In this note, we give a proof of the famous theorem of M. Morse dealing with
the cancellation of a pair of non-degenerate critical points of a smooth
function. Our proof consists of a reduction to the one-dimensional case where
the question becomes easy to answer

Laudenbach, Francois

Comptes Rendus Mathematique

English

arXiv

In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question becomes easy to answer

HAL-Univ-Nantes

A proof of Morse’s theorem about the cancellation ofcritical pointsFrancois LaudenbachTo cite this version:Francois Laudenbach. A proof of Morse’s theorem about the cancellation of critical points.Comptes Rendus de l Acade´mie des Sciences - Series I - Mathematics., Paris, 2013, 351, pp.483-488. <hal-00842792>HAL Id: hal-00842792https://hal.archives-ouvertes.fr/hal-00842792Submitted on 9 Jul 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A proof of Morse’s theorem about the cancellation of critical pointsFrançois LaudenbachAbstract. In this note, we give a proof of the famous theorem of M. Morse dealing withthe cancellation of a pair of non-degenerate critical points of a smooth function. Our proofconsists of a reduction to the one-dimensional case where the question becomes easy to answer.1. IntroductionLet us consider an n-dimensional closed manifold M equipped with a Morse function f :M → R. Marston Morse devoted two papers to the question of canceling a pair of criticalpoints. In [6] he considered the possibility of canceling a local minimum (index 0) with acritical point of index 1; and, similarly the cancellation of a pair of indices (n − 1, n); in thatpaper, a polar function means a function with no supernumerary local extrema. In [7] (see also[2]) Morse extended its cancellation criterion to the case of a pair of indices (k, k + 1). It isworth noticing that these two papers show the first time where Morse is considering gradientsglobally in the manifold M , in the spirit of the 1949 Note by René Thom [11].S. Smale generalized Morse’s criterion and got a more algebraic criterion for cancellation.This generalization became an important tool in Smale’s proof of the generalized Poincaréconjecture [10]. A detailed proof is offered by J. Milnor in [5] where the cancellation theoremof Morse occupies one chapter (i.e. Section 5) and is told to be “quite formidable”.Actually, Milnor’s proof deals with the cancellation of two zeroes of a gradient (or pseudo-gradient) allowing the deformation to run among vector fields which a priori are not gradients.The effective deformation of the function remains hidden and the crossing of the codimensionone stratum of functions with one cubic singularity is invisible. Our goal is to make thisdeformation visible; in particular the support of the deformation will be specified, as it is inthe notes by J. Cerf & A. Gramain [1] (actually, we specify a smaller support than in [1] wheresome saturation is not useful). Moreover, we intend to show that the question of cancellationreduces to the one-dimensional case where it is easy to solve. The same technique was alreadyused in [3].The statement we give here is slightly different from the one in [5] which will be derived as acorollary. Moreover, we work with descending pseudo-gradient, called pseudo-gradient for short.According to K. Meyer [4], it is a vector field X satisfying the two following conditions:– Lyapunov inequality: X · f < 0 apart from the critical points of f ;– Non-degeneracy condition: X · f has a non-degenerate maximum at each critical pointof f .Such a pseudo-gradient X may be built by partition of unity. It follows from the definition thatif p is a critical point of f , it is a hyperbolic zero of X and hence, there are stable and unstablemanifolds, respectively denoted by W s(p) and W u(p), which are formed by the points x ∈ Msuch that X t(x) tends to p as t tends to +∞ or −∞; here X t denotes the flow of X. The2dimension of the unstable manifold W u(p) and the codimension of the stable manifold W s(p)both equal the index of the critical point p. It is easily checked that the restriction of f toW u(p) (resp. W s(p)) has a non-degenerate maximum (resp. minimum).Theorem. Let us consider the Morse function f : Mn → R equipped with a pseudo-gradientX. Let p and q be two critical