Morse functions statistics

拓扑Morse函数的存在性与可变形性

Existence and deformability of topological Morse functions

本文研究拓扑Morse函数的存在性和可变形性，提出了一种简单构造连续拓扑Morse函数族的方法。

Morse theory on manifolds with boundary I. Strong Morse function, cellular structures and algebraic simplification of cellular differential

Main subject of the paper is a (strong) Morse function on a compact manifold with boundary. We construct a cellular structure and discuss its algebraic properties in this paper. Also we get an estimation on Arnold's question on a number of critical points of a Morse function with given boundary condition.

Moduli Spaces of Morse Functions for Persistence

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse--Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions o…

Conjugation of Morse Function on 3-Manifolds

The questions when two Morse function on closed manifolds are conjugated is investigated. Using the handle decompositions of manifolds the condition of conjugation is formulated. For each Morse function on 3-manifold the ordered generalized Heegaard diagram is built. The criteria of Morse function conjugation are given in the terms of equivalence of such diagrams.

在非局部极值原像上为 Morse 的光滑函数及在具有此类函数的流形连通和上构造自然函数

Smooth functions which are Morse on preimages of values not being local extrema and constructing natural functions of the class on connected sums of manifolds admitting these functions

本文研究在非局部极值原像上为 Morse 的光滑函数（I-Morse）及关于 Reeb 图整洁的 N-Reeb 函数，并在给定流形的连通和上构造尊重两个此类函数的 IN-Morse-Reeb 函数。

Oriented area is a perfect Morse function

We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral polygons (which appear as Morse points) are interpreted as independent generators of the homology groups of the (decorated) configuration space.

带边流形的莫尔斯函数族

Families of Morse functions for manifolds with boundary

本文研究带边流形上莫尔斯函数的单参数族，完整分类并列出一般族中可能出现的所有退化类型。

Connected components of the space of circle-valued Morse functions on surfaces

We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions $f, g: M\to S^1$ belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index.

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-torus

This paper is devoted to the study of special subgroups of the automorphism groups of Kronrod-Reeb graphs of a Morse functions on $2$-torus $T^2$ which arise from the action of diffeomorphisms preserving a given Morse function on $T^2$. In this paper we give a full description of such classes of groups.

Combinatorial realization of the Thom-Smale complex via discrete Morse theory

In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular realization by a complete matching on the Hasse diagram of a triangulation of the manifold.

Bordism of constrained Morse functions

We call a Morse function $f$ on a closed manifold $k$-constrained if neither $f$ nor $-f$ has critical points of indefinite Morse index $< k$. In this paper we study bordism groups of $k$-constrained Morse functions, and thus interpolate between the case $k = 1$ of bordism groups of Morse functions (computed by Ikegami) and the case $k \gg 1$ of bordism groups of special generic functions (computed by Saeki). We employ Levine's elimination of cusps, Stein factorization, the two-index theorem of Hatcher-Wagoner, and a handle extension theorem for fold maps due to Gay-Kirby to show that the noti…

Fiber connected, indefinite Morse 2-functions on connected n-manifolds

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of the range, in which case the Morse 2-function is "indefinite". This is foundational work for the long-range goal of defining smooth invariants from Morse 2-functions using tools analogous to classical Morse homology and Cerf theory.

Topological structure of Morse functions on the projective plane

To investigate the topological structure of Morse functions on the projective plane we use the Reeb graphs. We describe it properties and prove that it is a complete topological invariant of simple Morse function on $\mathbb{R} P^2$. We prove recurent formulas for the number of Reeb graphs with the given number of sadles (vertex).

流形上 Morse 函数的和谐

Concordance of Morse functions on manifolds

该研究定义流形上 Morse 函数的和谐关系，给出两函数和谐的充要条件，并与已有配边判据相比较。

Perfect Morse functions and exotic S^2 x S^2's

The main theorem of the paper shows that a smooth manifold which is homeomorphic to S^2xS^2 and has nonvanishing Ozsvath-Szabo invariant does not admit a perfect Morse function. I am withdrawing the paper because it is unclear to me if such a manifold exists.

Cusp cobordism group of Morse functions

By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all non-degenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine's cusp elimination techn…

由随机游走构造的 Morse 函数

Morse functions constructed by random walks

本文通过随机游走在曲面上构造随机 Morse 函数，并利用切割闭域曲面得到的带边曲面刻画函数空间，证明存在一小类 Morse 函数可在共享刻画曲面的意义上逼近任意其他 Morse 函数。

Cobordism group of Morse functions on unoriented surfaces

The computation of the cobordism group of Morse functions on unoriented surfaces using Stein factorizations.

