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# The arcs from ∂ + 2 X of the types A, B, and C.

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We notice that a generic nonsingular gradient field $v = \nabla f$ on a compact 3-fold $X$ with boundary canonically generates a simple spine $K(f, v)$ of $X$. We study the transformations of $K(f, v)$ that are induced by deformations of the data $(f, v)$. We link the Matveev complexity $c(X)$ of $X$ with counting the \emph{double-tangent} trajecto...

## Contexts in source publication

Context 1
... of γ belonging to ∂ + 2 X and to ∂ − 2 X alternate. In the first case, the contribution of γ to deg(h) is zero. In the second case, the contribution of each arc with the ends of opposite polarity is also zero. Each arc with two positive ends contributes a rotation of v by +π, while each arc with two negative ends contributes a rotation by −π (see Fig. 7). Hence the total rotation along γ ...
Context 2
... X is acyclic, a homological argument implies that ∂ 1 X ≈ S 2 . Thus ∂ + 1 X must be a contractible domain in S 2 , that is, a 2-disk. Therefore, the manifold X must be shaped as a lens, one face of which is that disk. Fig. 7 shows a typical behavior of a vector field v 1 in a neighborhood of ∂ 2 X. The arcs of ∂ + 2 X come in tree flavors: A is bounded by a pair of points from ∂ + 3 X, B is bounded by a pair of points from ∂ − 3 X and C is bounded by a pair of mixed ...

