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6. Morse Theory and Floer Homology

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6. Morse Theory and Floer Homology 6. Morse Theory and Floer Homology 6.1 Preliminaries: Aims of Morse Theory Let X be a complete Riemannian manifold, not necessarily of finite dimension 1 .Weshallconsiderasmoothfunctionf on X,i.e.f C∞(X, R)(actually ∈ f C3(X, R)usuallysuffices).TheessentialfeatureofthetheoryofMorse and∈ its generalizations is the relationship between the structure of the critical set of f, C(f):= x X : df (x)=0 { ∈ } (and the space of trajectories for the gradient flow of f) and the topology of X. While some such relations can already be deduced for continuous, not nec- essarily smooth functions, certain deeper structures and more complete re- sults only emerge if additional conditions are imposed onto f besides smooth- ness. Morse theory already yields very interesting results for functions on finite dimensional, compact Riemannian manifolds. However, it also applies in many infinite dimensional situations. For example, it can be used to show the existence of closed geodesics on compact Riemannian manifolds M by applying it to the energy functional on the space X of curves of Sobolev class H1,2 in M,asweshallseein 6.11 below. Let us first informally discuss§ the main features and concepts of the theory at some simple example. We consider a compact Riemannian manifold X diffeomorphic to the 2-sphere S2,andwestudysmoothfunctionsonX; more specifically let us look at two functions f1,f2 whose level set graphs are exhibited in the following figure, 1 In this textbook, we do not systematically discuss infinite dimensional Riemannian mani- folds. The essential point is that they are modeled on Hilbert instead of Euclidean spaces. At certain places, the constructions require a little more care than in the finite dimensional case, because compactness arguments are no longer available. 294 6. Morse Theory and Floer Homology f1 p1 f2 p1 p2 p3 p2 p4 Fig. 6.1.1. with the vertical axis describing the value of the functions. The idea of Morse theory is to extract information about the global topology of X from the critical points of f,i.e.thosep X with ∈ df (p)=0. Clearly, their number is not invariant; for f1,wehavetwocriticalpoints, for f2, four, as indicated in the figure. In order to describe the local ge- ometry of the function more closely in the vicinity of a critical point, we assign a so-called Morse index µ(p)toeachcriticalpointp as the number of linearly independent directions on which the second derivative d2f(p)is negative definite (this requires the assumption that that second derivative is nondegenerate, i.e. does not have the eigenvalue 0, at all critical points; if this assumption is satisfied we speak of a Morse function). Equivalently, this is the dimension of the unstable manifold W u(p). That unstable manifold is defined as follows: We look at the negative gradient flow of f,i.e.weconsider the solutions of x : R M → x˙(t)= grad f(x(t)) for all t R. − ∈ It is at this point that the Riemannian metric of X enters, namely by defining the gradient of f as the vector field dual to the 1-form df .Theflowlines x(t)arecurvesofsteepestdescentforf.Fort ,eachflowlinex(t) converges to some critical points p = x( ),q =→x( ±∞)off, recalling that in our examples we are working on a compact−∞ manifold.∞ The unstable manifold W u(p)ofacriticalpointp then simply consists of all flow lines x(t) with x( )=p, i.e. of those flow lines that emanate from p. −∞ 6.1 Preliminaries: Aims of Morse Theory 295 f1 p1 f2 p1 p2 p3 p2 p4 Fig. 6.1.2. In our examples, we have for the Morse indices of the critical points of f1 µf1 (p1)=2,µf1 (p2)=0, and for f2 µf2 (p1)=2,µf2 (p2)=2,µf2 (p3)=1,µf2 (p4)=0, as f1 has a maximum point p1 and a minimum p2 as its only critical points whereas f2 has two local maxima p1,p2,asaddlepointp3,andaminimump4. As we see from the examples, the unstable manifold W u(p)istopologicallya cell (i.e. homeomorphic to an open ball) of dimension µ(p), and the manifold X is the union of the unstable manifolds of the critical points of the function. Thus, we get a decomposition of X into cells. In order to see the local effects of critical points, we can intersect W u(p)withasmallballaroundp and contract the boundary of that intersection to a point. We then obtain a f1 p1 f2 p1 p2 p3 pp2 4 Fig. 6.1.3. pointed sphere (Sµ(p),pt.)ofdimensionµ(p). These local constructions al- ready yield an important topological invariant, namely the Euler character- istic χ(X), as the alternating sum of these dimensions, 296 6. Morse