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In this paper, we present our general results about non-vanishing gradient-like vector flows on $n$-dimensional manifolds with boundary in the context of the flows on $2$-dimensional surfaces with boundary. We take advantage of the relative simplicity of $2$-dimensional worlds to explain and popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.

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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.
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The invariants of filtered complexes, so-called "canonical forms", are introduced. The canonical form of a filtered complex is the partition of the set of critical values (indices) into pairs "birth-death" and a separate set representing homology of the complex. The canonical form is obtained by the action of upper-triangular matrices, which reduces the filtered complex to the simplest form. This invariant, ten years later, started to be widely used in applied mathematics under the name "Persistence Diagrams" or "Persistence Bar-codes". The main result of the paper uses this canonical form invariant to solve the following problem: given a function defined on a neighborhood of boundary of an n-dimensional ball, calculate optimal estimate from below for the number of Morse critical points that any generic extension of the function inside the ball must have.
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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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This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from \cite{K2}, for \emph{traversally generic flows} on $(n+1)$-manifolds $X$, we embark on a detailed and somewhat tedious study of universal combinatorics of their tangency patterns with respect to the boundary $\d X$. This combinatorics is captured by a universal poset $\Omega^\bullet_{'\langle n]}$ which depends only on the dimension of $X$. It is intimately linked with the combinatorial patterns of real divisors of real polynomials in one variable of degrees which do not exceed $2(n+1)$. Such patterns are elements of another natural poset $\Omega_{\langle 2n+2]}$ that describes the ways in which the real roots merge, divide, appear, and disappear under deformations of real polynomials. The space of real degree $d$ polynomials $\mathcal P^d$ is stratified so that its pure strata are cells, labelled by the elements of the poset $\Omega_{\langle d]}$. This cellular structure in $\mathcal P^d$ is interesting on its own right (see Theorem \ref{th4.1} and Theorem \ref{th4.2}). Moreover, it helps to understand the \emph{localized} structure of the trajectory spaces $\mathcal T(v)$ for traversally generic fields $v$, the main subject of Theorem \ref{th5.2} and Theorem \ref{th5.3}.
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Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theorem \ref{th3.5}). The definitions of boundary generic and traversally generic vector fields $v$ are inspired by some classical notions from the singularity theory of smooth Bordman maps \cite{Bo}. Like in that theory (cf. \cite{Morin}), we establish local versal algebraic models for the way a sheaf of $v$-trajectories interacts with the boundary $\d X$. For fields from the space $\mathcal V^\ddagger(X)$, the finite list of such models depends only on $\dim(X)$; as a result, it is universal for all equidimensional manifolds. In specially adjusted coordinates, the boundary and the $v$-flow acquire descriptions in terms of universal deformations of real polynomials whose degrees do not exceed $2\cdot \dim(X)$.
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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} trajectories of the $v$-flow, i.e. the trajectories that are tangent to the boundary $\d X$ at a pair of distinct points. Let $gc(X)$ be the minimum number of such trajectories, minimum being taken over all nonsingular $v$'s. We call $gc(X)$ the \emph{gradient complexity} of $X$. Next, we prove that there are only finitely many $X$ of bounded gradient complexity, provided that $X$ is irreducible and boundary irreducible with no essential annuli. In particular, there exists only finitely many hyperbolic manifolds $X$ with bounded $gc(X)$. For such $X$, their normalized hyperbolic volume gives an upper bound of $gc(X)$. If an irreducible and boundary irreducible $X$ with no essential annuli admits a nonsingular gradient flow with no double-tangent trajectories, then $X$ is a standard ball. All these and many other results of the paper rely on a careful study of the stratified geometry of $\d X$ relative to the $v$-flow. It is characterized by failure of $\d X$ to be \emph{convex} with respect to a generic flow $v$. It turns out, that convexity or its lack have profound influence on the topology of $X$. This interplay between intrinsic concavity of $\d X$ with respect to any gradient-like flow and complexity of $X$ is in the focus of the paper.
We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the trajectories that are tangent to the boundary in a particular canonical fashion. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension. These universal bounds are basically expressed in terms of the normed homology of the fundamental groups $\pi_1(X)$ and $\pi_1(DX)$, where $DX$ denotes the double of $X$. The norm here is the Gromov simplicial semi-norm in homology. It turns out 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 $X$.
Many of the results in this chapter appeared first in [GM3].
This chapter describes Seifert Fibered Spaces in 3-Manifolds. There exist finitely many disjoint, non-contractible, pairwise non-parallel, embedded 2-spheres in M, whose homotopy classes generate π2 (M) as a π2 (M)-module; and modulo the Poincaré conjecture, these 2-spheres are unique up to ambient homeomorphism. Thus, all singular 2-spheres in M, that is, maps of S2 into M, may be described, up to homotopy, in terms of a geometric picture in M. The strong version of the sphere theorem presented in the chapter gives a great deal of information about fundamental groups of compact 3-manifolds, for example that they are finite free products of torsion-free groups and finite groups. It also provides in a slightly refined version a reduction of the classification problem for compact, oriented 3-manifolds to the classification problem for compact, irreducible, 3-manifolds.
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this paper I show how the basic concept of angle leads naturally to the basic topological ideas of degree of mapping and of the Euler-Poincar'e Number. My story spans the history of mathematics. It concerns the, perhaps, most widely known non-obvious theorem of mathematics and it contains the same stunning generalization that characterizes the recent history of the Euler-Poincar'e number. In fact it concerns one of the most important and earliest of the applications of the Euler-Poincar'e number. It shows the fickleness of mathematical fame, and it shows the unreasonable power of unreasonable points of view, and it shows how easy it is for mathematicians to miss and forget beautiful and important theorems as well as simple and revealing points of view. This is a history of the Gauss-Bonnet theorem as I see it. I am not a mathematical historian. I am quoting only secondary sources or first hand papers which I quickly scanned and I did not conduct any thorough interviews. Nonetheless, I am writing this history because I have contributed the last sentence to it (for the moment). I especially want to acknowledge the help of Hans Samelson. His scholarship greatly altered the thrust of earlier versions of this paper. He discovered Satz VI. Mathematics Subject Classification. 55.
Complements of Discriminants of Smooth Maps: Topology and Applications, Translations of Mathematical Monographs On regular closed curves in the plane
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Gradient Flows, Concavity, and Complexity on Manifolds with Boundary, monograph, to be published by World Scientific
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Also translated into Russian and published by
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