Morse homology. Michael Landry. April 2014
|
|
- Rosamund Watts
- 5 years ago
- Views:
Transcription
1 Morse homology Michael Landry April 2014 This is a supplementary note for a (hopefully) fun, informal one hour talk on Morse homology, which I wrote after digesting the opening pages of Michael Hutchings great notes [2]. Please contact me with questions, comments, errors, confessions, etc. at michael.landry@yale.edu. 1 Some background Let X be a smooth, compact n-dimensional manifold, and f C (M, R). Let p be a critical point of f, that is df p = 0, and choose local coordinates x 1,..., x n around p. The Hessian matrix of f at p in these coordinates is the symmetric matrix [ 2 ] f H f (p) = x i x j (p). We say that p is a nondegenerate critical point of f if H f (p) is nonsingular; this notion does not depend on our choice of coordinates. If all the critical points of f are nondegenerate, then f is Morse. Since H f (p) is symmetric, its eigenvalues are real. The Morse index of p is the number of negative eigenvalues of H f (p), counted with multiplicity. For more detail, see [4]. Example 1. Let S be a surface embedded in R 3. After perturbing S if necessary, the height function f(x, y, z) = z defines a Morse function on S. The critical points of index 2 are local maxima, the saddle points have index 1, and the local minima have index 0. This is the example to keep in mind while reading this note. 2 The moduli space M(p, q) Equip X with a Riemannian metric g, and recall that the gradient of f, denoted grad f, is the vector field on X defined by df = grad f,. The gradient points in the direction of increase of f, orthogonally to the level curves of f. Let p be a critical point of f. We define the descending manifold D(p) and ascending manifold A(p) of p by D(p) = {x X flowing along grad f takes x to p }, A(p) = {x X flowing along grad f takes x to p }. 1
2 Figure 1: S 2, with critical points of the height function. In the example when X n is embedded in R n+1, f is the height function and g is the metric induced by the Euclidean one on R n, the index of p can be thought of as the maximum number of linearly independent directions in which grad f points out of p. It follows that in this case D(p) is a disk of dimension ind p and A(p) is a disk of dimension n ind p. This is true in the general case also, and for a proof Hutchings refers us to [1]. Example 2. Let X be the unit sphere in R 3, with f projection to the z coordinate. Then f has 2 critical points: the North Pole N = (0, 0, 1) has index 2, and the South Pole S = (0, 0, 1) has index 0. Furthermore D(N) = X \ {S} and A(S) = X \ {N}. A flow line of (f, g) is an equivalence class of integral curves of grad f, [α : R X] where α β if α(t) = β(t + a) for some a R and all t. We say [α] is a flow line from p to q if lim t α(t) = p and lim t α(t) = q. Definition 3. The pair (f, g) is Morse-Smale if f is Morse and for every pair of critical points p and q, A(q) D(p). It turns out that this condition is generic, and we will henceforth assume that (f, g) is Morse- Smale. Denote by M(p, q) the moduli space of flow lines from p to q. We can identify M(p, q) with (D(p) A(q))/R, where R acts on points in D(p) A(q) by the flow of grad f. By the Morse-Smale condition, D(p) A(q) is a submanifold of X of dimension (n ind q + ind p) n = ind p ind q. The action of R is smooth, free, and proper, so by the quotient manifold theorem (see [3], for example), M(p, q) has the structure of a (ind p ind q 1)-dimensional manifold (as long as that number makes sense). Remark 4. The Morse-Smale condition implies that if ind p ind q, then M(p, q) =. Indeed, if ind p ind q, then dim D(p) + dim A(q) = ind p + n ind q n, so dim(d(p) A(q)) 0 by transversality. If M(p, q) is nonempty then this dimension is 1 since D(p) A(q) would then contain a flow line. Hence M(p, q) =. 2
