Lecture Notes on Morse and Conley Theory { MANUSCRIPT IN PROGRESS { Joa Weber UNICAMP January 31, 2018 ii Contents 1 Introduction1 2 Preliminaries9 2.1 Topology...............................9 2.2 Algebra................................ 11 2.3 Analysis................................ 17 2.4 Manifolds and transversality..................... 18 2.5 Group actions............................. 19 2.6 Local and global dynamics...................... 21 I Morse theory 25 3 Classical Morse theory 27 3.1 Morse functions............................ 27 3.2 Regular intervals........................... 31 3.2.1 Application: Critical point theory............. 33 3.2.2 Application: Topology.................... 34 3.3 Attaching cells............................ 36 3.4 Morse inequalities.......................... 39 3.5 Generalizations............................ 51 3.5.1 Morse-Bott theory...................... 51 3.5.2 Equivariant Morse-Bott theory............... 55 3.5.3 Lusternik-Schnirelmann theory............... 61 4 Classical Morse theory revisited 75 4.1 Backward λ-Lemma and homotopy type.............. 75 5 Morse homology of manifold triads 99 5.1 Introduction { closed manifolds................... 99 5.2 The Morse-Witten complex..................... 104 5.2.1 Triad Morse functions.................... 104 5.2.2 Transversality and triad Morse-Smale pairs........ 105 5.2.3 Dynamical compactness................... 106 iii iv CONTENTS 5.2.4 Gluing............................. 109 5.2.5 Orientation transport and partner pairs.......... 112 5.3 Morse homology of triad Morse-Smale pairs............ 115 5.4 Continuation { changing the pair.................. 116 5.4.1 Monotone and admissible homotopies........... 117 5.4.2 Homotopies.......................... 120 5.4.3 Homotopies of homotopies.................. 125 5.5 Isomorphism to singular homology................. 127 5.6 Morse homology of manifold triads................. 131 5.7 Variations and generalizations.................... 132 5.8 Applications.............................. 132 6 Local Morse homology 133 6.1 Local Morse functions and gradient flows............. 133 6.2 Morse-Smale flows and isolating homotopies............ 134 6.3 Morse complex for isolated invariant sets.............. 135 6.4 Morse homology of an isolating Morse-Smale triple........ 138 6.4.1 Continuation { changing the triple............. 138 6.5 Isomorphism to relative singular homology............. 138 6.6 Continuation of isolated invariant sets............... 138 7 Mod two Morse homology theory 139 7.1 Morse (co)homology of a manifold................. 139 7.1.1 Homology........................... 141 7.1.2 Cohomology.......................... 149 7.1.3 Poincar´eduality....................... 154 7.1.4 Relative (co)homology.................... 156 7.1.5 Lefschetz duality....................... 159 7.2 Functoriality............................. 160 7.2.1 Induced chain maps..................... 160 7.2.2 Homotopy........................... 160 7.2.3 Composition......................... 160 7.3 Products in Morse (co)homology.................. 161 7.3.1 Thom isomorphism and Euler class............. 161 7.3.2 Umkehr map......................... 161 7.4 Morse field theory.......................... 161 II Conley theory 163 8 Introduction 167 8.1 Basic notions............................. 167 8.2 Overview............................... 170 CONTENTS v 9 Classical Conley theory 171 9.1 Conley pairs.............................. 171 9.1.1 Gradient flows........................ 171 10 Conley homology 177 10.1 Lyapunov functions and isolating blocks.............. 177 10.2 Gradient flows............................ 186 10.3 Morse homology of an isolating Morse-Smale triple........ 187 10.3.1 Continuation { changing the triple............. 187 10.4 Isomorphism to relative singular homology............. 187 10.5 Conley homology of an isolated invariant set............ 187 11 Classical Conley theory revisited 189 11.1 Continuation............................. 189 11.1.1 Applications......................... 190 11.2 Morse decompositions and connection matrices.......... 190 11.2.1 Morse-Conley inequalities.................. 190 III Appendices 191 A Hyperbolic gradient dynamics 193 A.1 Vector fields and limit sets...................... 194 A.2 Analytic setup near hyperbolic singularities............ 198 A.3 Invariant manifolds.......................... 206 A.3.1 Unstable manifold theorem................. 208 A.3.2 Stable manifold theorem................... 211 A.3.3 Convenient coordinates................... 214 A.4 The inclination or λ-Lemma..................... 217 A.4.1 Forward λ-Lemma...................... 219 A.4.2 Backward λ-Lemma..................... 219 A.5 Invariant foliations and induced flows................ 