N Unknown. More options. Find it at other libraries via WorldCat Limited preview. Bibliography Includes bibliographical references p. Contents Preface. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse theory over the reals. The second part consists of applications of Morse theory over the reals, while the last part describes the basics and some applications of complex Morse theory, a.
Picard-Lefschetz theory. This is the first textbook to include topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The exposition is enhanced with examples, problems, and illustrations, and will be of interest to graduate students as well as researchers. The reader is expected to have some familiarity with cohomology theory and with the differential and integral calculus on smooth manifolds.
Witten deformation and polynomial differential forms.
Floer, A.. Morse theory for Lagrangian intersections. Forman, R.. Frauenfelder, U.. The Arnold-Givental conjecture and moment Floer homology. Fukaya, K.. Goresky, M.. Stratified Morse theory, Ergeb. Grieser, D.. Grigor'yan, A.. Path homology theory of multigraphs and quivers. Haiden, F.. Semistability, modular lattices and iterated logarithms. Hatcher, A.. Algebraic Topology, Cambridge University Press. Hurtubise, D.. Multicomplexes and spectral sequences, Journal of Algebra and Its Applications 9 , Jost, J..
Riemannian Geometry and Geometric Analysis, Universitext. Joyce, D.. A generalization of manifolds with corners. Kirwan, F.. Knudson, K.. World Scientific. Morse theory, smooth and discrete. Kronheimer, P.. Monopoles and three-manifolds, New Math. Monographs 10, Cambridge University Press. Laudenbach, F.. MacLane, S.. Homology, Classics in Mathematics. Mazzeo, R.. Pseudodifferential operators on manifolds with fibred boundaries. Melrose, R.. Calculus of conormal distributions on manifolds with corners. Milnor, J.. Nanda, V.. Discrete Morse theory and classifying spaces.
Discrete Morse theory and localization. Pure Appl. Nicolaescu, L.. An invitation to Morse theory, Universitext. Novikov, S.. Multivalued functions and functionals: an analogue of Morse theory. Then there exist Morse—Smale pairs. We follow closely the strategy in [M4, Sect. We begin by describing the main technique that allows us to gradually modify f to a self-indexing Morse function.
This is possible since W. If y For example, the time at the point y is f. We can think of the family s! O b The rearrangement lemma works in the more general context, when instead of only two critical points, we have a partition C0 t C1 of the set of critical points in the region f0. We define a handlebody to be a three-dimensional manifold with boundary obtained by attaching one-handles to a three-dimensional ball. A Heegard decomposition of a smooth, compact, connected 3-manifold M is a quadruple. Any smooth compact connected 3-manifold admits a Heegard decomposition.
Fix a self-indexing Morse—Smale pair. The critical values of f are contained in f0; 1; 2; 3g. The case HC. We can encode this description as a graph. The endpoint s of an edge indicate how the attaching is performed. By attaching first the 1-handles corresponding to the edges of T we obtain a manifold H. In particular, the real numbers k C 12 are regular values of f. Z; p 2 Crf;k : We denote by hpj the generator of Hk. Observe that we have a natural morphism W Ck! This is called the Thom—Smale complex associated to the self-indexing Morse function f. We would like to give a more geometric description of the Thom—Smale complex.
More precisely, we will show that it is isomorphic to a chain complex which can be described entirely in terms of Morse data. Observe first that the connecting morphism k W Hk. The relative class hpj 2 Ck is represented by the fundamental class of the oriented manifold with boundary.
Assume for simplicity that the ambient manifold M is oriented. As explained in Remark 2. We will refer 3 For the cognoscienti. The E 1 term is precisely the chain the edge morphism induces an isomorphism Hp. Ep;0 complex 2. Thus hpjqi is a signed count of tunnelings from p to q. Note that the definitions of Ck. In view of Corollary 2. We have thus proved the following result. Corollary 2. For any Morse—Smale pair. We used it only for the ease of presentation.
Here is how one can bypass it. We can now define n. We then get an operator X O D n. The isomorphism is not induced by a morphism between the Morse—Floer complex and the singular chain complexes and thus does not highlight the geometric nature of this construction. The supports of such asymptotic simplices are invariant subsets of the Morse—Smale flow and thus must be unions of orbits of the flow.
The isomorphism between the Morse—Floer homology and the singular homology suggests that the subcomplex of the singular chain complex generated by asymptotic simplices might be homotopy equivalent to the singular chain complex. There is another equivalent way of visualizing the Morse flow complex, which goes back to Thom [Th].