3 If ind p = ind q + 1, then M(p, q) is 0-dimensional and, as the following theorem implies, compact. The nice thing about compact 0-dimensional manifolds is that they are finite! This trivial observation is the germ of the upcoming construction of the Morse chain complex. Theorem 5. The moduli space M(p, q) can be compactified by adding in broken flow lines from p to q, i.e. paths from p to q which are the concatenation of flow lines through intermediate critical points. The compactification M(p, q) is a smooth manifold with corners, and the k-times broken flowlines form the codimension k parts of the boundary. In particular, if ind p = ind q + 1 there are no possible intermediate critical points so M(p, q) = M(p, q) is compact and therefore finite. Figure 2: A sequence of flow lines converging to a broken flow line (dark). 3 The Morse chain complex We now orient M(p, q). First, choose an orientation of D(p) for all critical points p. Let x be a point on a flow line [α] M(p, q). Then at x im α there is an isomorphism T D(p) = T M(p, q) T α T D(q) and we orient T M(p, q) such that this isomorphism is orientation-preserving. This is confusing but it is easier to see what is going on in an example; I recommend trying to see the isomorphism for yourself in Example 1. Define C i to be the free abelian group generated by the critical points of index i, and i (p) = #M(p, q)q ind q=i 1 3
4 where the sum is taken over all critical points of index i 1 and #M(p, q) is the signed number of flowlines from p to q, the sign being determined by the orientation on M(p, q) as described above. Proposition 6. = 0. Proof. Let p, q be critical points of index i and i 2. The coefficient of q in 2 p is ind r=i 1 #M(p, r) #M(r, q) = # ind r=i 1 M(p, r) M(r, q) = # M(p, q) = 0 where the last equality follows from the fact that M(p, q) is an oriented 1-manifold with boundary, so its signed number of boundary points is 0. So we have a chain complex, and we can compute its homology groups. But this chain complex depends on a lot of things: namely the Morse function f, the metric g, and the orientations we chose for all but descending manifolds to orient the M(p, q) s. Unless we re working in the category of manifolds with a Morse function, Riemannian metric, and oriented descending submanifolds, = 0 does not seem like anything to jump over the moon about. However, as you probably suspected already, the following is true: Fact 7. The homology groups of this Morse complex are isomorphic to the singular homology groups of X. Perhaps you have now noticed a pattern of me citing big results in order to bail myself out of trouble. Oh well! This is a fact for another day, since one hour is a short amount of time. In the fall I might try to learn about Floer homology with others, and we would probably go over the proof of this fact then. Let me know if you are interested. 4 Examples: torus and Klein bottle Let s compute the Morse homology groups of some example surfaces. Example 8. Take the standard 2-torus T 2 and stand it on its end. This situation is not Morse- Smale since there are 2 flow lines between the saddle points, so we perturb it slightly as in Figure 3a. Consider 2 (p). The 2 flow lines from p to q have opposite sign so #M(p, q) = 0. Similarly #M(p, r) = 0 so 2 = 0. In fact, 1 = 0 also, so when we take the homology of the Morse chain complex we get H 2 (T 2 ) = Z, H 1 (T 2 ) = Z Z, and H 0 (T 2 ) = Z. Example 9. We can embed the Klein bottle K in R 4. Let f(w, x, y, z) = z; after perturbation, f and the metric R 4 induces on K are Morse-Smale. The situation is as shown in Figure 3b by cheating and projecting to R 3. We have C 2 = Zp, C 1 = Zq Zr, C 0 = Zs. Up to sign, we see that 2 (p) = 2r because the 2 elements in M(p, q) have opposite signs, and the 2 elements in M(p, r) have the same sign. Finally, 1 (q) = 1 (r) = 0. Hence H 2 (K) = 0, H 1 (K) = Zq Zr/(2r) = Z Z/(2), H 0 (K) = Z. The difference between our computations for the torus and Klein bottle lies in 2, and one can see that this stems from the fact that A(r) {p} represents a singular homology class of finite order. 4
5 (a) (b) Figure 3: The 2-torus and Klein bottle References [1] R. Abraham, J. Robbins. Transversal mappings and flows. W.A. Benjamin, [2] M. Hutchings. Lecture notes on Morse homology. Available on the author s website: math.berkeley.edu/ hutching/ [3] J. Lee. Introduction to smooth manifolds. Second edition. Springer, [4] J. Milnor. Morse theory. Princeton University Press,
MORSE HOMOLOGY. Contents. Manifolds are closed. Fields are Z/2.
