Intersection of stable and unstable manifolds for invariant Morse functions
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1 Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
2 Contents 1. Witten s Morse theory 2. GKM theory 3. Motivation 4. Results Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
3 1. Witten s Morse theory Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
4 Terminology from Morse theory M Compact Riemannian manifold. Φ Morse function. Cr(Φ) the set of critical points of Φ. λ(p) the Morse index of p Cr(Φ) Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
5 γ x (t) negative gradient flow w.r.t Φ i.e. the solution of d γ x (0) = x, dt γ x(t) = (grad Φ) γx(t) Since Φ is a Morse function, lim t ± γ x(t) Cr(Φ) Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
6 Definition The unstable manifold W u (p) and the stable manifold W s (p) for p Cr(Φ) are defined by { } W u (p) := x M lim γ x(t) = p, t { } W s (p) := x M lim γ x(t) = p t These are submanifold of M. dim W u (p) = λ(p), dim W s (p) = dim M λ(p). Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
7 Definition Φ Morse-Smale def W u (p) and W s (q) intersect transversally for all p, q Cr(Φ). If Φ is Morse-Smale, M(p, q) := W u (p) W s (q) is a (λ(p) λ(q))-dim submanifold of M. It is known that M(p, q) {p, q} is compact if λ(p) λ(q) = 1. In particular, there are only finitely many negative gradient flows which connect p and q. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
8 Witten s Morse theory We can calculate the homology of M from a picture. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
9 2. GKM theory Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
10 Let T = (S 1 ) n be a (compact) torus of rank n. Definition Let M be a smooth manifold with an effective T -action. Then M is called GKM manifold if the following conditions are satisfied M T is a finite set M has an T -invariant almost complex structure The weights of the isotropy representation of T on T p M are pairwise linearly independent for all fixed point p. These condition implies the followings The union of 0 and 1-orbits forms a balloon art in M The action of T on a 1-orbit is given by the rotation of sphere Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
11 We consider the universal T -bundle ET BT. Definition We say that M satisfies equivariant formality if the Serre spectral sequence for the fibration ET T M BT collapses, If H i (M) = 0 for all odd integer i, M satisfies the equivariant formality Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
12 Let M be an equivariantly formal GKM manifold By a theorem of Goresky-Kottwitz-MacPherson, the equivariant cohomology of M can be described in a combinatorial way. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
13 We denote by X(T ) the character group of T. Since the universal T -bundle is a principal T -bundle, we can associate the complex line bundle γ α on BT for all α X(T ). Using this construction, we have the following isomorphism of C-algebras Sym(X(T ) Z C) H (BT ) In particular, a character α defines an element of H (BT ). Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
14 Theorem (Goresky-Kottwitz-MacPherson) Let M be an equivariantly formal GKM manifold. The inclusion M T M induces the injection H T (M) H T (M T ) = C[x 1,, x n ] M T The image of the injection is given by {(f p ) p C[x 1,, x n ] M T f(p) f(q) mod α i (p, q S i )} Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
15 GKM theory We can calculate the T -equivariant cohomology of M from the picture. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
16 3. Motivation Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
17 In Witten s Morse theory, it is important to consider the negative gradient flows which connect two critical points p, q such that λ(p) λ(q) = 1. However there are too many examples of Morse functions which have no such pairs. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
18 Example (Flag manifold) G compact semisimple Lie group T maximal torus G C, T C complexification B Borel subgroup s.t. T C B B opposite Borel subgroup W Weyl group Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
19 G = SU(n) T = {diag(t 1,, t n ) t 1,, t n S 1 } G C = SL n (C), T C = { diag(t 1,, t n ) t 1 t n = 1 } C C C B = , C B = C C (det = 1) W = S n Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
20 The inclusion G G C induces the diffeomorphism G C /B G/T G/T can be identified with a coadjoint orbit O Lie(T ) (moment map) ξ Lie(T ) defines a function Φ ξ : G C /B R Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
21 Fact For generic ξ, Φ ξ is a Morse-Smale function. Cr(Φ ξ ) = W W u (p w ) = BwB/B, W s (p w ) = B wb/b (Bruhat cell) In particular, we have λ(p w ) = dim W u (p w ) = dim BwB/B 2Z. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
22 This phenomenon leads us to the study of the structure of M(p, q) for p, q Cr(Φ), λ(p) λ(q) = 2 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
23 Analogy Witten s Morse theory GKM theory Critical point Fixed point Negative gradient flow 2-sphere Homology equivariant cohomology I want to understand the Morse theoretic aspects of GKM theory Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
24 4. Results Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
25 Critical points Proposition Let G be a compact connected Lie group, M be a compact smooth G-manifold, and Φ : M R be a G-invariant Morse function on M. Assume that there exist only finitely many G-fixed points on M. Then we have Cr(Φ) = M G Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
26 Corollary Let p 0 be a point of M and H be its stabilizer. Assume the following three conditions: H is connected W H := N G (H)/H is a finite group The Fixed point set of the H-action on M is contained in the G-orbit of p 0. Then, we have Cr(Φ) = W H p 0 for any H-invariant Morse function Φ : M R. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
27 We apply the above corollary to flag manifolds. Corollary Let G be a compact Lie group and T be a maximal torus. Then, the critical point set of any T -invariant Morse function on the flag manifold G/T is given by its Weyl group. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
28 Structure of M(p, q) Theorem Let Φ be a G-invariant Morse-Smale function on M. Let p, q be critical points of Φ such that λ(p) λ(q) = 2. If M G is a finite set, every connected component of M(p, q) is diffeomorphic to S 1 R. We have the same picture which appeared in GKM theory Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
29 In the setting of the above theorem, let C be a connected component of M(p, q). We consider the group action on C. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
30 Let G be a compact connected Lie group which acts smoothly on S 1. We consider the following commutative diagram: Lie(G) Γ(T S 1 ) G Diff(S 1 ) Plante gave the classification of the image of the map Lie(G) Γ(T S 1 ). Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
31 Lemma (Plante) Let G be a Lie group. If G acts smoothly and transitively on S 1, the image of Lie(G) Γ(T S 1 ) is conjugate via a diffeomorphism to one of the following sub Lie algebras of Γ(T S 1 ) x (1 + cos x), (sin x), (1 cos x) x x x Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
32 Theorem There is a surjective group homomorphism α : G S 1 and a diffeomorphism C S 1 R which satisfies the following The action of G R on C = S 1 R is given by (g, t) (x, s) = (α(g)x, t + s) for all (g, t) G R, (x, s) S 1 R. This theorem says that the action of T on C is given by the rotation of sphere Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
33 Corollary The stabilizer of c C is independent of choice of c and is a codimension 1 closed normal Lie subgroup of G. Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
34 Thank you! Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and unstable manifolds for invariant Morse March functions 14, / 34
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