ) − j = ∅ for all j ≥ k . Therefore DX F − ( j +1) = ∅ for all j ≥ k , since the ( n − j )-dimensional strata from S • F ( DX ) are con-tributed only by the strata of F ( X ) of dimensions ( n − j ) and ( n − j − S • F ( DX ): they are contributedby the points from F ( X ) − n only.In contrast, only the ( n − j )-dimensional strata of F ( X ) contributes to the ( n + 1 − j )-dimensional strata from S ◦ F ( X ). So F ( X ) − j = ∅ implies X F ◦− j = ∅ . In turn, this implies C n +1 − j ℧ ( X, F ◦ ) = 0. Therefore, by (3.27), H ∆ j ( X ) = 0 for all j ≥ k . (cid:3) Definition 3.8.
Let X be a compact connected ( n + 1)-dimensional PL -manifold witha nonempty boundary, and A denotes an abelian group or a field. For any shadow F ∈ Shad ( X, E ⇒ T ) and each j ∈ [0 , n + 1], letΣ c j ( X, F ) = def rk A (cid:0) C j ℧ ( DX, F ) (cid:1) . We call this integer the j -th suspension ℧ -complexity of the shadow F .Let Σ c j shad ( X ) = def min F ∈ Shad ( X, E ⇒ T ) Σ c j ( X, F ) . We call Σ c j shad ( X ) the j -th suspension shadow complexity of X . ♦ These numbers can be organized into a sequence Σc shad ( X ) = def (cid:0) Σ c shad ( X ) , Σ c shad ( X ) . . . Σ c n +1 shad ( X ) (cid:1) . Note that this optimal sequence may not be realizable by a single shadow! To avoid thisfundamental difficulty, we will need a more manageable version of the previous definition.
Defintion 3.8.
Let X be a compact connected ( n + 1)-dimensional PL -manifold with anonempty boundary. For any shadow F ∈ Shad ( X, E ⇒ T ), form the sequence Σc ( X, F ) = def (cid:0) Σ c ( X, F ) , Σ c ( X, F ) , . . . , Σ c n +1 ( X, F ) (cid:1) . and take the lexicographic minimum Σc lexshad ( X ) = def lex.min F ∈ Shad ( X, E ⇒ T ) Σc ( X, F ) . We denote by Σ c lex , j shad ( X ) the ( j + 1) component of the vector Σc lexshad ( X ) and call it the j -th lexicographic suspension shadow complexity of X . ♦ Remark 3.2.
By its very definition, the lexicographically optimal sequence of complexitiesis delivered by some shadow F !Evidently, for each j , Σ c lex , j shad ( X ) ≥ Σ c j shad ( X ) . ♦ Theorem 3.4 has an immediate implication.
Corollary 3.4.
Let X be a compact connected ( n + 1) -dimensional PL -manifold with anonempty boundary. Assume that, for each connected component of the boundary ∂X , theimage of its fundamental group in π ( DX ) is an amenable group.Then, for any k ∈ [ − , n ] , the suspension shadow complexity of X satisfies the inequality Σ c n − k shad ( X ) ≥ rk (cid:0) H ∆ k +1 ( DX ) (cid:1) . ♦ Let us consider the sequence rk (cid:0) H ∆ ∗ ( DX ) (cid:1) = def (cid:16) rk (cid:0) H ∆ n +1 ( DX ) (cid:1) , rk (cid:0) H ∆ n ( DX ) (cid:1) , . . . , rk (cid:0) H ∆ ( DX ) (cid:1)(cid:17) . Then Corollary 3.4 can be expressed in its compressed form as Σc lexshad ( X ) ≥ Σc shad ( X ) ≥ rk (cid:0) H ∆ ∗ ( DX ) (cid:1) , (3.29)where the vectorial inequality is understood as the inequality among all the the correspond-ing components of the participating vectors.Let X be a compact connected ( n +1)-dimensional PL -manifold with a nonempty bound-ary, and F : X → K its shadow. Recall that, by Definition 3.1, the cardinality of the fiber F : X ( F, ω ) ∩ ∂X → K ( ω ) depends only on ω ∈ S : it is the difference µ ( ω ) − µ ′ ( ω ). Definition 3.9.
Consider a filtered CW -complex K ∈ Shad ( T , n ) and its filtration as in(3.1). Put κ j ( n ) = def max { ω | µ ′ ( ω )= n − j } (cid:0) µ ( ω ) − µ ′ ( ω ) (cid:1) . OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 27
The weighted j -th complexity of K is defined by the formula ♯ c j ( K ) = def κ j ( n ) · c j ( K ) . ♦ The next proposition helps to link the suspension complexities Σ c j ( X, F ) to the weightedones ♯ c j ( F ( X )) and thus motivates Definition 3.9. Corollary 3.5.
Let X be a compact connected ( n + 1) -dimensional PL -manifold with anonempty boundary. Assume that, for each connected component of the boundary ∂X , theimage of its fundamental group in π ( X ) is an amenable group.Then, for any shadow F ∈ Shad ( X, E ⇒ T ) and each k ∈ [ − , n ] , ♯ c n − k ( F ( X )) + 2 · ♯ c n − k − ( F ( X )) ≥ rk (cid:0) H ∆ k +1 ( DX ) (cid:1) , (3.30) ♯ c n − k − ( F ( X )) ≥ rk (cid:0) H ∆ k +1 ( X ) (cid:1) . Proof.