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## Citations

... That theorem claims that a closed 3-manifold X has a spine (see [6] for the brief description of spines) which is the image of an immersed 2-sphere in general position in X . Theorem 3.1 should be compared also with somewhat similar Theorem 5.2 in [8], the latter dealing with the flow-generated spines, not trajectory spaces. Theorem 3.1 (Trajectory spaces as the ball-based origami) For n ≥ 2, any compact connected smooth (n + 1)-manifold X with boundary admits a traversally generic vector field v such that: ...
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This paper describes a mechanism by which a traversally generic flow v on a smooth connected (n+1)-dimensional manifold X with boundary produces a compact n-dimensional CW-complex T(v), which is homotopy equivalent to X and such that X embeds in T(v)×R. The CW-complex T(v) captures some residual information about the smooth structure on X (such as the stable tangent bundle of X). Moreover, T(v) is obtained from a simplicial origami mapO:Dn→T(v), whose source space is a ball Dn⊂∂X. The fibers of O have the cardinality (n+1) at most. The knowledge of the map O, together with the restriction to Dn of a Lyapunov function f:X→R for v, make it possible to reconstruct the topological type of the pair (X,F(v)), were F(v) is the 1-foliation, generated by v. This fact motivates the use of the word “holography” in the title. In a qualitative formulation of the holography principle, for a massive class of ODE’s on a given compact manifold X, the solutions of the appropriately staged boundary value problems are topologically rigid.
... The article is an informal introduction into the philosophy and some key results from Katz (2009Katz ( , 2014aKatz ( , b, c, 2015Katz ( , 2016a, as they manifest themselves in the dimension two. Our strongest results deal with so called traversing vector fields on G. ...
... Then we employ these flow invariants to manufacture a variety of new smooth topological invariants of X itself. In Katz (2009Katz ( , 2014aKatz ( , b, c, 2015Katz ( , 2016a and Alpert and Katz (2015), we use a variety of standard algebraic and geometric topology tools to compute or to estimate from below the new invariants. However, for this exposition, the reader is expected to be familiar only with the homology of compact surfaces and with their fundamental groups. ...
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This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on n-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.
... Figure 1 depicts a traversing vector field on a 2-dimensional space, and the associated trajectory space. One of the authors has explored this general setup in multiple papers beginning with [5], and in the paper [6] he introduces the class of traversally generic vector fields, which have certain nice properties. In Theorem 3.5 of that paper, he proves that the traversally generic vector fields form an open and dense subset of the traversing vector fields. ...
... In particular, because Vol M is nonzero, there must be at least one maximummultiplicity trajectory. This theorem generalizes Theorem 7.5 of [5], which addresses the case where n + 1 = 3 and U is any finite disjoint union of balls, with constant 1/ Vol(∆ 3 ), where ∆ 3 denotes the regular ideal simplex in hyperbolic 3space. ...
... One immediate follow-up question is how large the constant should be in Theorem 2. The 3-dimensional case of Theorem 1 (Theorem 7.5 of [5]) suggests that we might hope for a constant of 1 for every n. However, the constant obtained in our proof is much weaker and is a little confusing to compute. ...
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For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of $X$ by adapting methods of Gromov, in particular his "amenable reduction lemma". We apply these bounds to vector fields on hyperbolic manifolds.
... We intend to present to the reader a version of the Morse Theory in which the critical points remain behind the scene, while shaping the geometry of the boundary! Some of the concepts that animate our approach can be found in [5], where they are adopted to the special environment a 3 D-gradient flows. These notions include stratified convexity or concavity of traversing flows in connection to the boundary of the manifold. ...
... by deforming the f -gradient-like field v (cf. Section 3 in [5]). ...
... Recall that, by Corollary 4.4 [5], for any 3-fold X and a boundary generic field 0 v ≠ on it, we get ...
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As has been observed by Morse [1], any generic vector field v on a compact smooth manifold X with boundary gives rise to a stratification of the boundary ∂X by compact submanifolds {\d_j^\pm X(v)}_j, where codim(\d_j^\pm X(v)) = j . Our main point is that this stratification reflects the stratified convexity/ concavity of the boundary ∂X with respect to the v -flow. We study the behavior of this stratification under deformations of the vector field v. We also investigate the restrictions that the existence of a convex/concave traversing v -flow imposes on the topology of X. Let v_1 be the orthogonal projection of v on the tangent bundle of ∂X . We link the dynamics of the v_1 -flow on the boundary with the property of v in X being convex/concave. This linkage is an instance of more general phenomenon that we call “holography of traversing fields”—a subject of a different paper to follow.
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We study {\sf traversing} vector flows $v$ on smooth compact manifolds $X$ with boundary. For a given compact manifold $\hat X$, equipped with a traversing vector field $\hat v$ which is {\sf convex} with respect to $\partial\hat X$, we consider submersions/embeddings $\alpha: X \to \hat X$ such that $\dim X = \dim \hat X$ and $\alpha(\partial X)$ avoids a priory chosen tangency patterns $\Theta$ to the $\hat v$-trajectories. In particular, for each $\hat v$-trajectory $\hat\gamma$, we restrict the cardinality of $\hat\gamma \cap \alpha(\partial X)$ by an even number $d$. We call $(\hat X, \hat v)$ a {\sf convex pseudo-envelop/envelop} of the pair $(X, v)$. Here the vector field $v = \alpha^\dagger(\hat v)$ is the $\alpha$-transfer of $\hat v$ to $X$. For a fixed $(\hat X, \hat v)$, we introduce an equivalence relation among convex pseudo-envelops/ envelops $\alpha: (X, v) \to (\hat X, \hat v)$, which we call a {\sf quasitopy}. The notion of quasitopy is a crossover between bordisms of pseudo-envelops and their pseudo-isotopies. In the study of quasitopies $\mathcal{QT}_d(Y, \mathbf c\Theta)$, the spaces $\mathcal P_d^{\mathbf c\Theta}$ of real univariate polynomials of degree $d$ with real divisors whose combinatorial types avoid the closed poset $\Theta$ play the classical role of Grassmanians. We compute, in the homotopy-theoretical terms that involve $(\hat X, \hat v)$ and $\mathcal P_d^{\mathbf c\Theta}$, the quasitopies of convex envelops which avoid the $\Theta$-tangency patterns. We introduce characteristic classes of pseudo-envelops and show that they are invariants of their quasitopy classes. Then we prove that the quasitopies $\mathcal{QT}_d(Y, \mathbf c\Theta)$ often stabilize, as $d \to \infty$.