Think of a Morse—Smale pair. The boundary of a face is a union with integral multiplicities of faces of one dimension lower. To better understand this point of view, it helps to look at the simple situation depicted in Fig. Let us explain this figure. The relative interior of the top face is the unstable manifold of F , and all the trajectories contained in this face will leave F and end up either at a vertex or in the center of some edge. The interiors of the edges are the corresponding one-dimensional unstable manifolds.
The arrows along the edges describe orientations on these unstable manifolds. The gradient flow trajectories along an edge point away from the center. In the picture there are two tunnelings connecting F with a1 , but they are counted with opposite signs. We refer to Chap. The tunneling approach has been used quite successfully in infinite dimensional situations leading to various flavors of the so-called Floer homologies. These are situations when the stable and unstable manifolds are infinite dimensional yet they intersect along finite dimensional submanifolds.
One can still form the operator using the description in Proposition 2. For more information on this aspect, we refer to [ABr, Sch]. R is a smooth function on the m-dimensional manifold M. Definition 2. A smooth submanifold S ,! M is said to be a nondegenerate critical submanifold of f if the following hold. M is a nondegenerate critical submanifold of f. Assume for simplicity that f jS D 0. Denote by TS M the normal bundle of S ,! The same arguments in the proof of Theorem 1.
There exist an open neighborhood U of S ,! Let F be a field. R defined on the compact manifold M is the polynomial Pf. Note that the Morse— Bott polynomial of a Morse function coincides with the Morse polynomial defined earlier. X of rank r over X , we denote by D.
Every real vector bundle over a simply connected space is Q-orientable. Equivalently, the transpose map HkCr. This implies PD. Suppose F is a field, and f W M! R is a Morse—Bott function. R is an F-orientable Morse—Bott function on the compact manifold. Then, we have the Morse—Bott inequalities Pf. R is a Morse—Bott function on a compact manifold M. R is a F-completable, F-orientable, Morse— Bott function on a compact manifold. Then, f is F-perfect, i. R is an orientable Morse—Bott function such that for every critical submanifold M we have.
Using the same notation as in the proof of Corollary 2. In this section we will turn the situation on its head. We will use topological methods to extract information about the critical points of a smooth function. To keep the technical details to a minimum so that the geometric ideas are as transparent as possible, we will restrict ourselves to the case of a smooth function f on a compact, connected smooth manifold M without boundary equipped with a Riemannian metric g.
We can substantially relax the compactness assumption, and the same geometrical principles we will outline below will still apply, but that will require additional technical work. Morse theory shows that if we have some information about the critical points of f we can obtain lower estimates for their number. For example, if all the critical points are nondegenerate, then their number is bounded from below by the sum of Betti numbers of M. What happens if we drop the nondegeneracy assumption? Can we still produce interesting lower bounds for the number of critical points?
We already have a very simple lower bound. Since a function on a compact manifold must have a minimum and a maximum, it must have at least two critical points. This lower bound is in some sense optimal because the height function on the round sphere has precisely two critical points.
This optimality is very unsatisfactory since, as pointed out by Reeb in [Re], if the only critical points of f are nondegenerate minima and maxima, then M must be homeomorphic to a sphere. We start with the basic structure of this theory. The min—max technology requires a special input. A collection of min—max data for the smooth function f WM! R is a pair. R, then the real number c D c. We argue by contradiction.
Assume that c is a regular value.
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We will spend the remainder of this section describing a few classical constructions of min—max data. Suppose x0 is a strict local minimum of f , i. This statement is often referred to as the mountain-pass lemma and critical points on the level set ff D cg are often referred to as mountain-pass points.
Observe that the Mountain Pass Lemma implies that if a smooth function has two strict local minima, then it must admit a third critical point. The search strategy described in the mountain-pass lemma is very intuitive if we think of f as a height function. The point x0 can be thought of as a depression and the boundary U as a mountain range surrounding x0. It is perhaps instructive to give another explanation of why there should exist a critical value greater than c0. Observe that the sublevel set M c0 is disconnected while the manifold M is connected.
The change in the topological type in going from M c0 to M can be explained only by the presence of a critical value greater than c0. Denote by CM the collection of closed subsets of M. For every x 2 M , there exists r D r. Hence, for every k the min—max value ck D inf max f. We want to prove that this conclusion holds even if some of these critical values are equal.
Then either c is an isolated critical value of f and Kc contains at least p C 1 critical points or c is an accumulation point of Crf , i. Assume that c is an isolated critical value. Suppose Kc contains at most p points. At this point we need a deformation result whose proof is postponed. Set Tr. The strategy is a refinement of the proof of Theorem 2. The homeomorphism will be obtained via the flow determined by a carefully chosen gradient-like vector field.