Let K = F ( X ) be a shadow of X . Any connected component of K ( ω ) of dimension n − µ ′ ( ω ) (by Definition 3.1) gives rise to µ ( ω ) − µ ′ ( ω ) strata σ ∈ S • F ( DX ) of dimension n − µ ′ ( ω ) and to 2( µ ( ω ) − µ ′ ( ω ) −
1) strata σ ∈ S • F ( DX ) of dimension n + 1 − µ ′ ( ω ).Therefore (see Fig. 3) the number of strata σ ∈ S • F ( DX ) of dimension j = n − k is givenby the formula: X ω | µ ′ ( ω )= k ( µ ( ω ) − k ) · (cid:16) π (cid:0) K ( ω ) (cid:1)(cid:17) + X ω | µ ′ ( ω )= k +1 µ ( ω ) − k − · (cid:16) π (cid:0) K ( ω ) (cid:1)(cid:17) . (3.31)By the definition of { κ j ( n ) } j , the latter number is smaller than or equal to κ n − k ( n ) · X ω | µ ′ ( ω )= k (cid:16) π (cid:0) K ( ω ) (cid:1)(cid:17) + 2 κ n − k − ( n ) · X ω | µ ′ ( ω )= k +1 (cid:16) π (cid:0) K ( ω ) (cid:1)(cid:17) , where the fist sum is the number of strata in F ( X ) of dimension n − k , and the secondsum is the number of strata of dimension n − k −
1. Thus the previous formula is an upperbound of the number of components in S • F ( DX ) of dimension n − k .Similar upper estimate of the number of connected components in X F ◦− k leads to the LHSof the second inequality in (3.30).Now Theorem 3.3 implies formulas (3.30). (cid:3) Corollary 3.5 has the following two immediate implications which reveal the non-trivialityof the groups/spaces H ∆ k +1 ( DX ) ⊂ H ∆ k +1 (Π) and H ∆ k ( X ) ⊂ H ∆ k ( π ) as obstructions tothe existence of shadows F of low/vanishing complexity. The first implication is just arepackaging of Corollary 3.3. Corollary 3.6.
Under the hypotheses of Corollary 3.5, if H ∆ k +1 ( DX ) = 0 , then either c n − k ( F ( X )) = 0 or c n − k − ( F ( X )) = 0 for any shadow F .If H ∆ k ( X ) = 0 , then c n − k − ( F ( X )) = 0 for any shadow F . ♦ Corollary 3.7.
Let X be a compact connected ( n + 1) -dimensional PL -manifold with anonempty boundary. Assume that, for each connected component of the boundary ∂X , theimage of its fundamental group in π ( X ) is amenable. Then, for each k ∈ [ − , n ] , ♯ c n − k shad ( X, E ⇒ T ) + 2 · ♯ c n − k − shad ( X, E ⇒ T ) ≥ rk (cid:0) H ∆ k +1 ( DX ) (cid:1) , (3.32) ♯ c n − k − shad ( X, E ⇒ T ) ≥ rk (cid:0) H ∆ k +1 ( X ) (cid:1) . ♦ Complexity of Traversing Flows
Now let us examine applications of these results about the complexities of shadows tothe traversing flows on smooth manifolds with boundary. In fact, the entire general settingin Section 3 was designed with these applications in mind. As we will show next, anyproposition about shadows of PL -manifolds with boundary and the simplicial semi-normshas an analogue for the smooth manifolds with boundary that carry a traversally genericvector field. In short, the traversally generic fields are a good source of shadows.We will apply Theorem 3.1, Theorem 3.3, and their corollaries to traversing vector fields v . In this setting, the poset S = def Ω • , µ := | ∼ | , µ ′ := | ∼ | ′ (see (2.2)), the role ofshadows is played by the obvious maps Γ : X → T ( v ) which belong to an appropriate set Shad ( X, E ⇒ T ). Theorem 4.1.
Let X be a smooth compact connected and oriented ( n + 1) -manifold withboundary. Any traversally generic vector field v gives rise to a shadow Γ : X → T ( v ) inthe sense of Definition 3.1. The the model projections { p ω : E ω → T ω } ω ∈ Ω •′h n ] are describedin [K3] , Theorems 7.4 and 7.5., utilizing special coordinates as in (2.4).Proof. Recall that, for a traversally generic v , the trajectory space T ( v ) is a Whineystratified space (see [K4], Theorem 2.2), which implies that Γ is a simplicial map withrespect to the appropriate triangulations of the source and target spaces. Moreover, thetriangulation of X can be chosen to be smooth.By the very definition of a traversing flow, each fiber Γ − ( γ ) , γ ∈ T ( v ) is either a closedsegment, or a singleton.By Corollary 5.1 from [K3], the map Γ : ∂ X → T ( v ), being restricted to the preimage ofeach proper stratum T ( v, ω ), is a cover with the trivial monodromy and fibers of cardinality ω )) = | ω | − | ω | ′ = µ ( ω ) − µ ′ ( ω ) . Moreover, by Theorem 2.1 from [K4], each pure stratum T ( v, ω ) is an open orientablesmooth manifold.By Lemma 3.4 from [K2], Theorems 5.2 and 5.3 from [K3], each point γ ∈ T ( v, ω ), hasa regular neigborhood V γ ⊂ T ( v ), so that Γ : Γ − ( V γ ) → V γ is PL -homeomorphic to themodel projection p ω : E ω → T ω .This completes the checklist of bullets from Definition 3.1. (cid:3) Let X be a smooth compact connected and oriented ( n + 1)-manifold with boundary,and v a traversally generic vector field. OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 29
Let us consider a filtration T ( v ) = T ( v ) − ⊃ T ( v ) − ⊃ · · · ⊃ T ( v ) − n of the trajectory space T ( v ) by the closed subcomplexes T ( v ) − j = def [(cid:8) ω ∈ Ω •′h n ] (cid:12)(cid:12) | ω | ′ ≥ j (cid:9) T ( v, ω )(4.1)of dimensions n − j .Let A be an abelian coefficient system on T ( v ) (equivalently, on X ). As a default, A = R . For each j ∈ [0 , n ], consider the relative homology groups { C ℧ j ( T ( v )) := H j (cid:0) T ( v ) − n + j , T ( v ) − n + j − ; A (cid:1) } j (4.2)associated with the filtration. As in the context of shadows, C ℧ j ( T ( v )) is the top homologyof the quotient T ( v ) − n + j / T ( v ) − n + j − .These homology groups can be organized into a differential complex C ℧ ∗ ( T ( v )) = def (4.3) (cid:8) → C ℧ n ( T ( v )) ∂ n → C ℧ n − ( T ( v )) ∂ n − → · · · ∂ → C ℧ ( T ( v )) → (cid:9) , where the differentials { ∂ j } are the boundary homomorphisms from the long exact homol-ogy sequences of the triples {T ( v ) − n + j ⊃ T ( v ) − n + j − ⊃ T ( v ) − n + j − } j . The modules C ℧ j ( T ( v )) are free and finitely generated, the number of generators beingthe number of connected components of the set T ( v ) ◦− n + j = def T ( v ) − n + j \ T ( v ) − n + j − . This observation is valid, since by Theorem 2.1 from [K4], the components of T ( v ) ◦− n + j are orientable smooth manifolds.The differential complex (4.3) gives rise to the ℧ - homology groups H ℧ j ( v ) = def H ℧ j ( T ( v ))of the traversally generic field v .As in the category of shadows, the traversing fields give rise to several notions of com-plexity. Definition 4.1.