Theory and Floer Homology χ(X)= ( 1)µ(p)µ(p). − p crit.! pt. of f We are introducing the signs ( 1)µ(p) here in order to get some cancellations between the contributions from− the individual critical points. This issue is handled in more generality by the introduction of the boundary operator ∂. From the point of view explored by Floer, we consider pairs (p, q) of critical points with µ(q)=µ(p) 1, i.e. of index difference 1. We then count the num- ber of trajectories from−p to q modulo 2 (or, more generally, with associated signs as will be discussed later in this chapter): ∂p = (# flow lines from p to q mod 2)q. q crit. pt of f { } µ(q)=µ(p) 1 ! − In this way, we get an operator from C (f,Z2), the vector space over Z2 generated by the critical points of f,toitself.Theimportantpointthenis∗ to show that ∂ ∂ =0. ◦ On this basis, one can define the homology groups Hk(X, f, Z2):=kernelof ∂ on Ck(f,Z2)/image of ∂ from Ck+1(f,Z2), where Ck(f,Z2) is generated by the critical points of Morse index k.(Because of the relation ∂ ∂ =0,theimageof∂ from Ck+1(f,Z2)isalwayscontained ◦ in the kernel of ∂ on Ck(f,Z2).) We return to our examples: In the figure, we now only indicate flow lines between critical points of index difference 1. f11p f 2pp12 p3 p2 p4 Fig. 6.1.4. For f1,therearenopairsofcriticalpointsofindexdifference1atall.Denoting the restriction of ∂ to Ck(f,Z2)by∂k,wethenhave 6.1 Preliminaries: Aims of Morse Theory 297 ker ∂ = p 2 { 1} ker ∂ = p , 0 { 0} while ∂1 is the trivial operator as C1(f1, Z2)is0.Allimagesarelikewise trivial, and so H2(X, f1, Z2)=Z2 H1(X, f1, Z2)=0 H0(X, f1, Z2)=Z2 Putting bk := dimZ2 Hk(X, f, Z2)(Bettinumbers), in particular we recover the Euler characteristic as χ(X)= ( 1)j b . − j j ! Let us now look at f2.Herewehave ∂2p1 = ∂2p2 = p3 , hence ∂2(p1 + p2)=2p3 =0 ∂1p3 =2p4 = 0 (since we are computing mod 2) ∂0p4 =0 Thus H2(X, f2, Z2)=ker∂2 = Z2 H1(X, f2, Z2)=ker∂1/image∂2 =0 H0(X, f2, Z2)=ker∂0/image∂1 = Z2. Thus, the homology groups, and therefore also the Betti numbers are the same for either function. This is the basic fact of Morse theory, and we also see that this equality arises from cancellations between critical points achieved by the boundary operator. This will be made more rigorous in 6.3 - 6.10. §§ As already mentioned, there is one other aspect to Morse theory, namely that it is not restricted to finite dimensional manifolds. While some of the considerations in this Chapter will apply in a general setting, here we can only present an application that does not need elaborate features of Morse theory but only an existence result for unstable critical points in an infinite dimensional setting. This will be prepared in 6.2 and carried out in 6.11. § § 298 6. Morse Theory and Floer Homology 6.2 Compactness: The Palais-Smale Condition and the Existence of Saddle Points On a compact manifold, any continuous function assumes its minimum. It may have more than one local minimum, however. If a differentiable func- tion on a compact manifold has two local minima, then it also has another critical point which is not a strict local minimum. These rather elementary results, however, in general cease to hold on noncompact spaces, for example infinite dimensional ones. The attempt to isolate conditions that permit an extension of these results to general, not necessarily compact situations is the starting point of the modern calculus of variations. For the existence of a minimum, one usually imposes certain generalized convexity conditions while for the existence of other critical points, one needs the so-called Palais-Smale condition. Definition 6.2.1 f C1(X, R) satisfies condition (PS) if every sequence ∈ (xn)n with ∈N (i) f(x ) bounded | n | (ii) df (x ) 0forn ∥ n ∥→ →∞ contains a convergent subsequence. Obviously, (PS) is automatically satisfied if X is compact. It is also sat- isfied if f is proper, i.e. if for every c R ∈ x X : f(x) c { ∈ | |≤ } is compact. However, (PS) is more general than that and we shall see in the sequel (see 6.11 below) that it holds for example for the energy functional on the space§ of closed curves of Sobolev class H1,2 on a compact Riemannian manifold M. For the sake of illustration, we shall now demonstrate the following result: Proposition 6.2.1 Suppose f C1(X, R) satisfies (PS) and has two strict relative minima x ,x X. Then∈ there exists another critical point x of f 1 2 ∈ 3 (i.e. df (x3)=0)with f(x3)=κ := inf max f(x) > max f(x1),f(x2) (6.2.1) γ Γ x γ { } ∈ ∈ with Γ := γ C0([0, 1],X):γ(0) = x ,γ(1) = x , the set of all paths { ∈ 1 2} connecting x1 and x2.
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