Discrete Morse Theory
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Summary of Morse Theory
Summary of Morse Theory One central theme in geometric topology is the classification of selected classes of smooth manifolds up to diffeomorphism . Complete information on this problem is known for compact 1 – dimensional and 2 – dimensional smooth manifolds , and an extremely good understanding of the 3 – dimensional case now exists after more than a century of work . A closely related theme is to describe certain families of smooth manifolds in terms of relatively simple decompositions into smaller pieces . The following quote from http://en.wikipedia.org/wiki/Morse_theory states things very briefly but clearly : In differential topology , the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold . According to the basic insights of Marston Morse , a differentiable function on a manifold will , in a typical case, reflect the topology quite directly . Morse theory allows one to find CW [cell complex] structures and handle decompositions on manifolds and to obtain substantial information about their homology . Morse ’s approach to studying the structure of manifolds was a crucial idea behind the following breakthrough result which S. Smale obtained about 50 years ago : If n a compact smooth manifold M (without boundary) is homotopy equivalent to the n n n sphere S , where n ≥ 5, then M is homeomorphic to S . — Earlier results of J. Milnor constructed a smooth 7 – manifold which is homeomorphic but not n n diffeomorphic to S , so one cannot strengthen the conclusion to say that M is n diffeomorphic to S . We shall use Morse ’s approach to retrieve some low – dimensional classification and decomposition results which were obtained before his theory was developed . -
Morse-Conley-Floer Homology
Morse-Conley-Floer Homology Contents 1 Introduction 1 1.1 Dynamics and Topology . 1 1.2 Classical Morse theory, the half-space approach . 4 1.3 Morse homology . 7 1.4 Conley theory . 11 1.5 Local Morse homology . 16 1.6 Morse-Conley-Floer homology . 18 1.7 Functoriality for flow maps in Morse-Conley-Floer homology . 21 1.8 A spectral sequence in Morse-Conley-Floer homology . 23 1.9 Morse-Conley-Floer homology in infinite dimensional systems . 25 1.10 The Weinstein conjecture and symplectic geometry . 29 1.11 Closed characteristics on non-compact hypersurfaces . 32 I Morse-Conley-Floer Homology 39 2 Morse-Conley-Floer homology 41 2.1 Introduction . 41 2.2 Isolating blocks and Lyapunov functions . 45 2.3 Gradient flows, Morse functions and Morse-Smale flows . 49 2.4 Morse homology . 55 2.5 Morse-Conley-Floer homology . 65 2.6 Morse decompositions and connection matrices . 68 2.7 Relative homology of blocks . 74 3 Functoriality 79 3.1 Introduction . 79 3.2 Chain maps in Morse homology on closed manifolds . 86 i CONTENTS 3.3 Homotopy induced chain homotopies . 97 3.4 Composition induced chain homotopies . 100 3.5 Isolation properties of maps . 103 3.6 Local Morse homology . 109 3.7 Morse-Conley-Floer homology . 114 3.8 Transverse maps are generic . 116 4 Duality 119 4.1 Introduction . 119 4.2 The dual complex . 120 4.3 Morse cohomology and Poincare´ duality . 120 4.4 Local Morse homology . 122 4.5 Duality in Morse-Conley-Floer homology . 122 4.6 Maps in cohomology . -
Persistent Homology of Unweighted Complex Networks Via Discrete
www.nature.com/scientificreports OPEN Persistent homology of unweighted complex networks via discrete Morse theory Received: 17 January 2019 Harish Kannan1, Emil Saucan2,3, Indrava Roy1 & Areejit Samal 1,4 Accepted: 6 September 2019 Topological data analysis can reveal higher-order structure beyond pairwise connections between Published: xx xx xxxx vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced fltration scheme based on the subsequence of the corresponding critical weights, thereby leading to a signifcant increase in computational efciency. We have employed our fltration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect diferences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between diferent model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks. In recent years, the feld of topological data analysis (TDA) has rapidly grown to provide a set of powerful tools to analyze various important features of data1. In this context, persistent homology has played a key role in bring- ing TDA to the fore of modern data analysis. -