The Morse Complex for a Morse Function on a Manifold with Corners

A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of classical Morse theory. It follows that M is homotopy equivalent to a CW-complex with one cell of dimension λfor each essential critical point of index λ. The goal of this article is to determine the boundary maps of this CW-complex, in the case where M is compact and orientable. First, the boundary maps are defined in terms of the modified gradient flow. Then …

Indefinite Morse 2-functions; broken fibrations and generalizations

A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions mapping to arbitrary compact, oriented surfaces. "Uniqueness" means there is a set of moves which are sufficient to go between two homotopic indefinite Morse 2-functions while remaining indefinite throughout. We extend the existence and uniqueness results to indefinite, Morse 2-f…

On Primes, Graphs and Cohomology

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincaré-Hopf indices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohomol…

Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from \cite{GrLaPo}, \cite{GrLaPo1}, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well dynamics of diffeomorphism. The paper is devoted to finding conditions to the existence of such an energy function, that is, a function whose set of critical points coincides with the non-wandering set of the considered diffeomorphism. We show that the necessary and sufficient conditions to the existence of a dynamically ordered energy f…

无奇点的圈值Morse函数扩展

Non-singular extensions of circle-valued Morse functions

研究闭合定向曲面上的圈值Morse函数非奇点扩展条件，给出在 collar 邻域存在子浸入时的必要充分条件。

Existence of perfect Morse functions on spaces with semi-free circle action

Let $M$ be a compact oriented simply-connected manifold of dimension at least 8. Assume $M$ is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove $M$ has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for the invariant Morse theory.

Heegaard splittings and singularities of the product map of Morse functions

We give an upper bound for the Reidemeister-Singer distance between two Heegaard splittings in terms of the genera and the number of cusp points of the product map of Morse functions for the splittings. It suggests that a certain development in singularity theory may lead to the best possible bound for the Reidemeister-Singer distance.

Reconstructing 4-manifolds from Morse 2-functions

Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions.

The Poincaré-Lefschetz pairing viewed on Morse complexes

Given a compact manifold with a non-empty boundary and equipped with a generic Morse function (that is, no critical point on the boundary and the restriction to the boundary is a Morse function), we already knew how to construct two Morse complexes, one yielding the absolute homology and the other the relative homology. In this note, we construct a short exact sequence from both of them and the Morse complex of the boundary. Moreover, we define a pairing of the relative Morse complex with the absolute Morse complex which induces the intersection product in homology, in the form due to S. Lefsc…

Morse同调与等变性

Morse homology and equivariance

本文开发了在有限群作用的闭流形上从等变Morse函数计算等变同调的方法，通过修改为稳定系统并证明其满足Morse-Smale条件，最终将其简化为Thom-Smale-Witten复形。

Special framed Morse functions on surfaces

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and $\mathbb{F}^1$ the space of framed Morse functions, both endowed with $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping $\mathbb{F}^0\hookrightarrow\mathbb{F}^1$ is a homotopy equivalence. In the case when at least $χ(M)+1$ critical points of each function of $F$ are labeled, homotopy equivalences $\mathbb{\widetilde K}\sim\widetilde{\cal M}$ and $F\sim\mathbb{F}^0\sim{\mathscr D}^0\times\mathbb{\widetilde K}$ are prove…

Cascades and perturbed Morse-Bott functions

Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades \cite{FraTheA}. Similar data, which also includes a parameter $ε> 0$ that scales the Morse-Smale functions $f_j$, can be used to define an explicit perturbation of the Morse-Bott function $f$ to a Morse-Smale function $h_ε:M \rightarrow \mathbb{R}$ \…

Morse-Bott functions on orthogonal groups

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B$. We give a simplified treatment of T. Frankel's analysis of the linear trace function on $SO(n)$, as well as a combinatorial explanation of the relationship between the mod $2$ Betti numbers of $SO(n)$ and those of the Grassmannians $\mathbb{G}(2k,n)$ obtained from this analysi…

Morse-Bott homology

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the homology of the chain complex is independent of the Morse-Bott-Smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a Floer-type continuation theorem.

Necessary and sufficient condition of Morse-Sard theorem for real valued functions

Necessary and sufficient condition is given for a set $A\subset R^1$ to be a subset of the critical values set for a $C^k$ function $f:R^m \to R^1$.

The geometric complex of a Morse-Bott-Smale pair and an extension of a theorem by Bismut-Zhang

We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 1992].

Gradient Vector Fields of Discrete Morse Functions and Watershed-cuts

In this paper, we study a class of discrete Morse functions, coming from Discrete Morse Theory, that are equivalent to a class of simplicial stacks, coming from Mathematical Morphology. We show that, as in Discrete Morse Theory, we can see the gradient vector field of a simplicial stack (seen as a discrete Morse function) as the only relevant information we should consider. Last, but not the least, we also show that the Minimum Spanning Forest of the dual graph of a simplicial stack is induced by the gradient vector field of the initial function. This result allows computing a watershed-cut fr…

Rigidity and gluing for Morse and Novikov complexes

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M,ω)$ with $c_{1}|_{π_{2}(M)}=[ω]|_{π_{2}(M)}=0$. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently $C^{0}$ close generic function/hamiltonian. The gluing result is a type of Maye…