With any traversally generic vector field v on a smooth compact connectedand oriented ( n +1)-manifold X with boundary, we associate the ordered collection of ranks n tc j ( v ) = def rk A (cid:0) C ℧ j (cid:0) T ( v ) (cid:1)(cid:1)o ≤ j ≤ n , where the groups C ℧ j ( T ( v ))) have been introduced in (4.2).We call tc j ( v ) the j -th traversing complexity (“ tc ” for short) of the field v . ♦ Remark 4.1.
Reviewing (4.2), we notice that tc n − k ( v ) = 0 implies that T ( v ) ◦− k = ∅ since T ( v ) ◦− k is an orientable manifold. Examining the local models { p ω : E ω → T ω } ω ,we observe that if a particular combinatorial type ω ′ is missing in a model, then all the in our notations, we suppress the dependance of these homology groups on the coefficients A . smaller combinatorial types ω ′′ ≺ ω ′ (of greater codimensions) are missing as well (see[K3]). Therefore T ( v ) ◦− k = ∅ implies that {T ( v ) − j = ∅} j ≥ k . As a result, if the complexity tc n − k ( v ) = 0, then the complexities tc n − j ( v ) = 0 for all j ≥ k . ♦ Definition 4.2.
Let X be a compact connected and oriented smooth ( n +1)-manifold withboundary. For each traversally generic field v ∈ V ‡ ( X ), we form the sequence of traversingcomplexities: tc ( v ) = def { tc ( v ) , tc ( v ) , . . . , tc n ( v ) } Consider the lexicographical minimum tc lex ( X ) = def lex.min { v ∈V ‡ ( X ) } tc ( v )(4.4)We call this vector the lexicographic traversing complexity of X .We denote by tc lex j ( X ) the ( j + 1)-component of the vector tc lex ( X ). ♦ Remark 4.2.
If a compact connected and oriented smooth manifold X of dimension n + 1 is globally k -convex in the sense of Definition 2.2, then evidently tc lex j ( X ) = 0 for all j ≤ n − k in accordance with Remark 4.1. ♦ Remark 4.3.
Thanks to Theorem 4.1, traversally generic fields produce a particular kindof shadows from the set
Shad ( X, E ⇒ T ). Therefore, c lexshad ( X ) ≤ tc lex ( X ) . ♦ As with the shadows, we will need “suspensions” of these notions and constructions.We consider the stratification Ω • ( X, v ) of X by the connected components of the strata (cid:8) X ◦ ( v, ω ) = def X ( v, ω ) \ ( ∂X ∩ X ( v, ω )) , X ∂ ( v, ω ) = def ∂X ∩ X ( v, ω ) (cid:9) ω ∈ Ω •′h n ] . (4.5)and the τ -invariant stratification Ω • ( DX, v ) of the double DX , which is induced by Ω • ( X, v ).Let X v − ( k +1) = def (cid:16) [(cid:8) ω (cid:12)(cid:12) | ω | ′ ≥ k +1 (cid:9) X ◦ ( v, ω ) (cid:17) [ (cid:16) [(cid:8) ω (cid:12)(cid:12) | ω | ′ ≥ k (cid:9) X ∂ ( v, ω ) (cid:17) , The stratifications Ω • ( X, v ) and Ω • ( DX, v ) give rise to the filtrations: X = def X v − ⊃ X v − ⊃ · · · ⊃ X v − ( n +1) ,DX = def DX v − ⊃ DX v − ⊃ · · · ⊃ DX v − ( n +1) by the union of strata of a fixed codimension, each pure stratum being an open manifold.Analogously to (3.7), we consider the homology and cohomology groups with coefficientsin A = R or A = Z : { C ℧ j (cid:0) DX, v (cid:1) = def H j (cid:0) DX vj − n − , DX vj − n ; A (cid:1) } j , (4.6) { C j ℧ (cid:0) DX, v (cid:1) = def H j (cid:0) DX vj − n − , DX vj − n ; A (cid:1) } j . OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 31
As in (3.8) and (3.9), they can be organized into a differential complex: C ℧ ∗ ( DX, v ) = def (cid:8) → C ℧ n +1 ( DX, v ) ∂ n +1 → C ℧ n ( DX, v ) ∂ n → · · · ∂ → C ℧ ( DX, v ) → (cid:9) , (4.7)where the differentials { ∂ j } are the boundary homomorphisms from the long exact homol-ogy sequences of triples { DX v − j +1 ⊃ DX v − j ⊃ DX v − j − } j . Similarly, we introduce the dual differential complex C ∗ ℧ ( DX, v ) = def (cid:8) ← C n +1 ℧ ( DX, v ) ∂ ∗ n +1 ← C n ℧ ( DX, v ) ∂ ∗ n ← · · · ∂ ∗ ← C ℧ ( DX, v ) ← (cid:9) (4.8)These differential complexes produce the suspension ℧ -homology H ℧ ∗ (cid:0) DX, v (cid:1) and ℧ -cohomology H ∗ ℧ (cid:0) DX, v (cid:1) of traversally generic v -flows on X . Definition 4.3.
Let X be a compact connected oriented and smooth ( n + 1)-dimensionalmanifold with a boundary. For any traversally generic field v ∈ V ‡ ( X ) and each j ∈ [0 , n + 1], let Σ tc j ( X, v ) = def rk A (cid:0) C j ℧ ( DX, v ) (cid:1) . We call this integer the j -th suspension ℧ -complexity of the field v . ♦ Definition 4.4.
Let X be a compact connected oriented and smooth ( n + 1)-dimensionalmanifold with a boundary. For any traversally generic field v ∈ V ‡ ( X ), we form thesequence Σtc ( X, v ) = def (cid:0) Σ tc ( X, v ) , Σ tc ( X, v ) , . . . , Σ tc n +1 ( X, v ) (cid:1) . and take the lexicographic minimum Σtc lextrav ( X ) = def lex.min v ∈V ‡ ( X ) Σtc ( X, v ) . We denote by Σ tc lex , j trav ( X ) the ( j + 1) component of the vector Σc lextrav ( X ) and call it the j -th lexicographic suspended traversing complexity of X . ♦ Remark 4.4.
