The moduli space of gradient flow lines and Morse homology Urs Frauenfelder, Robert Nicholls May 22, 2020 Contents 1 Introduction 4 2 Morse functions and Morse indices 4 2.1 Critical points . .4 2.2 The Hessian . .5 2.3 Invariants of a symmetric form and the Theorem of Sylvester . .5 2.4 Morse critical points and its index . .6 3 Examples of computations of Morse homology 7 3.1 A cooking recipe . .7 3.2 The heart . .8 3.3 The round sphere . 10 3.4 The torus . 10 3.5 Surfaces of higher genus . 11 3.6 The real projective plane . 13 3.7 Noncompact examples . 15 3.8 The Morse inequalities . 17 4 Analysis of gradient flow lines 18 4.1 Parametrised and unparametrised gradient flow lines . 18 4.2 Gradient flow lines flow downhill . 20 4.3 Energy . 20 arXiv:2005.10799v1 [math.AT] 21 May 2020 4.4 Local convergence . 21 4.5 Broken gradient flow lines . 22 4.6 Floer-Gromov convergence . 23 4.7 Floer-Gromov compactness . 25 4.8 Uniqueness of limits . 34 4.9 Floer-Gromov compactness for broken gradient flow lines . 39 4.10 Moduli spaces . 40 4.11 The Floer-Gromov topology . 42 4.12 Strategy to show that the boundary of the boundary vanishes . 43 1 5 Fredholm theory 46 5.1 Fredholm operators . 46 5.2 The linearized gradient flow equation . 48 5.3 The semi-Fredholm estimate . 50 5.4 Adjoints . 55 5.5 Computation of the indices . 57 6 Exponential decay 59 6.1 Exponential convergence to critical points . 59 6.2 Exponential decay in the Euclidean case . 59 6.3 The action-energy inequality . 61 6.4 Proof of exponential decay . 63 7 Transversality 65 7.1 The Hilbert manifold of paths . 65 7.2 Gradient flow lines as the zero set of a section . 66 7.3 The vertical differential . 66 7.4 Generic metrics . 68 7.5 The universal moduli space . 69 7.6 Regular values . 74 7.7 An argument by Taubes . 75 7.8 Morse-Smale metrics . 76 8 Stable and unstable manifolds 78 9 Gluing 83 9.1 The pregluing map . 83 9.2 Infinitesimal gluing . 84 9.3 The local gluing map . 88 9.4 Gluing gradient flow lines . 89 10 Invariance of Morse homology 93 10.1 Morse-Smale pairs . 93 10.2 Construction of the canonical isomorphism . 94 10.3 Floer-Gromov convergence for time dependent gradient flow lines 94 10.4 Generic breaking of time dependent gradient flow lines . 96 10.5 A chain map . 96 10.6 Homotopy of Homotopies . 97 10.7 Functoriality . 99 10.8 The trivial homotopy . 100 10.9 Proof that the canonical maps are isomorphisms . 101 11 Spectral numbers 101 11.1 Spectral numbers for Morse functions . 102 11.2 Lipschitz continuity . 102 11.3 Denseness of Morse functions . 106 2 11.4 Extension of spectral numbers to smooth functions . 108 11.5 Lipschitz continuity of the extension . 109 11.6 Spectrality . 110 11.7 The homological spectral bundle and the action gap . 112 3 1 Introduction Given a Morse function on a closed manifold one constructs a chain complex generated by the critical points of the Morse function and graded by their Morse index. The boundary operator is obtained by counting gradient flow lines be- tween critical points. A priori it is far from obvious that with this recipe one obtains actually a boundary operator, i.e., that the boundary of the boundary vanishes. That this is actually the case rests on a careful analysis of the moduli space of gradient flow lines. While the complex depends on the choice of the Morse function the resulting homology is independent of it. Critical points can be born and die but homology never dies nor is born. In particular, one has a stable lower bound on the number of critical points. An immediate consequence of this are the Morse inequalities. For people just interested in the Morse inequalities on a closed manifold the construction of Morse homology which involves quite heavy tools from functional analysis might be a bit too much and the wonderful book by Milnor [11] gives a quicker way to the Morse inequalities. On the other hand the amazing work of Floer [5, 6, 7] shows that there is a semi-infinite dimensional analogon of Morse homology referred to as Floer homology. For example periodic orbits of a Hamiltonian system can be obtained variationally as critical points of the action functional of classical mechanics and Floer showed how they can be used as generators of a chain complex. The Morse inequalities for Floer's chain complex than lead to topological lower