On the homotopy type of the spaces of Morse functions on surfaces

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$ having fixed number of critical points of each index, moreover at least $χ(M)+1$ critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a polyhedral complex, is introduced. The skew cylindric-polyhedral complex $\mathbb{\widetilde K}$ (the "complex of framed Morse functions"), associated with the space $F$, is defined. In the case when $M=S^2$, the polyhedron $\mathbb{\widetilde K}$ is finite; its Euler characteris…

Topology of the spaces of Morse functions on surfaces

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $χ(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $χ(M)$, and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Bet…

Thurston 脊的 Morse-Smale 性质

The Morse-Smale Property of the Thurston Spine

本文研究 Thurston 脊的局部性质及其光滑片段，证明其满足与带边流形上 Morse 函数的 Morse-Smale 复形类似的性质。

管状邻域的广义Morse理论

Generalized Morse theory for tubular neighborhoods

该研究针对欧氏空间紧集偏移集的距离函数正则值场景，定义广义Morse函数并构建类Morse理论，用非光滑分析等证明次水平集同伦型在临界值处按每临界点粘一个胞腔的方式改变。

Morse Inequalities for Orbifold Cohomology

This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show that a generic function on an orbifold is Morse. In obtaining these results we develop for differentiable Deligne-Mumford stacks those tools of differential geometry and topology -- flows of vector fields, the strong topology -- that are essential to the development of Morse theo…

非紧流形的莫尔斯不等式

Morse inequalities for noncompact manifolds

本文在离散群余紧真不连续作用于非紧流形的条件下，利用 $L^2$-贝蒂数及描述临界点粗略构型的群上函数，建立了莫尔斯函数不要求群不变时的莫尔斯不等式。

Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds

The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.

Realizability of singular levels of Morse functions by unions of geodesics

We list special graphs of degree 4 with at most 3 vertices (atoms from the theory of integrable hamiltonian systems) which could be represented by a union of closed geodesics on the one of the following surfaces with metric of constant curvature: sphere, projective plane, torus, Klein bottle.

First Betti numbers of orbits of Morse functions on surfaces

In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group $G\in\mathcal{G}$ the ranks of the center $Z(G)$ and the quotient by commutator subgroup $G/[G,G]$ coincide. Moreover, this rank is a first Betti number of the orbit of Morse function.

曲线模空间上的Morse理论

Morse theory on moduli of curves

本文从Teichmüller理论视角在稳定曲线模空间引入一族显式Morse函数，利用其良好性质与Weil-Petersson梯度流构建Morse理论，并给出适用于所有亏格与标记点数的首个自然胞腔分解示例。

Lifts of spherical Morse functions

In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate classes to other maps of other appropriate classes are fundamental and important. In this paper, we consider Morse functions such that inverse images of regular values are disjoint unions of spheres, which are extensions of Morse functions with just two singular points on homotop…

利用多参数离散Morse理论分析多过滤函数

Analyzing Multifiltering Functions Using Multiparameter Discrete Morse Theory

本文通过多参数离散Morse理论分析多过滤函数，证明其可被兼容函数近似，并定义了帕累托集与临界单纯形的关系。

从 Morse 函数到余切丛上的 Lefschetz 纤维化

From Morse Functions to Lefschetz Fibrations on Cotangent Bundles

本文证明紧流形上任意 Morse 函数与适配梯度可构造余切丛上同伦唯一的复值辛 Lefschetz 纤维化，并研究其正则纤维的拓扑与辛几何。

双曲6流形上的完美圆值Morse函数

Perfect circle-valued Morse functions on hyperbolic 6-manifolds

本文首次构造了具有完美圆值Morse函数的双曲6流形，并由此得到双曲群中一个类型为F2而非F3的子群新例子。

Deformations of circle-valued Morse functions on surfaces

Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1)$ with the $C^{\infty}$-topology. In this note the connected components of $F(M,S^1)$ are classified. This result extends the results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant on the boundary of $M$.

Extending a Morse function to a non-orientable $3-$manifold

Considering a solid 3-dimensional Klein bottle and a collaring of its boundary, can we extend a generic $C^\infty$ non-singular function defined on the collaring to the full solid Klein bottle without critical points? We give a condition on the Reeb graph of the given function that is necessary and sufficient for the existence of such a non-singular extension.

On Morse Index Retrieval

A smooth function f in a neighbourhood of the unit sphere $S^{n - 1}$ is said to admit index $λ$ if it can be extended to a function F in the unit ball $B^n$ such that F has a unique critical point p and the Morse index of p is equal to $λ$. It is easy to see that a function f cannot admit two indices of different parity. We prove that for any two indices that differ by two there exists a function f that admits both of them.

Perturbed gradient flow trees and $A_\infty$-algebra structures on Morse cochain complexes

We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse functions. The purpose of this article is to provide a coherent and detailed treatment of Abouzaid's approach including a discussion of all relevant analytic notions and results.