Morse Theory Flashcards

Question

Discuss refining Morse inequalities Proof?

Answer 1

Consider 2 polynomials: h(t) = sum dimH_k(P) t^k m(t) = sum |Crit_k(f)|t^k the Poincare polynomial and the Morse polynomial. Call coeffic of h b_k - betti and of m C_k Prop. (1) There exists a poly r(t) "remainder" with coef >= 0 s.t. m(t) = (1+t)r(t) + h(t) <-- in particular coeff of m(t) >= h(t) (2) Ck- Ck-1+Ck-2-... -+ C0 >= bk-bk-1+... Cor. Euler Char of complex = Euler Char of manifold. Plug in -1 pg 51-52

Answer 2

Idea: Use unit speed flow along gradient as we did for case without singularities. Use cut-off functions to ensure small neighborhood of singularity remains fixed. This unretracted neighborhood can be homotoped to a disc... pg 53-55

Answer 3

A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory. The cellular decomp we obtain in Morse theory is actually a handle body decomp - attach a k-handle for each critical pt of index k Cor. A closed smooth manifold has a "handlebody decomposition" pg 55-57

Answer 4

Already know that near a non-degen critical point the smooth and top picture is completely dominated by quadratic term - local dynamics pg 57

Answer 5

Start with some Morse function f. Want the simplest possible handle-body decomp - smallest number of singularities. f is PERFECT if d=0 in the associated complex: |crit_k(f)| = b_k Example surfaces. In reality perfect functions are rare - normally not possible to modify an in imperfect f to a perfect f. 2 steps always work to simplify: 1. Kill extra max's and min's 2.Can move lower index pts below higher index pts In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps M --> W and N --> W are homotopy equivalences. (Recall: Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher) The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds. The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture. pg 58-59

Answer 6

Hirsch pg 60

Answer 7

Milner pg 60

Answer 8

For every closed orientable 3-mfld P, exists a diffeo phi: S_g --> S_g s.t. P = union of 2 solid bodies with g handles glued via diffeo phi. Proof. Mores theory...nice pg 60-62

Answer 9

Form a chain complex with chains generated by critical points. Goal: describe differential explicitly without using algebraic topology or CW complex structure Idea: m(x,y) = number of trajectories from x to y. <-- this depends on choice of metric If index differs by 1, there are generically just a finite number of anti-gradient trajectories from x to y. pg 65-66

Answer 10

mu(x,y) = { z | phi_t(z) goes to x as t --> -inf and y as t --> inf} where phi_t is the anti-gradient flow of f and x, y critical points. Call mu(x,y) moduli space Thm. For a generic metric, mu(x,y) is a smooth manifold of dimension mu(x) - mu(y) <--difference in index. stable = incoming trajectories unstable = outgoing trajectories Each of these diffeo to open balls - by Morse Lemma locally just quadratic ... think of parabolas on different axises mu(x,y) = W^u(x) int W^s(y) and with a generic metric, we can ensure these sub-manifolds intersect transversely => m(x,y) is a manifold and dim mu(x,y) = n - dim W^u(x) - dim W^s(y) = ... = mu(x) - mu(y). pg 68-70

Answer 11

Generically, we shouldn't have trajectories between critical points on saddle. Tilt slightly. pg 70

Answer 12

Sliding handles. We start with an arbitrary Morse function f and modify it by moving critical points past each other pg 71-72

Answer 13

think of as set of solutions of antigradient equation instead of thinking of as submanifold of P. mu(x,y) = space of parameterized trajectories Note: we have a free R-action on mu(x,y). Taking quotient get space of unparameterized trajectories mu hat(x,y) = mu(x,y)/R This is a smooth manifold of dim mu(x)-mu(y) - 1. In particular, when m(x) - mu(y) = 1, this is a 0-dim manifold = discrete collection of pts pg 73

Answer 14

This is a key result which requires the modern view - m(x,y) not being subset. Thm. For a generic metric, mu hat has a compactification formed by broken trajectories -- form manifold with corners. In the case where index differs by 1, there are no broken paths. mu hat already compact --> finite set. Only finitely many trajectories from x to y. pg 74

Answer 15

Need to account for orientations. Fix orientations of stable and unstable manifolds by orienting tangent spaces. This gives orientation of mu(x,y) Trajectories also oriented by flow so we have 2 orientations on each trajectory. If they agree +1 if they disagree -1. pg 75-76

Answer 16

Explicitly computing we see that we need to sum over broken trajectories with one brake between critical points whose index differs by 2. mu hat (x,y) is a 1-dim manifold in this case. Compactification is either S^1 or closed interval. If S^1, no broken trajectories - already comapct. Otherwise, we have a pair of broken trajectories - two end points of interval... pg 77

Answer 17

Claims differential is the same <-- try to check. Note: we proved cellular = singular = topological invariant. So we see Morse homology is a topological invariant of manifold --- actually homotopy invariant. pg 78

Answer 18

Kunneth: P0, P1 closed manifolds. Then H_*(P0 x P1) = H_*(P0) (x) H_*(P1). Proof. Choose Morse functions f0, f1 on P0, P1 and generic metrics. Product metric on P0 x P1. pg 83

Answer 19

P smooth closed manifold - homology in dim k matches cohomology in codim k. Palindromic b_k = b_n-k Pf. Idea: Choose a Morse function f and look at f and -f. Gradient flows in opposite direction -- critical points flip index. pg 84

Answer 20

Thm. If P is a closed manifold, then Morse functions form an open dense subset in C^k(P). Corollary. Every manifold admits a Morse function. Moreover, every function can be C^k-approximated by Morse functions.

