G-MORSE THEORY REVISITED DAN BURGHELEA. Department of Mathematics The Ohio State University

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1 G-MORSE THEORY REVISITED DAN BURGHELEA Department of Mathematics The Ohio State University 1

2 ELEMENTARY MORSE THEORY. M closed manifold, h : M R smooth function X : M T M smooth vector field. h Morse function dh x = o (h, M, x) Ind(x) = q h : = h(x) 1/2 R q R n q 0 q n x 2 i + 1/2 i=1 i=q+1 x 2 i Cr(h) = {x M, dh x = 0}, Cr(h) = q Cr(h) q X h gradient like vector field a)x(h)(x) < 0, x M \ Cr(h) b)(x, M, x) ( grad g h, M, x) for some Riemannian metric g Cr(h) = X 2

3 Stable/Unstable set x critical point i.e x Cr(h) W ± x = {y M lim t ± γ y(t) = x} i ± x : W ± x M W ± x R Ind(x)/n Ind(x) (X, h) Morse pair a) h Morse function, b)x ( h)-gradient like vector field (X, h) Morse Smale pair a) (X, h) Morse pair b)x, y Cr(h) W x W + y Proposition Given f smooth function there exists h a Morse function arbitrary close from f in C r topology. 2. Given h a Morse function there exists X ( h)-gradient like vector field. 3. Given (X, h) a Morse pair there exists X arbitray close from X in C 0 topology so that (X, h) is a Morse Smale pair. One can actually choose X to be equal to X away from an arbitrary neighborhood of Cr(h) 3

4 Let (X, h) be a Morse-Smale pair x, y Cr(h) T (x, y) := Wx W y + /R smooth manifold of dimension ind(x) ind(y) 1 Consider ˆT (x, z), Ŵx and î x : Ŵx M defined by ˆT (x, z) := r T (x, y 1 ) T (y r 1, y r ) T (y r, z) y 1,,y r Ŵ x := r y 1,,y r T (x, y 1 ) T (y r 1, y r ) W y r î x y 1,,yr T (x,y 1) T (y r 1,y r ) W yr := i y r P r W y r. Theorem ˆT (x, z) is a smooth compact manifold with corners whose k corner is ˆT (x, z) r = y 1,,y r T (x, y 1 ) T (y r 1, y r ) T (y r, z). 2. Ŵx is a smooth compact manifold with corners whose k corner is (Ŵ x ) r = y 1,,y r T (x, y 1 ) T (y r 1, y r ) Wy r 3. î x is a smooth map. 4

5 APPLICATIONS 1) Smooth cell structure of M Corollary The partition of M into Wx, x Cr(h), is a smooth CW-complex whose closed cells are given by î x : Ŵx M. 2. There exists compatible smooth triangulations of M, ˆT (x, y), Ŵx. In particular the open simplexes of Wx define the collection of open simplexes of M 2)The geometric complex Fix orientations O := {o x, x Cr(h)} of T x (W x ) Define the complex (C (X), δh,o (t)) by: C q (X) := Maps(Cr(h) q, R) ((δ q h,o (t))(f))(y) := x Cr(h) q e t(h(x) h(y)) δ(x, y)f(x) y Cr(h) q+1, δ(x, y) = (T (y, x)), f C q (X) 5

6 3)Integration Theory Int q h (t)(ω) := Ŵ x (i x ) (e ( t(h h(x)) ω) Int h (t) : (Ω (M), d h (t) = d + tdh ) (C (X), δ h,o (t)) 4)WHS-theorem Theorem 2. Given a Riemannian metric g (which is admissible for (X, h)) there exists T > 0 and a canonical decomposition (Ω (M), d h(t)) = (Ω sm(m), d h(t)) (Ω la(m), d h(t)) so that Int h (t) : (Ω sm(m), d h(t)) (C (X, O), δh,o(t)) is an O(1/t) isometry provided t > T. 6

7 ELEMENTARY G-MORSE THEORY. M = (M, µ : G M M) closed G manifold, Let x M. H := G x = {g G, µ(g, x) = x} Σ(x) := µ(g, x) = G/H orbit of x M ρ x : H Gl(V = T x (M)/T x (Σ) isotropy repr. (U, Σ) (G H V, G/H) G H V G/H h : M R a smooth G invariant function X : M T M smooth G equivariant vector field. x Cr(h) Σ(x) Cr(h) x, x Σ ρ x ρ x 7

8 G Morse function h satisfies: a)ρ x = ρ x ρ x +, ρ x ± : H O(V ± ), V = V V + x Cr(h) Σ(x) Cr(h) b)σ Cr(h) (h, M, Σ) h : = h(x) 1/2 v 2 + 1/2 v 2 E(ρ ρ + ) Σ Ind(Σ) := dim(v ) 8

9 G normal Morse function h satisfies a) h is a G Morse function, b) ρ x trivial for any x Cr(h) X h gradient like vector field and g equivariant X := {x M X(x) = 0} X = Cr(h) Σ Cr(h) W ± Σ = {y M lim t ± γ y(t) Σ} = x Σ W ± x i ± Σ : W ± Σ M p± Σ : W ± Σ Σ p ± Σ is a G bundle eqv. to G Gx W ± x Σ(x) equivalent to G Gx V ± G/G x, for some (and then any) x Σ. (X, h) G-Morse pair... (X, h) G- normal Morse pair 9