Spring 2016 Tutorial Morse Theory
Spring 2016 Tutorial Morse Theory Description Morse theory is the study of the topology of smooth manifolds by looking at smooth functions. It turns out that a “generic” function can reflect quite a lot of information of the background manifold. In Morse theory, such “generic” functions are called “Morse functions”. By definition, a Morse function on a smooth manifold is a smooth function whose Hessians are non-degenerate at critical points. One can prove that every smooth function can be perturbed to a Morse function, hence we think of Morse functions as being “generic”. Roughly speaking, there are two different ways to study the topology of manifolds using a Morse function. The classical approach is to construct a cellular decomposition of the manifold by the Morse function. Each critical point of the Morse function corresponds to a cell, with dimension equals the number of negative eigenvalues of the Hessian matrix. Such an approach is very successful and yields lots of interesting results. However, for some technical reasons, this method cannot be generalized to infinite dimensions. Later on people developed another method that can be generalized to infinite dimensions. This new theory is now called “Floer theory”. In the tutorial, we will start from the very basics of differential topology and introduce both the classical and Floer-theory approaches of Morse theory. Then we will talk about some of the most important and interesting applications in history of Morse theory. Possible topics include but are not limited to: Smooth h- Cobordism Theorem, Generalized Poincare Conjecture in higher dimensions, Lefschetz Hyperplane Theorem, and the existence of closed geodesics on compact Riemannian manifolds, and so on. -
Graph Reconstruction by Discrete Morse Theory Arxiv:1803.05093V2 [Cs.CG] 21 Mar 2018
Graph Reconstruction by Discrete Morse Theory Tamal K. Dey,∗ Jiayuan Wang,∗ Yusu Wang∗ Abstract Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm. 1 Introduction Recovering hidden structures from potentially noisy data is a fundamental task in modern data analysis. A particular type of structure often of interest is the geometric graph-like structure. For example, given a collection of GPS trajectories, recovering the hidden road network can be modeled as reconstructing a geometric graph embedded in the plane. Given the simulated density field of dark matters in universe, finding the hidden filamentary structures is essentially a problem of geometric graph reconstruction. Different approaches have been developed for reconstructing a curve or a metric graph from input data. For example, in computer graphics, much work have been done in extracting arXiv:1803.05093v2 [cs.CG] 21 Mar 2018 1D skeleton of geometric models using the medial axis or Reeb graphs [10, 29, 20, 16, 22, 7]. -
MORSE-BOTT HOMOLOGY 1. Introduction 1.1. Overview. Let Cr(F) = {P ∈ M| Df P = 0} Denote the Set of Critical Points of a Smooth
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 8, August 2010, Pages 3997–4043 S 0002-9947(10)05073-7 Article electronically published on March 23, 2010 MORSE-BOTT HOMOLOGY AUGUSTIN BANYAGA AND DAVID E. HURTUBISE Abstract. We give a new proof of the Morse Homology Theorem by con- structing 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. 1. Introduction 1.1. Overview. Let Cr(f)={p ∈ M| df p =0} denote the set of critical points of a smooth function f : M → R on a smooth m-dimensional manifold M. A critical point p ∈ Cr(f) is said to be non-degenerate if and only if df is transverse to the ∗ zero section of T M at p. In local coordinates this is equivalent to the condition 2 that the m × m Hessian matrix ∂ f has rank m at p. If all the critical points ∂xi∂xj of f are non-degenerate, then f is called a Morse function. A Morse function f : M → R on a finite-dimensional compact smooth Riemann- ian manifold (M,g) is called Morse-Smale if all its stable and unstable manifolds intersect transversally. Such a function gives rise to a chain complex (C∗(f),∂∗), called the Morse-Smale-Witten chain complex. -
Appendix an Overview of Morse Theory