Answer 21

Identify the 0 section with P x is critical <=> x in graph(df) int P x is nondegenerate <=> graph(df) int P transverse at x f is Morse <=> graph(f) transverse to P pg 86-87

Answer 22

1. Start with f0 and approximate by Morse function 2. Transversality thms => df0 approximated by 1-forms whose graphs intersect P transversely. Difficulty: Need 1-forms exact i.e. = df pg 87

Answer 23

dF and TpZ together span TpY pg 88

Answer 24

X a closed manifold. Thm 1. F transverse to Z for open dense set in C^k(X,Y) Y = X x K > Z, F: X-->K Thm 2. Graph(f) transverse to Z for an open dense set in C^k(X,K) K --> Y --> X Fiber bundle, Z < Y, s a section. Thm 3. s transverse to Z for an open dense set of C^k sections First jet bundle J^1X = T*X x R --> X First jet of function j^1f = (df, f) Thm 4. j^1f transverse to Z for an open and dense set of f in C^k(X) k>= 2 pg 90 - 93

Answer 25

As a result of Jet transversality, we see that df transverse Z in T*X for an open and dense set f in C^k(X) k>= 2. Now just take Z= P = zero section and we're done. pg 93-94

Answer 26

Guillemin-Pollock pg 95-96

Answer 27

Look up on internet

Answer 28

Thm. For an open dense (full measure) subset of S^m-1, h_v is Morse. h_v is projection onto line spanned by v. This can be interpreted as a height function pg 97-98

Answer 29

Prop. H-v Morse <=> +-v are regular values of Gauss map G This implies thm, sense regular values form open dense subset of full measure by Sard's thm. Notice Crit(h_v) = points where tangent plane is horizontal = points where unit outer normal is vertical --> v or -v = G^-1(v) U G^-1(-v) pg 99-102

Answer 30

Lee Ch 6 pg 103

Answer 31

With the generalized version of Gauss map, have: h_v Morse <=> +- v regular values of G as before. PG 104 - 106

Answer 32

One way to think of cup product is wedge product of differential forms. If closed manifold, have Poincare duality - can use to define product in homology Want to think of intersection product as intersection of 2 chains yielding another chain --> think of as everything as submanifolds Also read in Hatcher pg 107

Answer 33

This requires modern perspective in terms of Morse complex, not just handlebody stuff. 1. Think of Morse homology groups as module over geometric homology groups. - product obtained by looking at trajectories from x to z that pass through B... 2. Use 3 Morse functions to define product... pg 107 - 111

Answer 34

Again we have f:P-->R and want to understand lower bound on |Crit(f)| but here we do not require f non-degenerate. Underlying idea LS and Morse the same: change in topology from sublevel Pa to Pb => critical pts between a and b. LS better adapted to inf dimensional settings -- think variational problems Morse deeper connection with topology in finite dims pg 112

Answer 35

In general, LS very far from sharp but any P^n admits f with |Crit(f)| <= n+1. By this, the examples given are sharp. pg 113

Answer 36

minimum size of an open cover by contractible sets (IN AMBIENT SPACE - S^1 can be contractible) , Contractible sets need not be connected - may have nontrivial topology cat(S^n) = 2 cat(surface) = 3 cat(CW) <= n+1 Difficult to compute - upper bounds by explicit constructions. Lower bounds?? Advantage: easy definition pg 114 - 115

Answer 37

max product of cohomology classes of positive degree before getting 0. Clearly between 0 and dim X depends on ground field Easier to calculate. Harder to define - need cohomology pg 116 -117

Answer 38

cat(X) >= cl(X) + 1 claim: A < X contractible to a point => H*(X,A) -->> H*(X) onto. Note: It need not be 1-to-1. Consider collapsing a circle to a point on S^2, creates a wedge of two S^2 pg 117-119

Answer 39

covering A, but contractible in ambient space X. With this notion, monotone, continuous, homotopy invariant, subadditive... pg 119