MORSE HOMOLOGY STUDENT GEOMETRY AND TOPOLOGY SEMINAR FEBRUARY 26, 2015 MORGAN WEILER 1. Morse Functions 1 Morse Lemma 3 Existence 3 Genericness 4 Topology 4 2. The Morse Chain Complex 4 Generators 5 Differential
More informationA Crash Course of Floer Homology for Lagrangian Intersections
A Crash Course of Floer Homology for Lagrangian Intersections Manabu AKAHO Department of Mathematics Tokyo Metropolitan University akaho@math.metro-u.ac.jp 1 Introduction There are several kinds of Floer
More informationMorse Theory for Lagrange Multipliers
Morse Theory for Lagrange Multipliers γ=0 grad γ grad f Guangbo Xu Princeton University and Steve Schecter North Carolina State University 1 2 Outline 3 (1) History of Mathematics 1950 2050. Chapter 1.
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the
More informationA Brief History of Morse Homology
A Brief History of Morse Homology Yanfeng Chen Abstract Morse theory was originally due to Marston Morse [5]. It gives us a method to study the topology of a manifold using the information of the critical
More informationHandlebody Decomposition of a Manifold
Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody
More informationMath 205C - Topology Midterm
Math 205C - Topology Midterm Erin Pearse 1. a) State the definition of an n-dimensional topological (differentiable) manifold. An n-dimensional topological manifold is a topological space that is Hausdorff,
More informationSelf-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds
Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu
More informationAn introduction to cobordism
An introduction to cobordism Martin Vito Cruz 30 April 2004 1 Introduction Cobordism theory is the study of manifolds modulo the cobordism relation: two manifolds are considered the same if their disjoint
More informationLagrangian Intersection Floer Homology (sketch) Chris Gerig
Lagrangian Intersection Floer Homology (sketch) 9-15-2011 Chris Gerig Recall that a symplectic 2n-manifold (M, ω) is a smooth manifold with a closed nondegenerate 2- form, i.e. ω(x, y) = ω(y, x) and dω
More informationAn introduction to discrete Morse theory
An introduction to discrete Morse theory Henry Adams December 11, 2013 Abstract This talk will be an introduction to discrete Morse theory. Whereas standard Morse theory studies smooth functions on a differentiable
More informationSmooth Structure. lies on the boundary, then it is determined up to the identifications it 1 2
132 3. Smooth Structure lies on the boundary, then it is determined up to the identifications 1 2 + it 1 2 + it on the vertical boundary and z 1/z on the circular part. Notice that since z z + 1 and z
More informationNotes by Maksim Maydanskiy.
SPECTRAL FLOW IN MORSE THEORY. 1 Introduction Notes by Maksim Maydanskiy. Spectral flow is a general formula or computing the Fredholm index of an operator d ds +A(s) : L1,2 (R, H) L 2 (R, H) for a family
More informationJ-holomorphic curves in symplectic geometry
J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 28, 2006 Since their introduction by Gromov [4] in the mid-1980 s J-holomorphic curves have been one of the most widely
More informationLECTURE 1: ADVERTISEMENT LECTURE. (0) x i + 1 2! x i x j
LECTUE 1: ADVETISEMENT LECTUE. PAT III, MOSE HOMOLOGY, 2011 HTTP://MOSEHOMOLOGY.WIKISPACES.COM What are nice functions? We will consider the following setu: M = closed 1 (smooth) m-dimensional manifold,
More informationMorse-Bott Homology. Augustin Banyaga. David Hurtubise
Morse-Bott Homology (Using singular N-cube chains) Augustin Banyaga Banyaga@math.psu.edu David Hurtubise Hurtubise@psu.edu s u TM s z S 2 r +1 q 1 E u f +1 1 p Penn State University Park Penn State Altoona
More informationCUT LOCI AND DISTANCE FUNCTIONS
Math. J. Okayama Univ. 49 (2007), 65 92 CUT LOCI AND DISTANCE FUNCTIONS Jin-ichi ITOH and Takashi SAKAI 1. Introduction Let (M, g) be a compact Riemannian manifold and d(p, q) the distance between p, q
More informationTHE POINCARE-HOPF THEOREM
THE POINCARE-HOPF THEOREM ALEX WRIGHT AND KAEL DIXON Abstract. Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under
More informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationModuli spaces and cw structures arising from morse theory
Wayne State University DigitalCommons@WayneState Wayne State University Dissertations 1-1-2011 Moduli spaces and cw structures arising from morse theory Lizhen Qin Wayne State University, Follow this and
More informationLINKING AND THE MORSE COMPLEX. x 2 i