Appendix An Overview of Morse Theory Morse theory is a beautiful subject that sits between differential geometry, topol- ogy and calculus of variations. It was first developed by Morse [Mor25] in the middle of the 1920s and further extended, among many others, by Bott, Milnor, Palais, Smale, Gromoll and Meyer. The general philosophy of the theory is that the topology of a smooth manifold is intimately related to the number and “type” of critical points that a smooth function defined on it can have. In this brief ap- pendix we would like to give an overview of the topic, from the classical point of view of Morse, but with the more recent extensions that allow the theory to deal with so-called degenerate functions on infinite-dimensional manifolds. A compre- hensive treatment of the subject can be found in the first chapter of the book of Chang [Cha93]. There is also another, more recent, approach to the theory that we are not going to touch on in this brief note. It is based on the so-called Morse complex. This approach was pioneered by Thom [Tho49] and, later, by Smale [Sma61] in his proof of the generalized Poincar´e conjecture in dimensions greater than 4 (see the beautiful book of Milnor [Mil56] for an account of that stage of the theory). The definition of Morse complex appeared in 1982 in a paper by Witten [Wit82]. See the book of Schwarz [Sch93], the one of Banyaga and Hurtubise [BH04] or the survey of Abbondandolo and Majer [AM06] for a modern treatment. -
Arxiv:2012.08669V1 [Math.CT] 15 Dec 2020 2 Preface
Sheaf Theory Through Examples (Abridged Version) Daniel Rosiak December 12, 2020 arXiv:2012.08669v1 [math.CT] 15 Dec 2020 2 Preface After circulating an earlier version of this work among colleagues back in 2018, with the initial aim of providing a gentle and example-heavy introduction to sheaves aimed at a less specialized audience than is typical, I was encouraged by the feedback of readers, many of whom found the manuscript (or portions thereof) helpful; this encouragement led me to continue to make various additions and modifications over the years. The project is now under contract with the MIT Press, which would publish it as an open access book in 2021 or early 2022. In the meantime, a number of readers have encouraged me to make available at least a portion of the book through arXiv. The present version represents a little more than two-thirds of what the professionally edited and published book would contain: the fifth chapter and a concluding chapter are missing from this version. The fifth chapter is dedicated to toposes, a number of more involved applications of sheaves (including to the \n- queens problem" in chess, Schreier graphs for self-similar groups, cellular automata, and more), and discussion of constructions and examples from cohesive toposes. Feedback or comments on the present work can be directed to the author's personal email, and would of course be appreciated. 3 4 Contents Introduction 7 0.1 An Invitation . .7 0.2 A First Pass at the Idea of a Sheaf . 11 0.3 Outline of Contents . 20 1 Categorical Fundamentals for Sheaves 23 1.1 Categorical Preliminaries . -
RGB Image-Based Data Analysis Via Discrete Morse Theory and Persistent Homology
RGB IMAGE-BASED DATA ANALYSIS 1 RGB image-based data analysis via discrete Morse theory and persistent homology Chuan Du*, Christopher Szul,* Adarsh Manawa, Nima Rasekh, Rosemary Guzman, and Ruth Davidson** Abstract—Understanding and comparing images for the pur- uses a combination of DMT for the partitioning and skeletoniz- poses of data analysis is currently a very computationally ing (in other words finding the underlying topological features demanding task. A group at Australian National University of) images using differences in grayscale values as the distance (ANU) recently developed open-source code that can detect function for DMT and PH calculations. fundamental topological features of a grayscale image in a The data analysis methods we have developed for analyzing computationally feasible manner. This is made possible by the spatial properties of the images could lead to image-based fact that computers store grayscale images as cubical cellular complexes. These complexes can be studied using the techniques predictive data analysis that bypass the computational require- of discrete Morse theory. We expand the functionality of the ments of machine learning, as