110 V Morse FunctionsV.2 Transversality ConditionGiven a Morse function, we can follow the gradient flow and decompose themanifold depending on where the flow originates and where it ends. To recoverthe homology of the manifold from this decomposition, we require that thefunction satisfies an additional genericity assumption.Integral lines. Recall the 1-parameter group of diffeomorphisms ϕ : R ×M → M defined by a Morse function f on a manifold M with a Riemannianmetric. The integral line that passes through a regular point x ∈ M is γ =γx: R → M defined by γ(t) = ϕ(t, x); see Figure V.5. It is the solution tovuwzFigure V.5: The upright torus with the four integral lines that end at the two saddles.the ordinary differential equation defined by ˙γ(t) = ∇f (γ(t)) and the initialcondition γ(0) = x. Because ϕ and therefore γ are defined for all t ∈ R, theintegral line necessarily approaches a critical point, both for t going to plus andto minus infinity. We call these critical points the origin and the destinationof the integral line,org(γ) = limt→−∞γ(t);dest(γ) = limt→∞γ(t).The function increases along the integral line which implies that org(γ) 6=dest(γ). The Existence and Uniqueness Theorems of ordinary differential equa-tions imply that the integral line that passes through another regular point yis either disjoint from or the same as the one passing through x, im γx= im γyV.2 Transversality Condition 111or im γx∩ im γy= ∅. This property suggests we decompose the manifold intointegral lines or unions of integral lines with shared characteristics.Stable and unstable manifolds. The stable manifold of a critical point uof f is the point itself together with all regular points whose integral lines endat u. Symmetrically, the unstable manifold of u is the point itself together withall regular points whose integral lines originate at u. More formally,S(u) = {u} ∪ {x ∈ M | dest(γx) = u};U(u) = {u} ∪ {y ∈ M | org(γy) = u}.The function increases along integral lines. It follows that f (u) ≥ f (x) for allpoints x in the stable manifold of u. This is the reason why S(u) is sometimesreferred to as the descending manifold of u. Symmetrically, f(u) ≤ f (y) for allpoints y in the unstable manifold of u and U(u) is sometimes referred to as theascending manifold of u.Suppose the dimension of M is d and the index of the critical point u is p.Then there is a (p − 1)-sphere of directions along which integral lines approachu. It can be proved that together with u these integral lines form an open ballof dimension p and that S(u) is a submanifold homeomorphic to Rpthat isimmersed in M. It is not embedded because distant points in Rpmay map toarbitrarily close points in M, as we can see in Figure V.5. For example, thesaddle v has a stable 1-manifold consisting of two integral lines that merge atv to form one open, connected interval. The two ends of the interval approachthe minimum, u, which does not belong to the 1-manifold. While the map fromR1to M is continuous its inverse is not.Morse-Smale functions. The stable manifolds do not necessarily form acomplex. Specifically, it is possible that the boundary of a stable manifold isnot the union of other stable manifolds of lower dimension. Take for examplethe upright torus in Figure V.5. The stable 1-manifold of the upper saddle, w,reaches down to the lower saddle, v, but the latter is not a stable 0-manifold.The reason for this deficiency is a degeneracy in the gradient flow. In particular,we have an integral line that originates at a saddle and ends at another saddle.Equivalently, the integral line belongs to the stable 1-manifold of w and to theunstable 1-manifold of v. Generically, such integral lines do not exist.Definition. A Morse function f : M → R is a Morse-Smale function ifthe stable and unstable manifolds defined by the critical points of f intersecttransversally.112 V Morse FunctionsRoughly, this requires that the stable and unstable manifolds cross when theyintersect. More formally, let σ : Rp→ M and υ : Rq→ M be two immersions.Letting z ∈ M be a point in their common image we say that σ and υ intersecttransversally at z if the derived images of the tangent spaces at preimagesx ∈ σ−1(z) and y ∈ υ−1(z) span the entire tangent space of M at z,Dσx(TRpx) + Dυy(TRqy) = TMz.We say that σ and υ are transversal to each other if they intersect transversallyat every point z in their common image.Complexes. Assuming transversality, the intersection of a stable p-manifoldand an unstable q-manifold has dimension p+q−d. Furthermore, the boundaryof every stable manifold is a union of stable manifolds of lower dimension. Theset of stable manifolds thus forms a complex which we construct one dimensionat a time.0-skeleton: add all minima as stable 0-manifolds to initialize the complex;1-skeleton: add all stable 1-manifolds, each an open interval glued at its end-points to two points in the 0-skeleton;2-skeleton: add all stable 2-manifolds, each an open disk glued along itsboundary circle to a cycle in the 1-skeleton;etc. It is possible that the two minima are the same so that the interval whoseends are both glued to it forms a loop. Similarly, the cycle in the 1-skeleton canFigure V.6: All integral lines of the height function of S2originate at the minimumand end at the maximum. We therefore have two stable manifolds, a vertex for theminimum and an open disk for the maximum.be degenerate, such as pinched or even just a single point. Similar situationsV.2 Transversality Condition 113are possible for higher-dimensional stable manifolds. An example is the heightfunction of the d-sphere. It has a single minimum, a single maximum, andno other critical points. The minimum has index 0 and forms a vertex in thecomplex. The maximum has index d and defines a stable d-manifold. It wrapsaround the sphere and its boundary is glued to a single point, the minimum,as illustrated for d = 2 in Figure V.6.Morse inequalities. If we take the alternating sum of the stable manifoldsin the above example we get 1 + (−1)d, which is the Euler characteristic of thed-sphere. This is not a coincidence. More generally, the alternating sum ofstable manifolds gives the Euler characteristic, and this equation is part of thecollection of strong Morse inequalities. We state both, the weak and the strongMorse inequalities, writing cpfor the number of critical points of index p.Morse Inequalities. Let M be
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