LINKING AND THE MORSE COMPLEX MICHAEL USHER ABSTRACT. For a Morse function f on a compact oriented manifold M, we show that f has more critical points than the number required by the Morse inequalities
More informationCup product and intersection
Cup product and intersection Michael Hutchings March 28, 2005 Abstract This is a handout for my algebraic topology course. The goal is to explain a geometric interpretation of the cup product. Namely,
More informationMORSE THEORY DAN BURGHELEA. Department of Mathematics The Ohio State University
MORSE THEORY DAN BURGHELEA Department of Mathematics The Ohio State University 1 Morse Theory begins with the modest task of understanding the local maxima, minima and sadle points of a smooth function
More informationIntersection of stable and unstable manifolds for invariant Morse functions
Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and
More informationTHE MORSE-BOTT INEQUALITIES VIA DYNAMICAL SYSTEMS
THE MORSE-BOTT INEQUALITIES VIA DYNAMICAL SYSTEMS AUGUSTIN BANYAGA AND DAVID E. HURTUBISE Abstract. Let f : M R be a Morse-Bott function on a compact smooth finite dimensional manifold M. The polynomial
More informationNOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY
NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY 1. Closed and exact forms Let X be a n-manifold (not necessarily oriented), and let α be a k-form on X. We say that α is closed if dα = 0 and say
More informationBordism and the Pontryagin-Thom Theorem
Bordism and the Pontryagin-Thom Theorem Richard Wong Differential Topology Term Paper December 2, 2016 1 Introduction Given the classification of low dimensional manifolds up to equivalence relations such
More informationTHREE APPROACHES TO MORSE-BOTT HOMOLOGY
THREE APPROACHES TO MORSE-BOTT HOMOLOGY DAVID E. HURTUBISE arxiv:1208.5066v2 [math.at] 3 Jan 2013 Dedicated to Professor Augustin Banyaga on the occasion of his 65th birthday Abstract. In this paper we
More informationSARD S THEOREM ALEX WRIGHT
SARD S THEOREM ALEX WRIGHT Abstract. A proof of Sard s Theorem is presented, and applications to the Whitney Embedding and Immersion Theorems, the existence of Morse functions, and the General Position
More informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
More information0. Introduction 1 0. INTRODUCTION
0. Introduction 1 0. INTRODUCTION In a very rough sketch we explain what algebraic geometry is about and what it can be used for. We stress the many correlations with other fields of research, such as
More informationMorse homology of RP n
U.U.D.M. Project Report 2013:17 Morse homology of RP n Sebastian Pöder Examensarbete i matematik, 15 hp Handledare och examinator: Ryszard Rubinsztein Juni 2013 Department of Mathematics Uppsala University
More informationMath 5BI: Problem Set 6 Gradient dynamical systems
Math 5BI: Problem Set 6 Gradient dynamical systems April 25, 2007 Recall that if f(x) = f(x 1, x 2,..., x n ) is a smooth function of n variables, the gradient of f is the vector field f(x) = ( f)(x 1,
More informationON A PROBLEM OF ELEMENTARY DIFFERENTIAL GEOMETRY AND THE NUMBER OF ITS SOLUTIONS
ON A PROBLEM OF ELEMENTARY DIFFERENTIAL GEOMETRY AND THE NUMBER OF ITS SOLUTIONS JOHANNES WALLNER Abstract. If M and N are submanifolds of R k, and a, b are points in R k, we may ask for points x M and
More informationLECTURE 16: CONJUGATE AND CUT POINTS
LECTURE 16: CONJUGATE AND CUT POINTS 1. Conjugate Points Let (M, g) be Riemannian and γ : [a, b] M a geodesic. Then by definition, exp p ((t a) γ(a)) = γ(t). We know that exp p is a diffeomorphism near
More informationLECTURE 9: THE WHITNEY EMBEDDING THEOREM
LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically, the word manifold (Mannigfaltigkeit in German) first appeared in Riemann s doctoral thesis in 1851. At the early times, manifolds are defined extrinsically:
More informationThe relationship between framed bordism and skew-framed bordism
The relationship between framed bordism and sew-framed bordism Pyotr M. Ahmet ev and Peter J. Eccles Abstract A sew-framing of an immersion is an isomorphism between the normal bundle of the immersion
More informationAn integral lift of contact homology
Columbia University University of Pennsylvannia, January 2017 Classical mechanics The phase space R 2n of a system consists of the position and momentum of a particle. Lagrange: The equations of motion
More informationMorse Theory and Applications to Equivariant Topology
Morse Theory and Applications to Equivariant Topology Morse Theory: the classical approach Briefly, Morse theory is ubiquitous and indomitable (Bott). It embodies a far reaching idea: the geometry and