By its very definition, the lexicographically optimal sequence of complexitiesis delivered by some traversally generic vector field!Evidently,
Σtc lextrav ( X ) ≥ Σc lexshad ( X ) , where “ shad ” refers to the shadows, based on the list of model maps { E ω → T ω } ω ∈ Ω •′h n ] exemplifying the local structure of traversally generic flows.A priori, both vectors, tc lextrav ( X ) and Σtc lextrav ( X ), are invariants of the smooth topologicaltype of X , while c lexshad ( X ) and Σc lexshad ( X ) are invariants of the PL -topological type. ♦ Consider the space V ‡ ( X ) of traversally generic vector fields on X and its subspace V ‡ fold ( X ), formed by fields for which the multiplicity m ( a ) ≤ v -trajectory γ ateach point a ∈ γ ∩ ∂X . Locally, for such fields v , Γ : ∂X → T ( v ) is a folding map. Definition 4.5.
Let X be a compact connected oriented and smooth ( n + 1)-dimensionalmanifold with a boundary. For any traversally generic field v ∈ V ‡ fold ( X ) we define the lexicographic fold complexity of X by tc lexfold trav ( X ) = def lex.min v ∈V ‡ fold ( X ) tc ( v ) . ♦ Clearly, tc lexfold trav ( X ) ≥ tc lextrav ( X ) , provided V ‡ fold ( X ) = ∅ . When X is a 3-fold, byTheorem 9.5 in [K], tc fold trav ( X ) = tc trav ( X ) . Besides the inequality above, we know little about the relation between tc lexfold trav ( X ) and tc lextrav ( X ).Now let us examine how the marriage of Amenable Localization and Poincar´e Dualityworks in the environment of traversing fields.For each k ∈ [ − , n ], by composing the Poincar´e duality map D with the localization tothe subsets DX v − ( k +1) ⊂ DX and X v ◦− ( k +1) ⊂ X , we get two localization transfer maps L vk +1 : H k +1 (cid:0) DX (cid:1) ≈D −→ H n − k (cid:0) DX (cid:1) i ∗ → H n − k (cid:0) DX v − ( k +1) (cid:1) , M vk +1 : H k +1 (cid:0) X (cid:1) ≈D −→ H n − k (cid:0) X, ∂X (cid:1) i ∗ → H n − k (cid:0) X v − ( k +1) , X v − ( k +2) ∪ ( ∂X ∩ X v − ( k +1) ) (cid:1) . whose targets can be identified with the quotients C n − k ℧ ( DX, v ) / B n − k ℧ ( DX, v ) , and C n − k ℧ ( X, v ◦ ) / B n − k ℧ ( X, v ◦ ) , respectively. We denote by L v, ℧ k +1 : H k +1 (cid:0) DX (cid:1) → C n − k ℧ ( DX, v ) / B n − k ℧ ( DX, v ) , (4.9) M v, ℧ k +1 : H k +1 (cid:0) X (cid:1) → C n − k ℧ ( X, v ◦ ) / B n − k ℧ ( X, v ◦ )the resulting operators.Now Theorem 3.1, being combined with Theorem 4.1, delivers Theorem 4.2 ( Amenable localization of the Poincar´e duality for traversingflows).
Let X be a compact connected oriented and smooth ( n + 1) -manifold X with boundary. • Assume that for each connected component of the boundary ∂X , the image of itsfundamental group in π ( DX ) is amenable. Then, for each for each k ∈ [ − , n ] ,there exists an universal constant Θ > such that, for any traversally genericvector field v , the space C n − k ℧ ( DX, v ) / B n − k ℧ ( DX, v ) admits a norm | [ ] | ℧ so that k h k ∆ ≤ Θ · (cid:12)(cid:12) [ L v, ℧ k +1 ( h )] (cid:12)(cid:12) ℧ (4.10) for any h ∈ H k +1 ( DX ) . Here the Poincar´e duality localizing operator L v, ℧ k +1 isintroduced in (4.9), and the universal constant Θ > depends only on the list ofmodel maps { p ω : E ω → T ω } ω ∈ Ω •′h n ] in the way that is described in formula (3.16). OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 33 • Let, for each connected component of the boundary ∂X , the image of its fundamentalgroup in π ( X ) be an amenable group. Then, for each k ∈ [0 , n ] , there exists anuniversal constant Θ ◦ > such that, for any traversally generic vector field v , thespace C n − k ℧ ( X, v ◦ ) / B n − k ℧ ( X, v ◦ ) admits a norm | [ ] | ℧ so that k h k ∆ ≤ Θ ◦ · (cid:12)(cid:12) [ M v, ℧ k +1 ( h )] (cid:12)(cid:12) ℧ (4.11) for any h ∈ H k +1 ( X ) . Here the Poincar´e duality localizing operator operator M v, ℧ k +1 is introduced in (4.9), and the constant Θ ◦ > depends only on the list of modelmaps { p ω : E ω → T ω } ω ∈ Ω •′h n ] in the way that is described in formula (3.23). ♦ Theorem 4.1, together with Corollary 3.1 and Corollary 3.2, produces
Corollary 4.1.
Let an integral homology class h ∈ H k +1 ( DX ) be realized my a singularoriented pseudo-manifold f : M → DX , dim( M ) = k + 1 .Under the hypotheses of Theorem 4.2, the number of intersections of the cycle f ( M ) withthe locus DX v − ( k +1) is greater than or equal to Θ − · k h k ∆ , provided that f is in generalposition with the subcomplex DX v − ( k +1) ⊂ DX .If a class h ∈ H k +1 ( X ) is represented by a singular pseudo-manifold f : M → X , then (Θ ◦ ) − · k h k ∆ gives a lower bound of the number of transversal intersections of the absolutecycle f ( M ) with the locus X v ◦− ( k +1) = def int ( X ) ∩ X v − ( k +1) . ♦ The following example is a product of the author’s conversations with Larry Guth.
Example 4.1.