well as lead to novel targets ANU code by introducing methods and software for analyzing for which key statistical properties of images should be under images encoded in red, green, and blue (RGB), because this study to boost traditional methods of predictive data analysis. image encoding is very popular for publicly available data. Our The methods in [10], [11] and the code released with them methods allow the extraction of key topological information from were developed to handle grayscale images. But publicly avail- RGB images via informative persistence diagrams by introducing able image-based data is usually presented in heat-map data novel methods for transforming RGB-to-grayscale. -
Morse Theory
Morse Theory Al Momin Monday, October 17, 2005 1 THE example Let M be a 2-torus embedded in R3, laying on it's side and tangent to the two planes z = 0 and z = 1. Let f : M ! R be the function which takes a point x 2 M to its z-coordinate in this embedding (that is, it's \height" function). Let's study what this function can tell us about the topology of the manifold M. Define the sets M a := fx 2 M : f(x) ≤ ag and investigate M a for various a. • a < 0, then M a = φ. 2 a • 0 < a < 3 , then M is a disc. 1 2 a • 3 < a < 3 , then M is a cylinder. 2 a • 3 < a < 1, then M is a genus-1 surface with a single boundary component. • a > 1, then M a is M is the torus. Notice how the topology changes as you pass through a critical point (and conversely, how it doesn't change when you don't!). To describe this change, look at the homotopy type of M a. • a < 0, then M a = φ. 1 a • 0 < a < 3 , then M has the homotopy type of a point. 1 2 a • 3 < a < 3 , then M , which is a cylinder, has the homotopy type of a circle. Notice how this involves attaching a 1-cell to a point. 2 • 3 < a < 1. A surface of geunus 1 with one boundary component deformation retracts onto a figure-8 (see diagram), and so has the homotopy type of a figure-8, which we can obtain from the circle by attaching a 1-cell. -
FLOER THEORY and ITS TOPOLOGICAL APPLICATIONS 1. Introduction in Finite Dimensions, One Way to Compute the Homology of a Compact
FLOER THEORY AND ITS TOPOLOGICAL APPLICATIONS CIPRIAN MANOLESCU Abstract. We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture. 1. Introduction In finite dimensions, one way to compute the homology of a compact, smooth manifold is by Morse theory. Specifically, we start with a a smooth function f : X ! R and a Riemannian metric g on X. Under certain hypotheses (the Morse and Morse-Smale conditions), we can form a complex C∗(X; f; g) as follows: The generators of C∗(X; f; g) are the critical points of f, and the differential is given by X (1) @x = nxy · y; fyjind(x)−ind(y)=1g where nxy 2 Z is a signed count of the downward gradient flow lines of f connecting x to y. The quantity ind denotes the index of a critical point, that is, the number of negative eigenvalues of the Hessian (the matrix of second derivatives of f) at that point. The homology H∗(X; f; g) is called Morse homology, and it turns out to be isomorphic to the singular homology of X [Wit82, Flo89b, Bot88]. Floer homology is an adaptation of Morse homology to infinite dimensions. It applies to certain classes of infinite dimensional manifolds X and functions f : X ! R, where at critical points of f the Hessian has infinitely many positive and infinitely many negative eigenvalues. -
Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds
Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds Maciej Borodzik & Mark Powell The Journal of Geometric Analysis ISSN 1050-6926 J Geom Anal DOI 10.1007/s12220-014-9538-6 1 23 Your article is protected by copyright and all rights are held exclusively by Mathematica Josephina, Inc.. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy JGeomAnal DOI 10.1007/s12220-014-9538-6 Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds Maciej Borodzik · Mark Powell Received: 11 October 2013 © Mathematica Josephina, Inc. 2014 Abstract We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codi- mension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Némethi and A. Ranicki. In the codimension one case, we provide a slightly weaker state- ment. We also give proofs of rearrangement and cancellation theorems for handles of embedded submanifolds with boundary.