More informationLecture 4: Knot Complements
Lecture 4: Knot Complements Notes by Zach Haney January 26, 2016 1 Introduction Here we discuss properties of the knot complement, S 3 \ K, for a knot K. Definition 1.1. A tubular neighborhood V k S 3
More informationBROUWER FIXED POINT THEOREM. Contents 1. Introduction 1 2. Preliminaries 1 3. Brouwer fixed point theorem 3 Acknowledgments 8 References 8
BROUWER FIXED POINT THEOREM DANIELE CARATELLI Abstract. This paper aims at proving the Brouwer fixed point theorem for smooth maps. The theorem states that any continuous (smooth in our proof) function
More informationEuler Characteristic of Two-Dimensional Manifolds
Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several
More informationWe have the following immediate corollary. 1
1. Thom Spaces and Transversality Definition 1.1. Let π : E B be a real k vector bundle with a Euclidean metric and let E 1 be the set of elements of norm 1. The Thom space T (E) of E is the quotient E/E
More informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationFunctoriality and duality in Morse Conley Floer homology
J. Fixed Point Theory Appl. 16 2014 437 476 DOI 10.1007/s11784-015-0223-6 Published online April 4, 2015 Journal of Fixed Point Theory 2015 The Authors and Applications This article is published with open
More informationDonaldson Invariants and Moduli of Yang-Mills Instantons
Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017 Moduli and Invariants
More informationTransversality. Abhishek Khetan. December 13, Basics 1. 2 The Transversality Theorem 1. 3 Transversality and Homotopy 2
Transversality Abhishek Khetan December 13, 2017 Contents 1 Basics 1 2 The Transversality Theorem 1 3 Transversality and Homotopy 2 4 Intersection Number Mod 2 4 5 Degree Mod 2 4 1 Basics Definition. Let
More informationDEVELOPMENT OF MORSE THEORY
DEVELOPMENT OF MORSE THEORY MATTHEW STEED Abstract. In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined
More informationMath 6455 Nov 1, Differential Geometry I Fall 2006, Georgia Tech
Math 6455 Nov 1, 26 1 Differential Geometry I Fall 26, Georgia Tech Lecture Notes 14 Connections Suppose that we have a vector field X on a Riemannian manifold M. How can we measure how much X is changing
More informationLecture 4: Stabilization
Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3
More informationNotes on Real and Complex Analytic and Semianalytic Singularities
Notes on Real and Complex Analytic and Semianalytic Singularities David B. Massey and Lê Dũng Tráng 1 Manifolds and Local, Ambient, Topological-type We assume that the reader is familiar with the notion
More informationDynamics of Group Actions and Minimal Sets
University of Illinois at Chicago www.math.uic.edu/ hurder First Joint Meeting of the Sociedad de Matemática de Chile American Mathematical Society Special Session on Group Actions: Probability and Dynamics
More informationDerived Differential Geometry
Derived Differential Geometry Lecture 1 of 3: Dominic Joyce, Oxford University Derived Algebraic Geometry and Interactions, Toulouse, June 2017 For references, see http://people.maths.ox.ac.uk/ joyce/dmanifolds.html,
More informationLECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS
LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationBredon, Introduction to compact transformation groups, Academic Press
1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions
More informationSolution: We can cut the 2-simplex in two, perform the identification and then stitch it back up. The best way to see this is with the picture:
Samuel Lee Algebraic Topology Homework #6 May 11, 2016 Problem 1: ( 2.1: #1). What familiar space is the quotient -complex of a 2-simplex [v 0, v 1, v 2 ] obtained by identifying the edges [v 0, v 1 ]
More informationSelf-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds
Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)
Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the
More informationLECTURE 11: TRANSVERSALITY
LECTURE 11: TRANSVERSALITY Let f : M N be a smooth map. In the past three lectures, we are mainly studying the image of f, especially when f is an embedding. Today we would like to study the pre-image
More informationNon-isolated Hypersurface Singularities and Lê Cycles
Contemporary Mathematics Non-isolated Hypersurface Singularities and Lê Cycles David B. Massey Abstract. In this series of lectures, I will discuss results for complex hypersurfaces with non-isolated singularities.