Consider a fibration p : E → M whose base is a closed oriented hyperbolicmanifold of dimension k +1 and whose fiber F is a closed manifold. Assume that the ( n +1)-dimensional manifold E is oriented and that p admits a section s : M → E . In the com-plement to s ( M ), pick a smooth codimension zero manifold V such that { π ( ∂V, pt ) } pt ∈ ∂V are amenable groups. Let X = E \ int( V ).Recall that k [ M ] k ∆ is proportional to the hyperbolic volume vol( M ) with an univer-sal positive proportionality constant [Gr]. Since s is a section, and the simplicial semi-norm does not increase under the continuous maps, it follows that the simplicial norm of s ∗ ([ M ]) ∈ H k +1 ( X ) is proportional to vol( M ) with the same proportionality constant.For any traversally generic vector field v on X , consider the locus X v − ( k +1) . We canperturb the section s so that the intersections of the cycle s ( M ) is transversal to X v − ( k +1) .According to Corollary 4.1, there exits a universal constant ρ > X and any traversally generic vector field v on X , the transversal intersections of theperturbed cycle s ( M ) with the locus X v − ( k +1) has ρ · vol( M ) intersections at least.Since by the choice of V , s ( M ) ∩ ∂V = ∅ , it follows that there exist at least ρ · vol( M )trajectories γ of the reduced multiplicity m ′ ( γ ) = k + 1, each of which has a nonemptyintersection with the section s ( M ) ⊂ int( X ) (and evidently with ∂X = ∂V ). ♦ It worth describing the important special case “ h = [ DX ]” of formula (3.16), beingapplied to the shadows that are produced by traversally generic flows. Theorem 4.3.
Let X be a smooth compact connected and oriented ( n + 1) -manifold withboundary. Assume that, for each connected component of the boundary ∂X the image ofits fundamental group in π ( DX ) is an amenable group.Let v be a traversally generic vector field on X . Then X ω ∈ Ω • | | ω | ′ = n θ ( ω ) · (cid:0) T ( v, ω ) (cid:1) ≥ (cid:13)(cid:13) [ DX ] (cid:13)(cid:13) ∆ , (4.12) where the universal constant θ ( ω ) = def (cid:13)(cid:13) [ D E ω , D ( δ E ω )] (cid:13)(cid:13) Ω •′h n ] ∆ is the Ω • ′ h n ] -stratified simplicial volume of the model pair ( D E ω , D ( δ E ω )) . ♦ The previous theorem represents a slight modification/improvement of Theorem 2 from[AK]; below we state it for the reader’s convenience.
Theorem 4.4. (Alpert, Katz)
Let X be a smooth compact connected and oriented ( n +1) -manifold with boundary. Assume that, for each connected component of the boundary, theimage of its fundamental group in π ( DX ) is an amenable group. Let v be a traversallygeneric vector field on X .Then there is an universal constant ρ ( n ) > such that, for any X and v , the cardinalityof the set T ( v ) − n , formed by the trajectories of the maximal reduced multiplicity n , satisfiesthe inequality (cid:0) T ( v ) − n (cid:1) ≥ ρ ( n ) · k [ DX ] k ∆ . (4.13) ♦ Example 4.2.
Take the surface X ⊂ R and the constant vertical field v = ∂ y on it as inFig. 1. Since X is a disk with 4 holes, the double DX is a closed surface of genus 4. Itadmits a hyperbolic metric. By [Gr], it follows that k [ DX ] k ∆ = 2(2 · −
2) = 12 . At the same time, DX v − ) = 30 and T ( v ) − ) = 12 (see Fig. 1 and Fig. 3). Since ρ (1) ≤ (cid:0) T ( v ) − n (cid:1) / k [ DX ] k ∆ for any connected compact surface X such that k [ DX ] k ∆ =0 and a traversally generic v on it, the universal constant ρ (1) in (4.13) gets a bound: ρ (1) ≤ /
12 = 1.However, the field v in Fig. 1 is not the “optimal” one for X . To optimize v , take theradial field on an annulus A . Delete tree convex disks from A to form X and restrict theradial field to X . For the restricted field v on X , the graph T ( v ) is trivalent with 6 verticies.So we get T ( v ) − ) = 6 and DX v − ) = 18. As a result, the universal constant mustsatisfy the inequality ρ (1) ≤ /
12 = 1 /
2. Similar calculations apply to any 2-disk with q ≥ ρ (1) ≤ /
2. Perhaps, ρ (1) = 1 / ♦ Although desirable, an exact computation of the universal constants { θ ( ω ) } in formula(4.12) is somewhat tricky. We can estimate the number of cells-strata of the top dimension | ω | ′ in the model space T ω . It looks that this number can be calculated by counting the OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 35 chambers in which the discriminant of each polynomial ℘ i ( u, ~x ) = def ( u − i ) ω i + ω i − X j =0 x i,j ( u − i ) j divides the space R ω i − .For ω ⋆ = def (1 2 . . . | {z } n T ω ⋆ is formed by attaching n copies of a half-disk D n + to a disk D n along the coordinate hyperplanes which divide D n into 2 n quadrants.So the number κ ( ω ⋆ ) of n -dimensional cells-strata in T ω ⋆ is 2 n + n (see [K3]). Therefore, thepull-back of the Ω • -stratification in T ω ⋆ divides the model space E ω ⋆ into 2 n + n chambers-cells of dimension n + 1. As a result, the stratified simplicial norms satisfy the inequalities (cid:13)(cid:13) [ E ω ⋆ , δ E ω ⋆ ] (cid:13)(cid:13) Ω •′h n ] ∆ ≥ n + n, (cid:13)(cid:13) [ D E ω ⋆ , D ( δ E ω ⋆ )] (cid:13)(cid:13) Ω •′h n ] ∆ ≥ n +1 + 2 n. It seems that κ ( ω ) < κ ( ω ⋆ ) for any minimal ω = ω ⋆ .Of course, it is more desirable to get lower bounds of these stratified simplicial norms...Combining Theorem 3.3 with Theorem 4.1, we get the following proposition. Theorem 4.5.
