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MORSE THEORY DAN BURGHELEA Department of Mathematics The MORSE THEORY DAN BURGHELEA Department of Mathematics The Ohio State University 1 • Morse Theory begins with the modest task of understand- ing the local maxima, minima and sadle points of a smooth function in relation with the topology of the ”space” the func- tion is defined on. 1 • It led to (or it was crucial in the proof of) some of the deepest mathematical results in: - differential, algebraic and infinite dimensional topology, - dynamics, - Riemannian and symplectic geometry, gauge theory, alge- braic geometry. (Added: There are also interesting and powerful results in coho- mology of groups (geometric group theory) due to Brady- Bestvina proven using ideas of Morse theory) • New aspects of the theory raise questions attractive by the simplicity of their formulation and promising for application. 1it works like the most obvious method to understand the shape of a curve in plane: intersect the curve with lines parallel to some direction and watch the change in cardinality of this intersection. 2 1. LECTURE 1 (ELEMENTARY MORSE THEORY) (Morse theory is one of the most powerful illustrations how mathematics works as a science. ) 1. It begins with very simple observations: O.1) (Morse lemma) Let f : U → R, U ⊂ Rn an open neigh- borhood of 0, with 0 a nondegenerate critical point. One can find a change of coordinates (diffeomorphism) ϕ :(U 0.0) → (U”, 0), U 0,U” ⊂ U open subsets of Rn so that k n X 2 X 2 f · ϕ(x1, ··· , xn) = c − xi + xi . i=1 i=k+1 O.2) (Morse observation) S2 has a smooth function with only two nondegenerate critical points of index 0 and 2 (the Betti num- 2 2 bers of S are β0 = 1, β1 = 0, β2 = 1) while on T this is not pos- sible. The best one can hope for is at least two more critical points 2 of index 1. (the Betti numbers of T are β0 = 1, β1 = 2, β2 = 1). 3 2. The observations raise an interesting question: Q 1 Is any meaningful relationship between the topology of the space the function is defined on and the critical points, at least in the case all the critical points are nondegenerate 2 ? 3. The question is answered by: Theorem 1. Let M be a compact smooth manifold. If f : M → R is a smooth function with all critical points nondegenerate (Morse function) then M is the underlying space of a cell complex whose k−cells are in bijective correspondence with the critical points of index k The following result insures that most of the functions are Morse functions. Theorem 2. The set of smooth functions whose critical points are nondegenerate form an open and dense set in the C2 topology 3. 2Liusternic-Schnirelmann theory was invented to relate the topology of a space with the number of critical points not necessary nondegenerate ( of a smooth function on a finite or infinite dimensional manifold). It led to a numerical invariant in homotopy theory, called ”Liusternick- Schnirelmann category”. Important contributions to this homotopy invariant and its generalizations were provided by the Romanian topologists: T. Ganea, I. Berstein, O. Cornea and (of romanian origin) J.Oprea 3the proof use basic results in Calculus and Linear Algebra 4 EXAMPLES: Let M n be a smooth submanifold of RN : Proposition 1. a) For almost any x ∈ RM the distance function from x restricted to M has all critical points nondegenerate. b) For almost all directions v in RN the orthogonal projection on v restricted to M has all critical points nondegenerate. 4. The above results have some immediate consequences and applications. Theorem 3. (Morse inequalities) Let M be a compact manifold and f : M → R be a smooth function with all critical points non- degenerate. Denote by Ci the cardinality of the set of critical points of index i and by βi the i − th Betti number. Then: 1) Ci ≥ βi , k k k X i k X i 2) (−1) (−1) Ci ≥ (−1) (−1) βi, in particular i=1 i=1 n X 3) χ(M) = (−1)Ci. i=1 5 Let M be a compact manifold. • AP 1: (dynamics) For any smooth vector field X with nonde- generate zeros (rest points) the number of rest points of index +1 minus the number of rest points of index −1 is exactly χ(M). More- over if (M, ω) is symplectic and X hamiltonian 4 then the rest points have a Morse index and the inequalities (1) and (2) in Theorem 1 hold with Ci denoting the number of rest points of Morse index i . • AP 2: (complex geometry) A complex analytic manifold of complex dimension k which can be analytically embedded as a closed subset of CN (in particular and algebraic variety given by a collection of polynomials in N complex variable which is a smooth submanifold of real dimension 2k ) has the homotopy type of a k−dimensional cell complex. • AP 3: (celestial mechanics)The number of relative equilibria which are colinear in the n− body plane problem is exactly n!