More informationABEL S THEOREM BEN DRIBUS
ABEL S THEOREM BEN DRIBUS Abstract. Abel s Theorem is a classical result in the theory of Riemann surfaces. Important in its own right, Abel s Theorem and related ideas generalize to shed light on subjects
More informationCohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationChapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves
Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction
More informationMany of the exercises are taken from the books referred at the end of the document.
Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the
More informationDIFFERENTIAL TOPOLOGY: MORSE THEORY AND THE EULER CHARACTERISTIC
DIFFERENTIAL TOPOLOGY: MORSE THEORY AND THE EULER CHARACTERISTIC DANIEL MITSUTANI Abstract. This paper uses differential topology to define the Euler characteristic as a self-intersection number. We then
More informationfy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))
1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical
More informationHomology Theory, Morse Theory, and the Morse Homology Theorem
Homology Theory, Morse Theory, and the Morse Homology Theorem Alex Stephanus June 8, 2016 Abstract This thesis develops the Morse Homology Theorem, first starting by motivation for and developing of the
More informationThe Differential Structure of an Orbifold
The Differential Structure of an Orbifold AMS Sectional Meeting 2015 University of Memphis Jordan Watts University of Colorado at Boulder October 18, 2015 Introduction Let G 1 G 0 be a Lie groupoid. Consider
More informationHomological mirror symmetry via families of Lagrangians
Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants
More informationM4P52 Manifolds, 2016 Problem Sheet 1
Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete
More informationThe d-orbifold programme. Lecture 5 of 5: D-orbifold homology and cohomology, and virtual cycles
The d-orbifold programme. Lecture 5 of 5: and cohomology, and virtual cycles Dominic Joyce, Oxford University May 2014 Work in progress, no papers yet. However, you can find a previous version of this
More informationExercise: Consider the poset of subsets of {0, 1, 2} ordered under inclusion: Date: July 15, 2015.
07-13-2015 Contents 1. Dimension 1 2. The Mayer-Vietoris Sequence 3 2.1. Suspension and Spheres 4 2.2. Direct Sums 4 2.3. Constuction of the Mayer-Vietoris Sequence 6 2.4. A Sample Calculation 7 As we
More informationLECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM
LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory
More information30 Surfaces and nondegenerate symmetric bilinear forms
80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces
More informationMATH DIFFERENTIAL GEOMETRY. Contents
MATH 3968 - DIFFERENTIAL GEOMETRY ANDREW TULLOCH Contents 1. Curves in R N 2 2. General Analysis 2 3. Surfaces in R 3 3 3.1. The Gauss Bonnet Theorem 8 4. Abstract Manifolds 9 1 MATH 3968 - DIFFERENTIAL
More informationLECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.
LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.
More informationMorse Theory and Supersymmetry
Morse Theory and Supersymmetry Jeremy van der Heijden July 1, 2016 Bachelor Thesis Mathematics, Physics and Astronomy Supervisors: prof. dr. Erik Verlinde, dr. Hessel Posthuma Korteweg-de Vries Instituut
More informationApril 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.
April 3, 005 - Hyperbolic Sets We now extend the structure of the horseshoe to more general kinds of invariant sets. Let r, and let f D r (M) where M is a Riemannian manifold. A compact f invariant set
More informationMetric spaces and metrizability
1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively
More informationCHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents
CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally
More informationTAUBES S PROOF OF THE WEINSTEIN CONJECTURE IN DIMENSION THREE
TAUBES S PROOF OF THE WEINSTEIN CONJECTURE IN DIMENSION THREE MICHAEL HUTCHINGS Abstract. Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? No,
More informationConjectures on counting associative 3-folds in G 2 -manifolds
in G 2 -manifolds Dominic Joyce, Oxford University Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, First Annual Meeting, New York City, September 2017. Based on arxiv:1610.09836.