Let X be a smooth compact connected and oriented manifold with boundary, dim( X ) = n + 1 . Let v be a traversally generic vector field on X . • Assume that, for each connected component of the boundary ∂X , the image of itsfundamental group in π ( DX ) is an amenable group. Then, for each k ∈ [ − , n ] , rk (cid:0) im( L v, ℧ k +1 ) (cid:1) ≥ rk (cid:0) H ∆ k +1 ( DX ) (cid:1) . (4.14) As a result, rk (cid:0) C n − k ℧ ( DX, v ) (cid:1) —the number of ( n − k ) -dimensional strata in thestratification Ω • ( DX, v ) of DX —is greater than or equal to rk (cid:0) H ∆ k +1 ( DX ) (cid:1) . • Assume that, for each connected component of the boundary ∂X , the image of itsfundamental group in π ( X ) is an amenable group. Then, for each k ∈ [ − , n − , rk (cid:0) im( M v, ℧ k +1 ) (cid:1) ≥ rk (cid:0) H ∆ k +1 ( X ) (cid:1) . (4.15) Thus, rk (cid:0) C n − k ℧ ( X, v ◦ ) (cid:1) —the number of ( n − k ) -dimensional strata in the stratifica-tion Ω • ( X, v ◦ ) —is greater than or equal to rk (cid:0) H ∆ k +1 ( X ) (cid:1) . ♦ The first bullet in Theorem 4.5, together with the definitions of suspension complexities,leads to
Theorem 4.6.
Let X be a smooth compact connected and oriented manifold with boundary, dim( X ) = n +1 . Assume that, for each connected component of the boundary ∂X , the imageof its fundamental group in π ( DX ) is an amenable group. Then, for each k ∈ [ − , n ] , Σtc lextrav ( X ) ≥ Σc lexshad ( X ) ≥ rk (cid:0) H ∆ ∗ (cid:0) DX (cid:1)(cid:1) , (4.16) see (11 . where the vectorial inequality is understood as the inequality among all the the correspondingcomponents of the participating vectors. The vector rk (cid:0) H ∆ ∗ (cid:0) DX (cid:1)(cid:1) has been introducedprior to formula (3.23). ♦ Recall that, for ( n + 1)-dimensional X and a traversally generic v on it, ω ) ≤ n + 2for all ω ’s ([K2]). By the construction of the stratification Ω • ( DX, v ) of the double DX (see Fig. 1 and Fig. 3), we get a formula which is similar to (3.31). It connects thecombinatorics of Ω • ( DX, v ) to the combinatorics of the stratification Ω • ( T ( v )) of T ( v ) bythe connected components of the strata T ( v, ω ): π (cid:0) DX v − j (cid:1) = X { ω | | ω | ′ = j } ω ) · π ( T ( v, ω ))+ 2 · X { ˆ ω | | ˆ ω | ′ = j +1 } ( ω ) − · π ( T ( v, ˆ ω )) ≤ ( n + 2) (cid:16) X { ω | | ω | ′ = j } π ( T ( v, ω )) + 2 · X { ˆ ω | | ˆ ω | ′ = j +1 } π ( T ( v, ˆ ω )) (cid:17) = ( n + 2) (cid:0) π (cid:0) T ( v ) ◦− j (cid:1) + 2 · π (cid:0) T ( v ) ◦− ( j +1) (cid:1)(cid:1) . Therefore, the suspended complexity of v can be estimated in terms of the v -complexities:Σ tc n +1 − j ( X, v ) ≤ ( n + 2) (cid:0) tc n − j ( v ) + 2 · tc n − j − ( v ) (cid:1) . By Theorem 4.5 and under its hypotheses, for each k , we getrk (cid:0) H ∆ k +1 ( DX ) (cid:1) ≤ ( n + 2) (cid:0) tc n − k ( v ) + 2 · tc n − k − ( v ) (cid:1) , (4.17) rk (cid:0) H ∆ k ( X ) (cid:1) ≤ ( n + 2) (cid:0) tc n − k ( v ) (cid:1) . We notice that if tc n − k ( v ) = 0, then by Remark 4.1, tc n − k − ( v ) = 0. Therefore, tc n − k ( v ) = 0 implies H ∆ k +1 ( DX ) = 0 and H ∆ k ( X ) = 0. Example 4.3.
Let { M i } ≤ i ≤ N be several closed orientable surfaces of genera g ( M i ) ≥ Y = def ( M × S ) M × S ) . . . M N × S ) , and let Z = Y \ B , the complement to a smooth 3-ball. Put Σ( M i ) = def M i × S .We may assume that the 1-handles { H i ≈ S × D } ≤ i ≤ N − , participating in the con-nected sum construction of Y , are attached to the complements to the surfaces M i × ⋆ i in M i × S and to the complement of B .Consider a map f j : Z → Σ( M j ) which is a homeomorphism on M j × ⋆ i . We construct f j in stages. First, we map each 1-handle H i ⊂ Y to its core D i so that Y is mapped tothe union W of Σ( M i )’s to which the 1-cores D i ’s are attached; each core is attached at apoint a i ∈ Σ( M i ) and at a point b i +1 ∈ Σ( M i +1 ). Under f j , each of the two 3-disks, D a i and D b i +1 , from D × ∂D i is mapped to its center a i or b i +1 . Finally for each j ∈ [2 , N − Y \ (cid:0) Σ( M j ) \ ( D b j ∪ D a j ) (cid:1) to a j ` b j . For j = 1 and j = N , theconstructions of f and f N are similar. OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 37
For each j , by an isotopy argument, we may assume that the ball B belongs to Y \ Σ( M j )and then restrict f j to Z .We claim that the 2-dimensional classes { I ∗ [ M i ] ∈ H ∆ ( Z ) } i , where I : ` i M i → Z isthe obvious embedding, are linearly independent. Indeed, assume that some combination h = def P i r i I ∗ [ M i ] has the property k h k ∆ = 0. Then k ( f j ) ∗ ( h ) k ∆ = 0. On the other hand,( f j ) ∗ ( h ) = r j · ( I j ) ∗ [ M j ] ∈ H (Σ( M j )) ≈ R , where the imbedding I j : M j ⊂ Σ( M j ), being composed with f j , produces the identitymap of M j . Therefore by the property of the simplicial semi-norm not to increase undercontinuous maps [Gr], k ( I j ) ∗ [ M j ] k ∆ = k [ M j ] k ∆ = 0. Thus r j = 0.Let X be a compact smooth 3-fold which is homotopy equivalent to Z and has a sphericalboundary. Let v be a traversally generic field on X . Since π ( ∂X ) = 0, by Theorem 4.5and following the arguments that lead to (3.31) and (4.17), we count the 1-dimensionalconnected strata in the stratification Ω • ( DX, v ) to get the inequality6 · T ( v, · T ( v,
13) + 2 · T ( v, · π ( T ( v, π ( T ( v, ≥ rk (cid:0) H ∆ ( DZ ) (cid:1) ≥ N. Here the coefficients 6, 2, 2, 3, and 1 next to the cardinalities are determined by the car-dinalities of the support of the corresponding combinatorial types ω = 1221 , , , , k [ DX ] k ∆ = 0. So, for k + 1 = 3, Theorem 4.5 provides a trivial estimate for6 · T ( v, · T ( v,
13) + 2 · T ( v, , the number of 0-dimensional strata in Ω • ( DX, v ). ♦ Let π = def π ( X ) and Π = def π ( DX ) . Utilizing formula (4.17), Theorems 4.2 and 4.5expose the groups H ∆ k +1 ( DX ) ⊂ H ∆ k +1 (Π) and H ∆ k ( X ) ⊂ H ∆ k ( π ) as obstructions to theexistence of globally k -convex traversing flows on X . Corollary 4.2.