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This paper describes a mechanism by which a traversally generic flow v on a smooth connected manifold X with boundary produces a compact CW-complex T (v), which is homotopy equivalent to X and such that X embeds in T (v) × R. The CW-complex T (v) captures some residual information about the smooth structure on X (such as the stable tangent bundle of X). Moreover, T (v) is obtained from a simplicial origami map O : D n → T (v), whose source space is a disk D n ⊂ ∂X of dimension n = dim(X)−1. The fibers of O have the cardinality (n + 1) at most. The knowledge of the map O, together with the restriction to D n of a Lyapunov function f : X → R for v, make it possible to reconstruct the topological type of the pair (X, F(v)), were F(v) is the 1-foliation, generated by v. This fact motivates the use of "holography" in the title.
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Let $M$ be a compact connected smooth Riemannian $n$-manifold with boundary. We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic geodesic flows on $SM$, the space of the spherical tangent bundle. Such flows generate stratifications of $SM$, governed by rich universal combinatorics. The stratification reflects the ways in which the geodesic flow trajectories interact with the boundary $\d(SM)$. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension $k$. These universal bounds are expressed in terms of the normed homology $H_k(M; \R)$ and $H_k(DM; \R)$, where $DM = M\cup_{\d M} M$ denotes the double of $M$. The norms here are the Gromov simplicial semi-norms in homology. The more complex the metric on $M$ is, the more numerous the strata of $SM$ and $S(DM)$ are. So one may regard our estimates as analogues of the Morse inequalities for the geodesics on manifolds with boundary. It turns out that some close relatives of the normed homology spaces form obstructions to the existence of globally $k$-convex traversally generic metrics on $M$.
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For a given smooth compact manifold $M$, we introduce a massive class $\mathcal G(M)$ of Riemannian metrics, which we call \emph{metrics of the gradient type}. For such metrics $g$, the geodesic flow $v^g$ on the spherical tangent bundle $SM \to M$ is traversing. Moreover, for every $g \in \mathcal G(M)$, the geodesic scattering along the boundary $\d M$ can be expressed in terms of the \emph{scattering map} $C_{v^g}: \d_1^+(SM) \to \d_1^-(SM)$. It acts from a domain $\d_1^+(SM)$ in the boundary $\d(SM)$ to the complementary domain $\d_1^-(SM)$, both domains being diffeomorphic. We prove that, for a \emph{boundary generic} metric $g \in \mathcal G(M)$ the map $C_{v^g}$ allows for a reconstruction of $SM$ and of the geodesic flow on it, up to a homeomorphism (often a diffeomorphism). Also, for such $g$, the knowledge of the scattering map $C_{v^g}$ makes it possible to reconstruct the homology of $M$, the Gromov simplicial semi-norm on it, and the fundamental group of $M$. We aim to understand the constraints on $(M, g)$, under which the scattering map allows for a reconstruction of $M$ and the metric $g$ on it. In particular, we consider a closed Riemannian $n$-manifold $(N, g)$ which is locally symmetric and of negative sectional curvature. Let $M$ is obtained from $N$ by removing an $n$-ball such that the metric $g|_M$ is \emph{boundary generic} and of the gradient type. Then we prove that the scattering map $C_{v^{g|_M}}$ makes it possible to recover $N$ and the metric $g$ on it.
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We study the non-vanishing gradient-like vector fields v on smooth compact manifolds X with boundary. We call such fields traversing. The traversally generic vector fields form an open and dense subset in the space of all traversing vector fields on X. In Theorem 2.2, we show that the trajectory spaces T (v) of such traversally generic v-flows are Whitney stratified spaces and therefore admit triangulations, amenable to their natural stratifications. Despite being spaces with singularities, T (v) retain some residual smooth structure of X. Let F(v) denote the oriented 1-dimensional foliation on X, produced by a traversing v-flow. With the help of a generic field v, we divide the boundary ∂X of X into two complementary compact manifolds, ∂ + X(v) and ∂ − X(v). Then we introduce the causality map Cv : ∂ + X(v) → ∂ − X(v), a distant relative of the Poincaré return map. Our main result, Theorem 3.1, claims that, for boundary generic traversing vector fields v, the knowledge of the causality map Cv is allows for a reconstruction of the pair (X, F(v)), up to a homeomorphism Φ : X → X which is the identity on the boundary ∂X. In other words, for a massive class of ODE's, we show that the topology of their solutions, satisfing a given boundary value problem, is rigid. We call these results " holographic " since the (n + 1)-dimensional X and the un-parameterized dynamics of the flow on it are captured by a single correspondece Cv between two n-dimensional screens ∂ + X(v) and ∂ − X(v). This holography of traversing flows has numerous applications to the dynamics of general flows. Some of them are described in the paper. Others, like geodesic flows on manifolds with boundary, are just outlined.
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Full-text available
We study the non-vanishing gradient-like vector fields v on smooth compact manifolds X with boundary. We call such fields traversing. The traversally generic vector fields form an open and dense subset in the space of all traversing vector fields on X. In Theorem 2.2, we show that the trajectory spaces T(v) of such traversally generic v-flows are Whitney stratified spaces and therefore admit triangulations, amenable to their natural stratifications. Despite being spaces with singularities, T(v) retain some residual smooth structure of X Let F(v) denote the oriented 1-dimensional foliation on X, produced by a traversing v-flow. With the help of a generic field v, we divide the boundary d X of X into two complementary compact manifolds, d^+X(v) and d^-X(v). Then we introduce the causality map C_v: d^+X(v) \to d^-X(v), a distant relative of the Poincare return map. Our main result, Theorem 3.1, claims that, for boundary generic traversing vector fields v, the knowledge of the causality map C_v allows for a reconstruction of the pair (X, F(v)), up to a homeomorphism \Phi: X \to X which is the identity on the boundary d X. In other words, for a massive class of ODE's, we show that the topology of their solutions, satisfing a given boundary value problem, is rigid. We call these results "holographic" since the (n+1)-dimensional X and the un-parameterized dynamics of the flow on it are captured by a single correspondece C_v between two n-dimensional screens d^+X(v) and d^-X(v). This holography of traversing flows has numerous applications to the dynamics of general flows. Some of them are described in the paper. Others, like geodesic flows on manifolds with boundary, are just outlined.