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 function 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, algebraic geometry. (Added: There are also interesting and powerful results in cohomology 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 neighborhood 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 nondegenerate. 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 nondegenerate 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 homeomorphic 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 ”homeomorphic” or ”diffeomorphic”. This result is a based on a result about the canonical compactifications of the space of trajectories and of the stable and unstable sets of a ”gradient like” vector field with considerable implications and applications.

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 numbers 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 manifold 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 trajectory is an embedding γ : R → M. If M is closed then a maximal 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.

A Morse-Smale pair is a pair (f, g) consisting of a Morse function f and a good Riemannian metric g with respect to f. Theorems 2 and 3 imply that most of the pairs are Morse- Smale pairs.

7cf S.Smale, Annals of Math. 11 Gradient like vector ﬁeld. Instead of (f, g) consiting of a Morse function f and a good Riemannian metric g it is convenient, and in fact equivalent, to consider pairs (f, X), f a Morse function, X a vector ﬁeld which satisfy: i : X(f)(x) < 0 iff x not a critical point of f. ii : For any critical point x there exist a chart ϕ : U → Rn so that

k n −1 X 2 X 2 f · ϕ =c − 1/2 xi + 1/2 xi i=1 i=k+1 k n X X X = xi∂/∂xi − xi∂/∂xi. i=1 i=k+1

For such pair the vector field X is called f − gradientlike, while f is called Lyapunov function for X. If f is a Morse function and g a Riemannian metric which satisfies (ii) then −gradgf is a vector field which satisfies (i) and (ii).

12 We have :

Proposition 2. If f is a Morse function and X a gradient like vector ﬁeld there exist a riemannian metric g so thatX = −gradgf. Moreover one can choose g to essentially agree with a given metric 0 g which satisﬁes X = −gradg0f on U subset of M

The same ”goodness condition= Morse-Smale property” can be formulated for f− gradient like vector fields and one also refer to the pair (f, X) with f a Morse function and X an −f gradient like vector field which is Morse-Smale as a Morse- Smale pair. The cell complex stated in Theorem 1 is built up of the unstable sets of this vector field −gradgf or X and all cells are diffeomorphic to the Euclidean space and their closure are images by smooth maps of compact manifolds with corners 8. Change in the metric induces change in the cell structure (however not in the number of cells).

8cf D.Burghelea, Lectures on Witten Helffer Sjostrand¨ Theory 13 The second question is:

Q.2 How the changes of the Morse -Smale pair reﬂects in the changes of the associated cell structure? Can this be used to the beneﬁt of topology? A few easy observations were already made by M. Morse:

OB1: One can insert a pair of two critical points of index q and q + 1 by modifying the function only in an arbitrary ”small” open set.

OB2: One can eliminate a pair of critical points x of index q and y of index q−1 provided there exists only one trajectory from x to y. One can do this by changing the function in a given neighborhood of the trajectory.

OB3: Small changes on the critical values of f can be achieved by keeping the rest points , the metric g and the instantons un- changed. However there are topological constraints for a change of the critical values ( if the rest points and the instantons of −gradgf are supposed to be kept ﬁxed).

14 These simple observations sufﬁce for the proof of the clas- siﬁcation theorem for compact 2-dimensional manifolds.

• AP5 (B. Riemann,··· T.Rado) Two orientable (nonorientable) two dimensional manifolds are diffeomorphic iff their Euler Poincare characteristic are equal.

A considerable progress was achieved by Smale in early 60’s:

He has shown that for a ﬁxed Morse function f, by changing the good Riemannian metric, almost any other conﬁguration of instantons between the critical points of f can be realised provided that no homotopical obstruction prevents such change.

Combined with thee above obserrvations this was the be- gining of surgery theory. 9

As a consequence Smale proved the famous Poincare´ conjecture in dimension lager or equal to 5.

9partially explained in the book of J.Milnor, Lectures on h-cobordism theorem, Princeton Univ. press. 15 • AP6 (S.Smale)On any compact manifold M n, n ≥ 5 of the homotopy type of a sphere, starting with an arbitrary Morse pair (f, g) , one can produce a Morse pair (f 0, g0) whose function f 0 has only two nondegenerate critical points, a maximum and a minimum. hence M n is homeomorphic to the sphere Sn.

and then so called h−Cobordism theorem, the key tool in differential topology, which led to the classiﬁcation up to diffeomorphism of all compact (one connected) manifolds.

• AP6’(S.Smale)If W n+1 is a smooth one connected compact manifold with boundary whose boundary ∂W is the disjoint union n n n of two compact manifolds M1 and M2 and the inclusions Mi → n W are homotopy equivalences, then W is diffeomorphic to M1 × [0, 1]

The above theorem (freed from the one connectivity hypoth- esis) is the basic tool for the classiﬁcation of all manifolds, the central problem of differential topology.

16 Generalizations and extensions.

i)- replace manifolds by manifolds with boundaries and manifolds with corners. (routine but particularly important)

ii)- replace manifolds ... by stratiﬁed spaces (also relatively routine but with many applications in algebraic geometry (topology of algebraic varieties over C, cf Goresky MacPherson)

iii)- replace function f whose all critical points are nondegenerate by function f whose critical locus is a disjoint union of manifolds with hessian nondegenerate in normal directions. (Morse -Bott functions). As already noticed by Bott it has considerably many applications in topology.

iv)- replace smooth function by closed one form= action (Novikov). This is a very active research area with potentially many applications.

v)- replace ﬁnite dimensional manifolds with inﬁnite dimensional manifolds.

17 A few Notes and warnings. Most of the smooth functions on a given manifod are Morse A Morse function and some extra data provides a partition of the manifold in cells This explains why Morse theory is instrumental in topology. There is a wrong perception that the Morse function provides the partition which is wrong In fact not the function but a vector field for which the function is Lyapunov provides the partition. Different vector fields having the same Morse function as a Lyapunov function provide different partitions. A less known theorem claims that the cochain complexes associated to different such vector fields are ((not canonically) isomorphic (however not the same). There is also the impression that a function whose all critical points are nondegenerted is good for describing the topology. This is the case when the manifold is compact. When this is not this is far from the truth. Some compacity property is necessary like ”properness” in finite dimesion or property ”C”.

18 Problems (list 1) . 1, 2. Show that there are no smooth function with less than three nondegenerate critical points on RP 2. Construct a smooth function with exactly three non degenerate critical points on RP 2. Specify their indices. 3. Show that any smooth function with non degenerate critical points on T 2 has at least 4 critical points. Can you produce smooth functions with 3, resp. 2, resp. 1 critical points? 4. Improve Morse inequalities to involve the torsion numbers of the homology with integral coefﬁcients. 5. Show the equivalence of ”multi valued functions” and ”closed differential 1− forms”.