Let X be a smooth compact connected and oriented ( n + 1) -manifold withboundary. Assume that X admits a globally k -convex traversally generic vector field v .If, for each connected component of the boundary ∂X , the image of its fundamentalgroup in π ( DX ) is amenable, then the simplicial semi-norm on H j +1 ( DX ) is trivial forall j ≥ k .If, for each connected component of the boundary ∂X , the image of its fundamental groupin π ( X ) is amenable, then he simplicial semi-norm is trivial on H j ( X ) for all j ≥ k .In particular, if X admits a vector field such that T ( v ) − n = ∅ , then k [ DX ] k ∆ = 0 and H ∆ n ( X ) = 0 . ♦ The next theorem, proven in [AK], should be compared with Theorem 7.5 from [K], itsolder 3 D -relative. In a way, this theorem is the source of motivation for developing themachinery of amenable localization in [AK] and in the present paper. For 3-folds M witha simply-connected boundary, Theorem 4.7 below is known with c (2) = 1 / Vol(∆ ), theinverse of the volume of the ideal tetrahedron in the hyperbolic space (see [K]). Theorem 4.7 ( Alpert, Katz).
Let M be a closed, oriented hyperbolic manifold of di-mension n + 1 ≥ . Let X be the space obtained by deleting a domain U from M , suchthat U is contained in a ball B n +1 ⊂ M . Let v be a traversally generic vector field on X .Then the cardinality of the set of v -trajectories of the maximal reduced multiplicity n satisfies the inequality (cid:0) T ( v ) − n (cid:1) ≥ c ( n ) · Vol( M ) , (4.18) where c ( n ) > is an universal constant, and Vol( M ) denotes the hyperbolic volume of M . ♦ Corollary 4.3.
The inequality (4.18) of Theorem 4.7 is valid for any X , obtained bydeleting a number of ( n + 1) -balls from a closed hyperbolic manifold M .Proof. When the domain U from Theorem 4.7 is a union of ( n +1)-balls, we can incapsulatethem into a single ( n + 1)-ball B . Therefore, Theorem 4.7 is applicable to X = M \ U . (cid:3) Let us apply Corollary 4.3 to the landscape of Morse functions on closed hyperbolicmanifolds. Moving towards this goal, we formulate the following
Conjecture 4.1.
Let f : M → R be a Morse function on a closed manifold M of dimension ( n + 1) , and v its gradient-like field. Assume that v satisfies the Morse-Smale transversalitycondition ( [S] , [S1] ). Then, for all sufficiently small ǫ > , the field v on X = def M \ a x ∈ Σ f B ǫ ( x ) is traversally generic. The combinatorial types of the v -trajectories on X are drawn fromthe list: (11) , (121) , (1221) , . . . , (1 2 . . . | {z } n . Moreover, there exists a universal constant c ( n ) > such that the number of broken v -trajectories on M , comprising ( n + 1) segments, and the number of n -tangent v -trajectoriesin X are related by the formula (cid:0) T ( v, ω ⋆ ) (cid:1) = c ( n ) · (cid:0) broken ( n +1) ( v ) (cid:1) , (4.19) where ω ⋆ = (1 2 . . . | {z } n . ♦ In fact, we can show that c (2) = 4 and c (3) = 4.Combining Conjecture 4.1 with Corollary 4.3, one could arrive to an estimate of broken ( n +1) ( v )) from below for Morse-Smale gradient fields v on closed hyperbolic ( n +1)-manifolds.Fortunately, regardless of the validity of the conjecture, broken ( n +1) ( v )) has a lowerboundary, delivered by the normalized hyperbolic volume! The beautiful proposition belowhas been recently proven by Hannah Alpert [A]. The proof involves the same circle ofamenable localization techniques as in [Gr1], [AK], being applied to intricate stratificationsof certain compactifications of the unstable manifolds of the gradient v -flow. OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 39
Theorem 4.8 ( Alpert).
Let f : M → R be a Morse function on a closed hyperbolicmanifold M of dimension ( n + 1) and v its gradient-like field. Assume that v satisfies theMorse-Smale transversality condition. Then broken ( n +1) ( v )) ≥ Vol( M ) . ♦ We just have internalized the crucial role that non-amenable fundamental groups π ( X )and π ( DX ) play in delivering some lower bounds of the traversing complexities tc lextrav ( X )and tc lextrav ( DX ).Now we will connect the complexity of another fundamental group π ( X/∂X ) with thenumber of minimal connected components in the Ω • -stratification { X ( v, ω ) } ω of X . Theconsiderations to follow are rather elementary; in particular, the amenability properties donot play any role here.Each v -trajectory γ of the combinatorial type ω defines a loop or a bouquet of loops inthe quotient space X/∂X . Similarly, each γ defines its double D ( γ ) ⊂ DX . The double D ( γ ) is a chain of loops (like “ ∞ ” or “ooo”), the number of loops in the chain being equalto γ ∩ ∂X ) − γ produces an element [ γ ] ∈ π ( X/∂X ) and a subgroup [[ γ ]] of π ( X/∂X ),equipped with the ordered set of ( | sup( ω ) | −
1) generators—the v -ordered loops of thebouquet [ γ ]. Here | sup( ω ) | is the cardinality of the set γ ∩ ∂X . The element [ γ ] and thesubgroup [[ γ ]] are constant within each connected component X ( v, σ ) of the pure stratum X ( v, ω ). This follows from the fact that the finite coveringΓ : X ( v, σ ) ∩ ∂X → Γ (cid:0) X ( v, σ ) ∩ ∂X (cid:1) is trivial ([K3], Corollary 5.1). Let us denote by S • ( v ) the poset whose elements are theconnected components X ( v, σ ).Therefore, we get a system of groups { [[ γ σ ]] } σ ∈S • ( v ) , linked by homomorphisms ψ σ,σ ′ :[[ γ σ ]] → [[ γ σ ′ ]] for any pair σ ≻ σ ′ in S • ( v ).This construction leads to the following Lemma 4.1.