points of f satisfying the following conditions:i) W u(p) and W s(q) intersect transversely and the intersection is made of one orbit ℓ of Xonly.ii) For some ε > 0, each orbit of X in W u(p) distinct from ℓ crosses the level set f−1(f(q)−ε).Then the pair (p, q) is cancelable. More precisely, if U denotes an open neighborhood of theclosure of W u(p) ∩ {f ≥ f(q)− ε}, there is a path of smooth functions (ft)t∈[0,1] , such that:1) f0 = f ;2) for every t ∈ [0, 1], ft coincides with f on M r U ;3) the function ft|U is Morse with two critical points when 0 ≤ t < 1/2; it has a cubic singu-larity when t = 1/2 and it has no critical point when 1/2 < t ≤ 1.By computing the dimensions one has the following formula for the Morse indices: ind(p) =ind(q) + 1. There are other statements by varying assumption ii) which can be derived fromthe above theorem.2. Proof of the theoremSay index(p) = k + 1 = index(q) + 1. We are given an open neighborhood U of the closureof W u(p) ∩ {f(q) ≤ f ≤ f(p)}. After looking at U ∩ {f = f(q)}, we may assume that thefrontier of U in M traces a cylinder C tangent to X in the domain {f(q)− ε′ ≤ f ≤ f(q)+ ε′},where ε′ > 0 satisfies assumption ii) of our theorem. We then choose Morse charts U(p) andU(q) about p and q respectively, both contained in U . They are equipped with coordinates(y˜, z) ∈ Rk+1 × Rn−k−1 (resp. (y, z˜) ∈ Rk × Rn−k) such that f |U(p) = f(p)− |y˜|2 + |z|2 (resp.f |U(q) = f(q) − |y|2 + |z˜|2). Moreover, it is possible to require that (y˜, 0) are coordinates ofW u(p) ∩ U(p) (thus, z must be coordinates in the orthogonal complement to TpWu(p) withrespect to the Hessian of f). Similarly, the coordinates of U(q) may be chosen so that (0, z˜)are coordinates of W s(q) ∩ U(q).For 0 < ε < ε′ small enough there is a Morse model M(q) ⊂ U(q) disjoint from C whose topand bottom are respectively at level f(q) + ε and f(q) − ε; and also, there is a Morse modelM(p) ⊂ U(p) with top and bottom at level f(p) + ε and f(p)− ε.Denote by ℓ˜ the trace of ℓ in the domain {f(q) + ε ≤ f ≤ f(p)− ε}. One end point a of ℓ˜ isW u(p)∩ {f = f(p)− ε}∩W s(q) and the other end point b is W u(p)∩ {f = f(q) + ε}∩W s(q).Consider the diameter ∆a ⊂ U(p) of the ball {|y˜|2 ≤ ε, z = 0} ending at a. Split the coordinates(y˜, z) as (t, y, z) so that t is a coordinate of ∆a andf |U(p) = f(p)− t2 − |y|2 + |z|2.Similarly, split (y, z˜) = (y, t, z) in U(q) so that b ∈ ∆b := {y = 0, z = 0} andf |U(q) = f(q)− |y|2 + t2 + |z|2.It is easy to find a C∞ parametrized arc α : [0, 1]→M , u ∈ [0, 1] 7→ α(u) ∈M , whose imageis denoted by A, and a sequence 0 < u0 < u′0 < u′1 < u1 < 1 with the following properties:3- p = α(u0), a = α(u′0), b = α(u′1), q = α(u1) and α([u′0, u′1]) = ℓ˜;- the image α([0, u′1]) is contained inWu(p) and the image α([u′1, 1]) is contained inWs(q);- the function h defined by h := f ◦α has two critical points, one maximum in p, one localminimum in q (both non-degenerate) and varies from h(0) = f(q)−ε to h(1) = f(q)+ε.One observes that there exists an increasing smooth function h1 without critical points, whichcoincides with h near the extremities of [0, 1], and satisfies(∗) h1 ≤ h .This is the solution of the cancellation problem in the one-dimensional case.We now construct a (k + 1)-dimensional sub-manifold W ⊂ U containing A such that pand q are the only critical points of f |W . This W will contain the part of W u(p) above thelevel f = f(q) + ε that is, W u(p) ∩ {f ≥ f(q) + ε} ∼= Dk+1. Let S be the boundary of thisdisc. Let T denote the top of M(q); we have T ∼= Dk × Sn−k−1 and the product structure ofT is determined by the Morse coordinates. The point b belongs to the so-called belt sphere{0}×Sn−k−1 ⊂ W s(q). The radius of Dk measured with the norm of the y-coordinates is a freechoice for the Morse model M(q). If this radius is small enough, the transversality assumptionimplies that S ∩ T is isotopic in T to the standard meridian Dk × {b0} by an isotopy keepingthe belt sphere fixed. Therefore, by a first level preserving isotopy of the coordinates of M(q)one makes S ∩ T be contained in {z = 0}. By a second isotopy of this type one makes X and∇f coincide near the top of M(q); here ∇f stands for the descending gradient of f in M(q)associated to the Euclidean metric of the coordinates (t, y, z). These two isotopies