236 A.5.1 Invariant unstable foliations and induced flow....... 236 A.5.2 Invariant stable foliations and induced flow........ 236 A.6 Grobman-Hartman Theorem.................... 242 A.6.1 Perturbing hyperbolic linear vector fields......... 244 A.6.2 Perturbing hyperbolic linear isomorphisms......... 250 A.6.3 Grobman-Hartman via λ-Lemma.............. 251 A.6.4 Appendix on general hyperbolicity............. 254 B (Co)homology theory 259 B.0.5 Outline of contents...................... 261 B.0.6 List of symbols, conventions, notation........... 262 B.1 (Co)homology of chain complexes.................. 264 B.1.1 (Co)chain complexes..................... 265 B.1.2 Kronecker pairing...................... 269 vi CONTENTS B.1.3 Exact sequences....................... 271 B.1.4 Relative (co)homology.................... 275 B.1.5 Mayer-Vietoris sequences.................. 277 B.1.6 Poincar´epolynomial and Euler characteristic....... 280 B.1.7 Universal and local coefficients............... 281 B.2 Mod two (co)homology groups................... 282 B.2.1 Simplicial (co)homology................... 282 B.2.2 Singular (co)homology.................... 294 B.2.3 Cellular (co)homology.................... 303 B.2.4 Cechˇ versus Steenrod (co)homology............ 303 B.3 Mod two (co)homology products.................. 304 B.3.1 Cup product......................... 304 B.3.2 Cap product......................... 311 B.3.3 Tensor product of modules and algebras.......... 316 B.3.4 Cross product and K¨unnethTheorem........... 319 B.3.5 Leray-Hirsch Theorem.................... 326 B.3.6 Thom isomorphism and Euler class............. 336 B.3.7 Poincar´eduality....................... 346 B.3.8 Submanifolds and Poincar´eduality............. 348 C Transversality theory 359 Bibliography 361 Index 373 List of Symbols 383 Chapter 1 Introduction To put it in a nutshell, Morse and Conley theory connect the areas of analysis and topology through the theory of dynamical systems. Morse theory lives in the well known realm of hyperbolic dynamics and can be viewed as a special case of Conley theory. Analysis Dynamical Systems Topology ∞ Conley theory 0 rkH (Σ) R Morse theory ∗ S n χ(Σ) P Hyperbolic dynamics lims →∞ Figure 1.1: Morse and Conley theory connect analysis and topology Morse theory The early days 1920s-40s { Topology of sublevel sets Consider a smooth manifold M. In the 20s of the last century it was the insight of Marston Morse [Mor34] that the topology of M is related to non-degenerate critical points of smooth functions f : M ! R. By definition a critical point p of f satisfies the identity dfp = 0, that is it is an extremum of f. By Crit we denote the set of critical points of f. A critical point p is called non- degenerate if, given a local coordinate system φ = (x1; : : : ; xn): M ⊃ U ! Rn around p, the (symmetric) Hessian matrix @2f Hpf := i j (p) (1.0.1) @x @x i;j=1;:::;n 1 2 CHAPTER 1. INTRODUCTION M aj M aj 1 Dind(pj ) − ∪χ a R M 4 ∼ p4 f M a 3 ∼ p3 a2 ∼ p2 a 1 ∼ a0 p1 ∅ k Figure 1.2: Height function f and attaching of disks D where k = ind(pj) of f at p is non-singular, that is invertible or, equivalently, zero is not an eigenvalue of Hpf. Hence critical points are isolated by the inverse function theorem 2.3.1. The number ind(x) of negative eigenvalues of Hpf, counted with multiplicities, is called the Morse index of a non-degenerate critical point. The key idea to relate topology and analysis is to study the family of sublevel sets, also called half-spaces, which are the closed subsets of M defined by a M := ff ≤ ag := fq 2 M j f(q) ≤ ag ; a 2 R: (1.0.2) A regular value of f is a constant b such that its pre-image f −1(b) = ff = bg contains no critical point. All other constants are called critical values. Note that according to this definition any constant whose pre-image is empty, that is which lies outside the range of f, is a regular value. The geometric significance of regular values lies in the fact that by the regular value theorem 2.4.1 their pre- images under f are hypersurfaces in M, that is codimension-1 submanifolds. Consequently, if a is a regular value, then M a is a codimension-0 submanifold of M whose boundary is the hypersurface f −1(a). Main results of the epoch are the following; cf. the beautyful survey [Bot88]. For compact M the first theorem tells that the submanifolds M a vary smoothly as long as a does not cross a critical value. The second theorem asserts that the topological change which occurs when a crosses a simple critical level c, that is f −1(c) contains precisely one critical point p and p is non-degenerate, amounts to attaching a disk whose dimension k is the Morse index of p; see Figure 1.2. Theorem 1.0.1 (Regular interval theorem). Assume f : M ! R is of class C2 and the pre-image f −1[a; b] is compact1 and contains