bounds on the number of periodic orbits conjectured before by Arnold. Although we do not discuss Floer homology in these notes they are written having Floer homology in mind. Therefore we use concepts which can be gener- alized to the semi-infinite dimensional case. In the finite dimensional case they might not always be the most efficient ones but on the other hand the finite dimensional case has the advantage that these concepts can be visualized much more easily than in the semi-infinite dimensional one giving the reader the right intuition how Floer homology works. These notes grew out of lectures the first named author gave at Ludwig- Maximilian University in Munich, Seoul National University, and the University of Augsburg. He would like to thank all the participants of his lectures which gave him all a lot of inspiration. Of course there are many excellent textbooks and articles on these topics from which these notes profited a lot, like [3, 4, 13, 17, 18, 22]. 2 Morse functions and Morse indices 2.1 Critical points Assume that M is a manifold and f : M R is a smooth function. A critical point of the function f is a point u M !such that the differential of f at this point vanishes, i.e., df(u) = 0. Alternatively2 suppose that we are in a local 4 coordinate chart around u, then we can rephrase this condition that all partial derivatives at u vanish @f (u) = 0; 1 i n @ui ≤ ≤ where n is the dimension of the manifold. Note that the concept of a critical point does not depend on the choice of coordinates. Indeed, if u = u(v) is a coordinate change we obtain by the chain rule @f @f @u (v) = (u) i (v) = 0; 1 j n: @vj @ui @vj ≤ ≤ 2.2 The Hessian The Hessian at a point u in a coordinate chart is the symmetric n n-matrix given by × @2f Hu = (u) : @ui@uj 1 i;j n ≤ ≤ Let us see how the Hessian transforms under coordinate change. By the chain rule we obtain @2f @2f @u @u @f @2u (v) = (u) k (v) ` (v) + (u) k (v): @vi@vj @uk@u` @vj @vi @uk @vj@vi This looks a bit scary. However, note that if u is a critical point, the last term just vanishes so that the above formula simplifies to @2f @2f @u @u (v) = (u) k (v) ` (v): @vi@vj @uk@u` @vj @vi If we abbreviate by @u Φ = j (v) @vi 1 i;j n ≤ ≤ the Jacobian of the coordinate change we obtain the following transformation behaviour for the Hessian at a critical point T Hu = Φ HvΦ where ΦT denotes the transpose of the Jacobian. That means that the Hessian transforms as a symmetric form under coordinate change. 2.3 Invariants of a symmetric form and the Theorem of Sylvester We recall a classical theorem due to Sylvester [21] about invariants of a quadratic form. Suppose that H is a n n-matrix. We can associate to H the following × 5 three nonnegative integers n0 := dim ker H; n o n := max dimV : V Rn subvectorspace; vT Hv > 0; v V 0 ; + ⊂ 8 2 n f g n o n := max dimV : V Rn subvectorspace; vT Hv < 0; v V 0 : − ⊂ 8 2 n f g If H is symmetric, i.e., H = HT we can find an orthogonal matrix Φ which diagonalizes H, i.e., 1 Φ− HΦ = D where D is diagonal. Because Φ is orthogonal we can write this equation equiv- alently as ΦT HΦ = D: By scaling we can arrange that all entries of the diagonal matrix D are either 1, 1 or 0. In this last step we usually loose the property that Φ is orthogonal, however,− Φ is still invertible. In view of this we obtain the following relation between the three invariants n + n0 + n+ = n: (1) − Moreover, we see that there are no other invariants of a quadratic form. 2.4 Morse critical points and its index Suppose that p M is a critical point of a smooth function f : M R and let 2 ! Hf (p) be the Hessian of f at p. We say that the critical point is Morse if ker H (p) = 0 f f g or equivalently if n0(Hf (p)) = 0: A smooth function is called Morse, if all its critical points are Morse. Suppose that p is a Morse critical point of f. Then we define its Morse index µ(p) := n Hf (p) − namely the number of negative eigenvalues of the Hessian counted with multi- plicity. If n is the dimension of the manifold M, then we infer from (1) that n H (p) = n µ(p): + f − In fact apart from the value the function f attains at a Morse critical point, its Morse index is the only local invariant of the function as the following Lemma tells 6 µ = 2 µ = 2 µ = 1 f µ = 0 Figure 1: Distorted sphere with Morse critical points and Morse indices.