leave W s(q)invariant and are stationary near q.Let B ⊂ M(q) be defined by B := {z = 0}; it is C∞ tangent to W u(p) along B ∩ S. Bypartition of unity one finds a pseudo-gradient ξ such that:- ξ = X on W u(p) above the level f = f(q)+ ε and on the cylinder C defined previously;- ξ = ∇f in M(q).Let W uξ (p) be the unstable manifold of p for ξ. Each orbit of ξ in Wuξ (p) which does not meetℓ˜ reaches the level set {f = f(q) − ε} and remains in U since C prevents it from getting out.We define W as the union of B and W uξ (p) truncated by removing {f < f(q)− ε}.Lemma 1. There exist a tubular neighborhood N of A in U and coordinates (u, y, z) ∈ [0, 1]×Rk × Rn−k−1 of N , such that:i) A = {y = 0, z = 0};ii) N0 := N ∩W = {z = 0};iii) f(u, y, z) = h(u)− |y|2 + |z|2 at every point of N .Proof. The first two items are easy to realize: take a provisory pair of small tubular neigh-borhoods (N,N0) of A in (M,W ) whose fibres are contained in {t = const} with respect tothe coordinates of M(p) and M(q). Choose the coordinates y in the fibre of N0 over A and zin the fibre of N over N0 such that (y, z) are the so-named coordinates in M(p) and M(q). Forproving item iii), we need to invoke a general fact stated below.4Claim. Let V be a smooth manifold and V ′ ⊂ V be a sub-manifold. Two germs of smooth func-tions f and g along V ′ whose restrictions to V ′ coincide and have no critical points are isotopicrelative to V ′ (meaning that they become equal after moving one of them by an isotopy of V fix-ing V ′); moreover, if f = g near a compact set K ⊂ V ′, the isotopy may be the identity near K.For proving this claim the path method of Moser is available; it may be applied to the patht ∈ [0, 1] 7→ ft := (1 − t)f + tg, that is: there is a time-dependent vector field Zt whose in-tegration solves the problem of conjugating each ft to f0 near V′ smoothly in t ∈ [0, 1]. It issufficient to find Zt satisfying dft(x).Zt(x) = f(x)− g(x) near V′ (see [8]); for that, apply theimplicit function theorem for finding local solutions which are then glued by partition of unity.We are going to apply this claim twice. First, apply it to the data: V = W , V ′ = A,K = A ∩(M(p) ∪M(q)), the restriction f |N0 and g(u, y) = h(u) − |y|2. The isotopy yieldedby the claim moves the coordinates (u, y), without changing u on A, so that the equality of iii)holds true in a neighborhood N1 ⊂ N0 of A in W . Then, apply the claim to the data: V = M ,V ′ = N1, K = N1 ∩(M(p) ∪M(q)), f and g(u, y, z) = h(u) − |y|2 + |z|2. The new isotopyfinishes to put the coordinates in a position which makes the equality of item iii) hold true nearA in M yielding the desired N . In W ∩N , the vector field Y := y∂y is tangent to W and it is a pseudo-gradient for f |(W ∩(N rA)). Still denoted Y , it extends to the whole W as a pseudo-gradient for f |(W rA) (bypartition of unity). Each orbit of Y converge to a point of A in the past. This defines a k-diskfibration W → A which is pinched at α(0) ∈ A. Indeed, this fibration is clear for W ∩N → Aand the orbits of Y yield an ambient isotopy from W to W ∩ N . Denote Du the fibre of Wover α(u).Let W˜ be a tubular neighborhood of W in {f ≥ f(q) − ε}. The fibration W → A extendsto W˜ as a (n− 1)-disk fibration:D˜u →֒ W˜ → A ,pinched at α(0) and coinciding with the projection (u, y, z) → α(u) in N . The functionfu := f |D˜u is a Morse function with α(u) as unique critical point and its index is k. Ithas a pseudo-gradient Y˜u, tangent to D˜u, which coincide with Y on W and with y∂y − z∂z onN .Lemma 2 (Decreasing of a critical value.) Let g : V → R be a Morse function with a pseudo-gradient Z. Let p be a critical point of index k and let a < g(p). Assume that the unstablemanifold W u(p) contains a compact k-disk D whose boundary lies in the level set {g = a}. LetU be a neighborhood of D in {g ≥ a}.Then, for every ε > 0, there exists a one parameter familyof Morse functions (gt)t∈[0,1], such that:- g0 = g, a < g1(p) < a+ ε,- Z is a pseudo-gradient of gt for every t ∈ [0, 1],- and gt = g out of U .5The same statement holds true with parameters.Proof. The point p is a hyperbolic zero of the vector field Z. Then, the orbits of Z close toD but not tangent to D are crossing the level set {g = g(p) + η}, for a small η > 0 and, ofcourse, the level set {g = a}. Consider the foliation F defined by g = const on U rD. Find ann-dimensional compact domain K in U rD made of pieces of Z-orbits from {g = g(p) + η} to{g = a} surrounding D. It is easy to replace F|K by a foliation whose leaves are still transverseto Z in order to have the level sets of the wanted function g1. We refer to [3] for more details(see also [5] Section 4 or [9] Section 2). End of the proof of the theorem. The critical point of fu is pu := α(u) = (u, 0, 0)and its value is h(u). The vector field Y˜u is a pseudo-gradient for fu and the disk Du fulfillsthe requirement of Lemma 2, which we apply smoothly with respect to the parameter u. So,it is possible to decrease the critical value of fu, smoothly in u ∈ [0, 1], from h(u) to h1(u),still keeping Y˜u as a pseudo-gradient. Therefore, the deformation of f over A from h to h1(compare (∗)), extends to W˜ without creating new critical points away from A. Moreover, thedeformation is supported in W˜ , hence in U , as wanted. 3. ApplicationsIn this section, there are given two more classical statements, starting with the one fromMilnor’s book ([5], Section 5).Corollary 1. Let (W,L0, L1) be a compact cobordism and f : (W,L0, L1)→ ([0, 1], 0, 1) be aMorse function with two critical points p and q. Assume there is a pseudo-gradient such thatW u(p) and W s(q) intersect in one orbit and transversely. Then the cobordism is a product:W ∼= M0 × [0, 1].Proof. Certainly 0 < f(q) < f(p) < 1. By compactness, every orbit of the pseudo-gradient inW u(p) distinct from the connecting orbit reaches L0. Hence, the assumptions of our theoremare fulfilled. After canceling the critical points, W is a product by gradient lines. Corollary 2. Let M be a closed manifold, f : M → R be a Morse function, (p, q) be apair of critical points whose respective indices are k + 1 and k. Let X be a pseudo-gradient off . Assume that there are compact disks D(p) ⊂ W u(p) and D(q) ⊂ W s(q) with the followingproperties:- D(p) and D(q) are neighborhoods of p and q respectively in W u(p) and W s(q));- their boundaries lie in a regular level set {f = a}, f(q) < a < f(p) in which theyintersect in one point only and transversely.Then, the pair (p, q) is cancelable.6Proof. Since a is a regular value, the disk D(q) can be extended (keeping its name) so thatits boundary lies in {f = a+ ε}, for some small ε > 0. Lemma 2 can be applied for decreasingthe critical value f(p) so that a < f(p) < a+ ε. Similarly, by considering −f , Lemma 2 can beapplied for increasing the critical value f(q) so that a < f(q) < f(p) < a+ ε. In this situation,as in Corollary 1, the assumptions of our theorem are fulfilled. The cancellation follows. References[1] J. Cerf, A. Gramain, Le théorème du h-cobordisme (Smale), cours Orsay 1966, Secr. math. Éc. NormaleSup., Paris, 1968 (www.maths.ed.ac.uk/ aar/surgery/cerf-gramain.pdf).[2] W. Huebsch, M. Morse, The bowl theorem and a model non-degenerate function, Proc. Nat. Acad. Sc.U.S.A., vol. 51 (1964), 49-51.[3] F. Laudenbach, A proof of Reidemeister-Singer’s theorem by Cerf’s methods, arXiv(math-GT): 1202.1130.[4] K. Meyer, Energy functions for Morse-Smale systems, Amer. J. Math. 90 (1968), 1031-1040.[5] J. Milnor, Lectures on the h-cobordism theorem, Princeton Univ. Press, 1965.[6] M. Morse, The existence of polar non-degenerate functions on differentiable manifolds, Annals of Math. 71(1960), 352-383.[7] M. Morse, Bowls of a non-degenerate function on a compact differentiable manifold, 81-103 in: Differentialand Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, 1965.[8] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294.[9] S. Smale, On gradient dynamical systems, Annals of Math. 74 (1961), 199-206.[10] S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Annals of math. 74 (2) (1961),391-406.[11] R. Thom, Sur une partition en cellules associée à une fonction sur une variété, Comptes Rendus Acad. Sc.Paris, t. 228, (1949), 973-975.Laboratoire de Mathématiques Jean Leray, UMR 6629 du CNRS, Faculté des Sciences etTechniques,Université de Nantes, 2, rue de la Houssinière, F-44322 Nantes cedex 3, France.E-mail address: francois.laudenbach@univ-nantes.fr

A proof of Morse's theorem about the cancellation of critical points

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