More informationMorse Theory and Stability of Relative Equilibria in the Planar n-vortex Problem
Morse Theory and Stability of Relative Equilibria in the Planar n-vortex Problem Gareth E. Roberts Department of Mathematics and Computer Science College of the Holy Cross SIAM Conference on Dynamical
More informationVector, Matrix, and Tensor Derivatives
Vector, Matrix, and Tensor Derivatives Erik Learned-Miller The purpose of this document is to help you learn to take derivatives of vectors, matrices, and higher order tensors (arrays with three dimensions
More informationMUMFORD-TATE GROUPS AND ABELIAN VARIETIES. 1. Introduction These are notes for a lecture in Elham Izadi s 2006 VIGRE seminar on the Hodge Conjecture.
MUMFORD-TATE GROUPS AND ABELIAN VARIETIES PETE L. CLARK 1. Introduction These are notes for a lecture in Elham Izadi s 2006 VIGRE seminar on the Hodge Conjecture. Let us recall what we have done so far:
More informationInstanton Floer homology with Lagrangian boundary conditions
Instanton Floer homology with Lagrangian boundary conditions Dietmar Salamon ETH-Zürich Katrin Wehrheim IAS Princeton 13 July 2006 Contents 1. Introduction 1 2. The Chern Simons functional 6 3. The Hessian
More informationCorrigendum to: Geometric Composition in Quilted Floer Theory
Corrigendum to: Geometric Composition in Quilted Floer Theory YANKI LEKILI MAX LIPYANSKIY As we explain below, an error in Lemma 11 of [5] affects the main results of that paper (Theorems 1, 2 and 3 of
More informationThe Geometrization Theorem
The Geometrization Theorem Matthew D. Brown Wednesday, December 19, 2012 In this paper, we discuss the Geometrization Theorem, formerly Thurston s Geometrization Conjecture, which is essentially the statement
More informationMONDAY - TALK 4 ALGEBRAIC STRUCTURE ON COHOMOLOGY
MONDAY - TALK 4 ALGEBRAIC STRUCTURE ON COHOMOLOGY Contents 1. Cohomology 1 2. The ring structure and cup product 2 2.1. Idea and example 2 3. Tensor product of Chain complexes 2 4. Kunneth formula and
More information58 CHAPTER 2. COMPUTATIONAL METHODS
58 CHAPTER 2. COMPUTATIONAL METHODS 23 Hom and Lim We will now develop more properties of the tensor product: its relationship to homomorphisms and to direct limits. The tensor product arose in our study
More informationA Field Extension as a Vector Space
Chapter 8 A Field Extension as a Vector Space In this chapter, we take a closer look at a finite extension from the point of view that is a vector space over. It is clear, for instance, that any is a linear
More informationManifolds in Fluid Dynamics
Manifolds in Fluid Dynamics Justin Ryan 25 April 2011 1 Preliminary Remarks In studying fluid dynamics it is useful to employ two different perspectives of a fluid flowing through a domain D. The Eulerian
More informationNoncommutative geometry and quantum field theory
Noncommutative geometry and quantum field theory Graeme Segal The beginning of noncommutative geometry is the observation that there is a rough equivalence contravariant between the category of topological
More informationMath 230a Final Exam Harvard University, Fall Instructor: Hiro Lee Tanaka
Math 230a Final Exam Harvard University, Fall 2014 Instructor: Hiro Lee Tanaka 0. Read me carefully. 0.1. Due Date. Per university policy, the official due date of this exam is Sunday, December 14th, 11:59
More information32 Proof of the orientation theorem
88 CHAPTER 3. COHOMOLOGY AND DUALITY 32 Proof of the orientation theorem We are studying the way in which local homological information gives rise to global information, especially on an n-manifold M.
More informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent
More information(iv) Whitney s condition B. Suppose S β S α. If two sequences (a k ) S α and (b k ) S β both converge to the same x S β then lim.
0.1. Stratified spaces. References are [7], [6], [3]. Singular spaces are naturally associated to many important mathematical objects (for example in representation theory). We are essentially interested
More information