For a traversally generic field v on X , the subgroups (cid:8) [[ γ σ ]] (cid:9) σ ∈S• min( v ) generate π ( X/∂X ) .Proof. We need to show that any loop ρ : I/∂I → X/∂X through the point ⋆ = ∂X/∂X is a product of loops of the form the groups [[ γ σ ]], where σ is a minimal element of theposet S • ( v ). Equivalently, it will suffice to show that any path ( ρ, ∂ρ ) : ( I, ∂I ) → ( X, ∂X )can be homotoped, relatively to the boundary ∂X , to an ordered union several segments of oriented trajectories γ , labeled with the elements of the minimal set S • min ( v ). Here thesegments of trajectories γ are bounded by two points from γ ∩ ∂X and the orientations ofsegments are not necessarily the ones induced by v .Note that each oriented segment of γ belongs to the subgroup [[ γ ]].First, with the help of the ( − v )-flow, we homotop ( ρ, ∂ρ ) to a path that is realized byseveral segments of trajectories, intermingled with some paths residing in ∂X . The rest of argument is an induction based on the order ≻ in S • ( v ). We can replaceany segment of γ ⊂ X ( v, σ ) with the corresponding segment of any other trajectory fromthat stratum. This replacement is possible since Γ : X ( v, σ ) ∩ ∂X → T ( v, σ ) is a trivialcovering. Moreover, we can homotop such segment of γ to a segment of some trajectory γ ′ , residing in any stratum X ( v, σ ′ ) that belongs to the closure of X ( v, σ ). In other words,we can replace any segment of γ ⊂ X ( v, σ ) with some segment of γ ′ ⊂ X ( v, σ ′ ), where σ ′ ≺ σ . Following these replacements, eventually we will arrive to a ordered finite collectionof segments of { γ σ ⊂ X ( v, σ ) } σ ∈S • min ( v ) ; the collection will represent the relative homotopyclass of the path ρ (some of the the segments of trajectories γ σ ’s in this representation mayappear with integral multiplicities). (cid:3) We denote by c gen ( π ) the minimal number of generators in finite presentations of thegroup π . Theorem 4.9.
For a traversally generic field v on a connected compact smooth manifold X with boundary, c gen (cid:0) π ( X/∂X ) (cid:1) ≤ X σ ∈S • min ( v ) (cid:16) | sup( ω ( σ )) | − (cid:17) , where ω ( σ ) is the combinatorial type of a typical v -trajectory passing through the connectedcomponent labeled by σ .Proof. By Lemma 4.1, any element β ∈ π ( X/∂X ) produces a word in the alphabet whoseletters are the loops, generated by the segments of special trajectories that label the minimalstrata of S • min ( v ); each minimal stratum X ( v, σ ), σ ∈ S • min ( v ), contains a unique specialtrajectory. The number of loops-letters in each trajectory is | sup( ω ( σ )) | −
1. Thereforethe total number of letters in the alphabet is given by the RHS of the formula above. (cid:3)
Corollary 4.4.
For a traversally generic field v on a connected compact smooth ( n + 1) -dimensional manifold X with boundary, the number of minimal connected components in T ( v ) satisfies the inequality: (cid:0) S • min ( v ) (cid:1) ≥ c gen (cid:0) π ( X/∂X ) (cid:1) / ( n + 1) . In particular, if ∂X consists of m components, then (cid:0) S • min ( v ) (cid:1) ≥ ( m − / ( n + 1) . Proof.
Note that for a traversally generic v , | sup( ω ( σ )) | ≤ n + 2 , so that the number ofsegments in which a typical trajectory γ σ is divided by ∂X is n + 1 at most. Utilizing thearguments from Lemma 4.1, Theorem 4.9 implies the first inequality of the corollary.To prove the second one, note that X/∂X admits a continuous map F onto a connectedgraph G with two vertices, a and b , and m edges. Indeed, let U be a collar of ∂X in X . We send X \ int( U ) to a ∈ G , ∂X to b , and each component of the collar U to thecorresponding edge. Each basic loop in G lifts against F to a loop in X/∂X . Therefore theexists an epimorphism π ( X/∂X ) → F m − , where F m − denotes the free group in m − c gen (cid:0) π ( X/∂X ) (cid:1) ≥ m − (cid:3) OMPLEXITY OF SHADOWS & TRAVERSING FLOWS IN TERMS OF THE SIMPLICIAL VOLUME 41
Remark 4.5.
Note that, by definition, (cid:0) S • min ( v ) (cid:1) ≥ (cid:0) T ( v ) − n (cid:1) . Thus, under thehypotheses of Theorem 4.7 and by that theorem, there exists an universal positive constantsuch that (cid:0) S • min ( v ) (cid:1) ≥ const ( n ) · Vol( M ), where the hyperbolic volume Vol( M ) dependsonly on π ( M ).If X is obtained from M by deleting a single ball, π ( M ) ≈ π ( X/∂X ). In such a case, (cid:0) S • min ( v ) (cid:1) ≥ c gen (cid:0) π ( M ) (cid:1) / ( n + 1) . So, for the hyperbolic X = M \ D n +1 , both lowerbounds for (cid:0) S • min ( v ) (cid:1) are expressed essentially in terms of π ( M ). ♦ Acknowledgments.
The author is grateful to Larry Guth and Hannah Alpert for theenjoyable in-depth discussions that have led to this work. He also likes to thank the refereewhose advice helped to improve substantially the quality of this text.
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