Fix a Riemannian metric g on M. For r sufficiently small, Nr. For r 2. We will achieve this in several steps. We will prove that if t 2 Tx , then dist. Loosely speaking, we want to show that there exists a moment of time t when the energy f. Below this level the rate of decrease in the energy f will pickup. We argue by contradiction, and thus we assume f. Step 3. We now have the following consequence of Theorem 2. It turns out that the Lusternik—Schnirelmann category of a space is such a theory.
M is homotopic to the constant map. If such a cover does not exist, we set catM. If M is a compact smooth manifold, then the correspondence CM 3 C 7! It is very easy to check that catM satisfies all the axioms of an index theory: normalization, topological invariance, monotonicity, and subadditivity, and we leave this task to the reader.
The lower estimate of cat. In particular, the long exact sequence of the pair. Unfortunately, we cannot assume this. There are two ways out of this technical conundrum. Either we modify the definition of catM to allow only covers by closed, contractible, and excisive sets, or we work with a more supple concept of cohomology. Since CL. Every even smooth function f W S n! R has at least 2. Observe that f descends to a smooth function fN on RPn , which has at least cat.
Every critical point of fN is covered by precisely two critical points of f. Most applications of Morse theory that we are aware of share one thing in common. More precisely, they rely substantially on the special geometric features of a concrete situation to produce an interesting Morse function, and then squeeze as much information as possible from geometrical data.
Often this process requires deep and rather subtle incursions into the differential geometry of the situation at hand. The end result will display surprising local-to-global interactions. The applications we have chosen to present follow this pattern and will lead us into unexpected geometrical places that continue to be at the center of current research.
More precisely, consider a robot arm with arm lengths r1 ; : : : ; rn , where the initial joint J0 is fixed at the origin. As explained in Example 1. We declare two positions or configurations of the robot arm to be equivalent if one can be obtained from the other by a rotation of the plane about the origin.
Following [Fa], we will refer to Wn as the work space of the robot arm. In particular, we can equip Mr with a topology as a closed subspace of Wn. This shows that the configuration space Cn discussed in Example 1. Observe that the configurations in Mr can be identified with n-gons whose side lengths are r1 ; : : : ; rn. For this reason, the topological space Mr is called the moduli space of planar polygons with length vector r. In this section we want show how clever Morse theoretic techniques lead to a rather explicit description of the homology of Mr.
Proposition 3. The work space Wn is homeomorphic to a. Consider the diagonal action of S 1 on T n D. Wn is a homeomorphism. Note that we have a homeomorphism Wn. Thus, in order to understand the topology of Mr we can assume that r is ordered, i. We will also assume that the genericity assumption 1. We have thus established the following result. If the length vector r satisfies the genericity assumption 1. Due to the genericity assumption, we see that r.
Let us briefly outline the strategy which at its core is based on a detailed analysis of a Morse function on Wn. The work space Wn is equipped with a natural continuous function hn W Wn! Using hn , we can construct the smooth function f D fr W Wn! Hence it suffices to understand the co homology of N". On the other hand, N" is an oriented. From the excision isomorphism we see that this is isomorphic to the co homology of the pair. We will determine the cohomology of the pair. Obtain detailed information about the morphisms entering into the long exact sequence of the pair.
For simplicity we write h instead of hn. The function. The computations in Example 1. Lemma 3. Suppose I is a r-long subset. Let c be a critical value of f , c 0 sufficiently small the following hold. The class vI. To prove a it suffices to show that uI. We let y denote the vectors in PI and z denote the vectors in P? In this notation we have f. It also coincides with uI. R is a proper Morse function on a smooth manifold M , and p is a critical point of f of index , and f. Using unstable disks as in Sect.
In particular, hpj D 0. Note that Lemma 3. Hk W cC"! H0 W cC"! Thus W c1 C" has the homotopy type of a point, and its homology is generated by the point WIn. Using 3. This completes Step A of our strategy. More precisely, we have the following result. Then the torus TI has angular coordinates.
We leave to the reader as an exercise Exercise 6. This concludes the proof of Theorem 3. Thus WI can be identified with the configuration space of this robot arm as defined in Example 1. Example 3. Then, the length vector r D. This agrees with the conclusion of Example 2. The Grassmannian Gk;n is a complex manifold of complex dimension k. Denote by Pk;n. In this section we will present a Morse theoretic computation of Pk;n. Moreover, Pk;nC1. We carry out an induction on D k C n. The statement is trivially valid for D 2, i.