/2. 5 4if noot hamitlonian the Betti numbers should be replaced by the dimension of the cohomology vector spaces Hi(M, ξ) whrre ξ is the cohomology class rrepresented by thhe closed form ”symplecttic grradient of X. 5This result was established by J.Multon more that 120 years ago in a memoir of more than 150 pages. A Morse theoretic proof was provided by S.Smale in ”Topology and Mechanics II”, Inventiones Mathematicae, 11, pp 45-64, 1970 6 • AP 4(topology): If M is a compact smooth manifold which admits a function f with only two critical points, both of which are nondegenerate, then M is homeomorphic to a sphere 6 . (This remark was first made by G. Reeb and it was used by J. Milnor to show that his examples of manifolds nondiffeo- morphic to the standard spheres were actually homeomor- phic to the standard sphere.) About the proof of Theorem 1: - Easy, if the statement ”M is the underlying space of a cell complex” means ”M is homotopy equivalent to the underlying space of a cell complex”. - More difficult if ”homotopy equivalent” is replaced by ”home- omorphic” or ”diffeomorphic”. This result is a based on a re- sult about the canonical compactifications of the space of tra- jectories and of the stable and unstable sets of a ”gradient like” vector field with considerable implications and applica- tions. 6A famous theorem due to S.Smale, the truth of Poincare conjecture in dimension larger than 4 was proven by showing that on a closed manifold homoltopy equivalent to a shere Sn, n ≥ 5, one can produce a Morse function with onlly two critical points 7 About Morse inequalitis: 1. There are stronger inequalities, involving the torsion num- bers of the homology groups with integral coefficients. 2. An interesting generalisation is due to Novikov and it extends the Morse inequalities and in fact most of the Morse theory, to ”multivalued functions” or closed one form. A multivalued function= closed one form 1 (ω ∈ Ω (M), dω = 0), is an equivalece class of systems (Uα, f : Uα → R) with {Uα} an open covering of the underlying mani- fold M, fα smooth functions such that i : fα|Uα∩Uβ − fβ|Uα∩Uβ is constant. Two systhems (Uα, fα : Uα → R) and (Uβ, fβ : Uβ → R) which satisfy (i) are are equivalent if (Uα,Uβ, fα : Uα → R), fβ : Uβ → R satisfy (i). The inequalities remain the same the meanig of the Betti numbers changes appropriately. 8 References: 1. J. Milnor, Morse Theory, Princeton Univ. Press, 2. J. Milnor, Lecture on H-cobordism, Princeton Univ. Press, 3. D. Burghelea, Th Hangan, H.Mooscovici, A. Verona, In- troducere in Topologia diferentiala a. J.Milnor, On manifolds homeomorphic th the 7-sphere, Ann, Math, 1965. b. D.Burghelea, N.Kuiper , Hilbert manifolds, Annals of Math, 1969 9 2. LECTURE 2 (ELEMENTARY MORSE THEORY AND TOPOLOGY) Morse Smale pairs: • Contrary to a superficial impression Morse theory is not about a Morse function only, but about a Morse-Smale pair which is, primarily, a pair (f, g) consisting of a Morse function f and a Riemannian metric g, or equivalent data. • A Riemannian metric g and the Morse function f give rise to a vector field X := −gradgf . For such vector field: 1. The rest points are exactly the critical points of f, 2. The function f is strictly decreasing on trajectories, there- fore there are no closed trajectories, and any maximal trajec- tory is an embedding γ : R → M. If M is closed then a max- imal trajectory goes from one rest points x to an other y, i.e. limt→±∞ γ(t) = x/y. When isolated such trajectory is called INSTANTON. 3.Each rest point x ∈ M has a stable and unstable set which is a smooth submanifold diffeomorphic to the Euclidean space Rn−indx, resp.Rindx. 10 • To develope the Morse theory one has to suppose that the metric g is good with respect to the Morse function f which, essentially, means that the set of trajectories between the rest points x and y is a smooth manifold of dimension ind(x) − ind(y). This is not a serious restriction. One can show that 7 Theorem 4. Let f : M → R be a smooth function with all critical points nondegenerate and g a Riemannian metric on M. Given a neighborhood of the critical points of f one can modify the metric g into g0 by an arbitrary small C1− perturbation, with g equal to g0 outside a neighborhood U of the critical points and g0 good for f.
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