Answer 40

Topological approach (LS approach) to proof. P closed manifold. Thm. Assume that Crit(f) are isolated. Then |crit values of f| >= cat(P). Cor. |Crit(f)| > = cat(P) >= cl(P) + 1 Inequalities established for some of our favorite examples are sharp. Pf. Idea: what is responsible for presence of critical points is change in topology. Proceed by induction. Since critical points isolated, can cover every critical point on a critical level by little geodesic ball which is contractible. Using anti-gradient flow we see the rest is diffeomorphic to sublevel from previous induction step. pg 120 - 122 Min/Max Approach By "LS Inequality" since critical pts isolated, intersecting with non-unit homology class strictly reduces the value of the critical value selector. Writing a maximal intersection = cup length, we get k+1 distinct critical values. pg 138

Answer 41

n+1 critical pts. pg 123-125

Answer 42

Says that inf sup f|A is a critical value min f = given by choosing F = collection of all pts of F max f = given by choosing F = {P} pg 126-127

Answer 43

pg 128-129

Answer 44

Let alpha in H_k(P) and define F_alpha = { A : [A] = alpha} = {cycles representing alpha} Def 1. C_alpha(f) = inf_F_alpha max_A f Def 2. Assume f is Morse. Then A = linear combo of critical points (in Morse homology) <-- if P compact as we assume throughout, there are finitely many critical points so inf becomes min and we only have to calculate on finitely many chains. (just max/min over critical points having non-zero coeff in chain) If f not Morse, approximate by Morse and take limit Def 3. c_alpha(f) = inf {a <-- regular value : alpha in image(H_*(P_a) --> H_*(P) } Properties: 1. Criticality 2. monotonicity 3. C^0 continuity 4. subadditivity pg 130 - 136

Answer 45

Assume critical points are isolated and beta != fundamental class, then: C_alpha int beta (f) < c_alpha(f) KEY: inequality strict. IDEA: intersecting A with B we get a submanifold of positive codim in A. No reason for it to pass through the isolated max point in A. measure 0 chance pg 137

Answer 46

Thm, Let Q(x) be a quadratic form, f = Q|S^n, and l1 <= l2 <= ... <= ln the eigenvalues of Q. Then lk = inf max f|Lk Pf. Crit(f) = unit eigenvectors cv(f) = eigenvalues This is an immediate consequence of Lagrange multipliers... pg 141-144

Answer 47

Original motivation for LS theory. Any metric on S^2 has >= 3 simple closed geodesics. Pf. Difficult. Cleaned up by Ballman -- required tool of curvature forms to develop. Outline: 1. Look at space Lambda of smooth embedded loops in S^2 2. Lambda ~ RP^2 ~ Space of Great circles 3. Critical points of the length functional L:RP^2 --> R = simple closed geodesics 4. LS theory => |Crit(L)| >= cl(Lambda) +1 >= 2+1 = 3 pg 145-146

Answer 48

P = f.d. or Hilbert/Banach manifold f: P --> R satisfies PS if every sequence {xi} < P s.t. |f(xi)| < C < inf and df(xi) --> 0 contains a convergent subsequence. IMPORTANT: This condition is very fragile. Easily breaks down. Without PS, change of topology from Pa to Pb does not imply existence of critical points. Loose connection between topology and critical points. pg 147 - 148

Answer 49

1. f bounded from below 2. f satisfies PS 3. The flow for anti-gradient-like vector field is defined for all t >= 0. => f has a critical pt, actually min is attained Pf. Start with x in P, Create sequence using anti-gradient-like flow. PS + bounded from below ==> converges to a critical point y. pg 149

Answer 50

Works out very similarly to min/max in compact case and example of PS pg 150-152

Answer 51

Let M be a closed manifold with fixed Riemannian metric. Def 1. A geodesic is a curve locally minimizing length Def 2. A geodesic is a critical point of the length functional on space of all curves between p and q. Def 3. A geodesic is a critical point of the energy functional. Def 4. 0 acceleration curves 3 => 1,2 pg 153-155, 156, 158

Answer 52

The change in function over any variation = 0. pg 155-156

Answer 53

an r > 0 s.t. any pts in a r-ball are connected by a unique geodesic. radius where exponential map is injective pg 158

Answer 54

Thm. For any non-trivial free homotopy class [alpha] in [S^1, M] there is a closed geodesic representative of [alpha]. Idea: take shortest loop in [alpha] or minimize energy over all geodesics Pf. 1. Introduce idea of geodesic polygons with edge lengths < injectivity radius 2. Every loop in [alpha] can be approximated by geodesic polygons in [alpha]. Every connected component of loop space contains geodesic polygon ==> P_k int Connected component of alpha = union of components of P_k = smooth manifold with boundary 3. Replace E by cursive E (write CE). See both have same critical points. (broken geodesic actually a geodesic if critical point of CE) 4. Find minimum pg 159-168 175-179

Answer 55

Thm. If pi_1 = 1, there exists a non-constant closed geodesic. Pf. 1. Topology of loop space: e: Gamma --> M. Use LES of fibration... 2. State exactly how P_k approximates Lambda --> can get isomorphisms of homotopy groups for all pi_n with n < N <-- arbitrary, by taking k big enough. pg 169 - 179