Suppose that U is a complex n-dimensional vector space equipped with a Hermitian metric. The metric on U defines a metric on V , its direct sum with the standard metric on C. For every complex Hermitian vector space W , we denote by Gk. Note that we have a natural map Gk. W denotes the orthogonal projection on L. This map is a smooth embedding. See Exercise 6. Then, A 2 S.
Observe that we have natural embeddings Gk. Observe that. The only critical values of f are 0 and 1. Let L 2 Gk. If we choose v. Observe that S0 is a complex submanifold of Gk. For every u 2 U denote by Xu W V! V the skew-Hermitian operator defined by Xu. Xu 2 HomC. The operator Xu defines a one-parameter family of unitary maps etXu W V! PL Xu. Let us compute f. We have 3. Let L 2 S1. Again we set P. The above computations can be refined to prove that the normal bundle of S0 D Gk. Hence the function f is a perfect Morse—Bott function, and we deduce PGk.
The above analysis can be further refined and generalized. We leave most of the details to the reader as an exercise Exercise 6. Suppose E is a finite dimensional real Euclidean space, and A 2 End. Denote by Grk.
An Invitation to Morse Theory, 2nd Edition
For every L 2 Grk. The map Grk. PL 2 End E embeds Grk. On End. This inner product induces a smooth Riemann metric on Grk. The function fA W Grk.
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R; fA. Its gradient flow has an explicit description, Grk.
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Thus, the set of minima of fA consists of all k-dimensional subspaces containing U. We denote this set with Grk. L; Grk. C is a proper map. Suppose M ,! C is a Stein manifold. Modulo a translation of M we can assume that the function f W C! The following theorem due to Andreotti and Frankel [AF] is the main result of this section. Theorem 3. The Morse indices of critical points of f jM are not greater than m. A Stein manifold of complex dimension m has the homotopy type of an m-dimensional C W complex.
Suppose M is a complex manifold of complex dimension m. TM 1 With a bit of extra work one can prove that if X is affine algebraic, then f has only finitely many critical points, so X is homotopic to a compact C W complex. There exist, however, Stein manifolds for which f has infinitely many critical values. If we set TM 1;0 WD ker. Given V 2 Vect. R is a smooth real valued function on the complex manifold M and p0 is a critical point of M.
Denote by H the Hessian of f at p0. R; Cf. Fix complex coordinates. Set f0 D f. Near p0 we have a Taylor expansion f. Consider the function f W C! Let M ,! C be a Stein manifold of complex dimension m and suppose f W C! Suppose p0 is a critical point of f jM and denote by H the Hessian of f jM at p0. We want to prove that. TM the associated almost complex structure. Then, H. Modulo a linear change in coordinates, we can assume that it is described by the equation z0 D 0. Notice that M is a submanifold of C described as the zero set of the collection of polynomials Pj.
In particular, M is a Stein manifold. By Theorem 3. Only V n V1 needs to be smooth; V1 is automatically tautly embedded, since it is triangulable. If V is a projective algebraic manifold and V1 is the intersection of V with a hyperplane, then the natural restriction morphism H j. The nondegeneracy means that the induced linear map I!
A symplectic space is a pair. An almost complex structure tamed by! We denote by J! Suppose that. Then, J! Denote by MV the space of Euclidean metrics on V , i. Then, MV is a contractible space. V uniquely determined by!. We denote by M!
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We have thus produced a homeomorphism M! Ah ; and it suffices to show that M! More precisely, we will show that M! Fix a metric h 2 MV. For every linear operator B W V! Observe that Ah commutes with Th. This shows that the correspondence h 7! Observe that i i 1 n! A symplectomorphism of. We deduce that M is orientable. We will refer to dv! Note that if f W M! Consider the vector space Cn with Euclidean coordinates zj D x j C iy j. We will refer to. Equivalently, the standard model is the pair. Suppose M is a smooth manifold. M the canonical projection.
In particular, its dual. We will refer to it as the tautological 1-form on the classical phase space. If we choose local coordinates. The tautological 1-form is described in the coordinates. The pair. Let us point out a confusing fact. Suppose M is a complex manifold. Now define gx ;! R by setting gx. Note that! X 1;0 ;. Conversely, if! In particular, any complex submanifold of CPn has a symplectic structure.
The complex submanifolds of CPn are precisely the projective algebraic manifolds, i. A symplectic structure on a manifold may seem like a skewsymmetric version of a Riemannian structure. As is well-known, two Riemann structures can be very different locally. In particular, there exist Riemann metrics which cannot be rendered Euclidean in any coordinate system. The Riemann curvature tensor is essentially the main obstruction. More precisely, if. To understand this example, we will need a few basic facts concerning homogeneous spaces.
For proofs and more information we refer to [Helg, Chap.