10 ... G-Morse Smale pair (normal Morse Smale pair) (X, h) satisfies a) (X, h) a G-Morse pair (G- normal Morse pair) b) For any Σ, Σ, x Σ Wx W + Σ It suffices to check for one x. Let (X, h) be a G Morse-Smale pair Σ, Σ Cr(h) T (Σ, Σ ) := WΣ W + Σ /R smooth manifold of dimension dim(σ) + ind(σ) ind(σ ) 1 Σ, Σ Cr(h), x Σ T (x, Σ ) := WΣ W + Σ /R smooth manifold of dimension ind(σ) ind(σ ) 1 on which G x acts. u : T (Σ, Σ ) Σ, l : T (Σ, Σ ) Σ 10

11 smooth maps, u a smooth G bundle isomorphic to G Gx T (x, Σ ) G/G x ρ x trivial p Σ and u are trival bundles. Define: ˆT (Σ, Σ ), Ŵ Σ G-spaces, û : ˆT (Σ, Σ ) Σ, l : ˆT (Σ, Σ ) Σ G-equiv. maps, ˆp Σ : Ŵ Σ Σ, : î Σ : Ŵ Σ M G- equiv.maps. ˆT (Σ, Σ ) := r Σ 1,,Σ r T (Σ, Σ 1 ) Σ1 T (Σ r 1, Σ r ) Σr T (Σ r, Σ ) Ŵ Σ := r Σ 1,,Σ r T (Σ, Σ 1 ) Σ1 Σr 1 T (Σ r 1, Σ r ) Σr W Σ r î Σ Σ 1,,Σr T (Σ,Σ 1) Σ1 Σr 1 T (Σ r 1,Σ r ) Σ r W Σr := i Σ r P r W Σ r. 11

12 Theorem ˆT (Σ, Σ ) is a smooth compact manifold with corners whose k corner is ˆT (Σ, Σ ) r = T (Σ, Σ 1 ) Σ1 Σr 1 T (Σ r 1, Σ r ) Σr T (Σ r, Σ ). y 1,,y r The maps û and ˆl are smooth with û a G bundle with fiber T x := r T (x, Σ 1 ) Σ1 Σr 1 T (Σ r 1, Σ r ) Σr T (Σ r, Σ ). Σ 1,,Σ r This bundle is isomorphic to G Gx T x G/G x and is trivial if ρ is trivial. 2. ŴΣ is a smooth compact manifold with corners whose k corner is (Ŵ Σ ) r = T (Σ, Σ 1 ) Σ1 Σr 1 T (Σ r 1, Σ r ) Σr WΣ r. y 1,,y r The map ˆp Σ a smooth G bundle with fiber F x = T (x, Σ 1 ) Σ1 Σr 1 T (Σ r 1, Σ r ) Σr WΣ r. y 1,,y r This bundle is isomorphic to G Gx F x G/G x and is trivial if ρ is trivial. 3. î Σ is a smooth map. 12

13 Suppose (X, h) is G-normal Morse Smale. Choose for any Σ Cr(h) a point x Σ By passing to the orbit spaces i Σ : W Σ M i x : Wx M/G Corollary 2. The manifolds Wx have canonical compactifications to smooth manifolds with corners F x and the maps i x have (smooth) completions î x : F x M/G providing a (smooth) cell complex structure for M/G. Theorem (K.H.Meyer) Given a G invariant smooth function f there exists h a normal G-Morse function arbitrary close to f in C 0 topology. 2. Given h a G Morse function there exists ( h)-gradient like vector fields. 3. ([B])Given (X, h) a normal G Morse pair, there exists X arbitray close to X in the C 0 topology so that (X, h) is a G-normal Morse Smale pair. One can choose X to be equal to X away from an arbitrary G-invariant neighborhood of Cr(h). 13

14 APPLICATIONS: 1) Smooth cell structure Corollary 3. Suppose (X, h) is a G- Morse Smale pair. 1. The partition of M into W Σ is a smooth G handle body whose G- handles are î Σ : Ŵ Σ M where Ŵ Σ are the total spaces of the bundles ˆp Σ : ŴΣ Σ. 2. If the pair is normal this partition induces a smooth cell structure for M/G and compatible smooth triangulations of F x, T x lead to compatible smooth triangulations of M/G; equivalently to a smooth G-triangulation of M. The open simplexes of M/G are open simplexes of Wx. 2)The geometric complex The smooth maps i Σ combined with integration along the fibers ( F ) provide for any critical orbit Σ the linear maps Int Σ(t) : Ω (M) Ω Ind(Σ) (Σ, o Σ ) by Int Σ(t)(ω) := F (î x ) (e t(h h(σ)) ω) 14

15 Choose orientations O : {O x orientation int x (Wx ). O orientations o Σ of the bundles ˆp Σ : WΣ Σ. The integration along the fibers induces the linear maps Σ,Σ : Ω (Σ, o Σ) Ω Ind(Σ)+Ind(Σ )+1 (Σ, o Σ). Define: Σ,Σ : Ω (Σ, o Σ ) Ω +1 (Σ, o Σ ) = d Σ C q (X) = Σ Ω q Ind(Σ) (Σ, o Σ ) and for ω Ω q (Σ, o Σ ). δ q h,o (t)(ω) := Σ e t(h(x ) h(x)) δ Σ,Σ (ω) 4 Integration Theory Int h(t) : (Ω (M), d h(t) = d + tdh ) (C (X, O), δ h,o(t)) Int h (t)(ω) = Σ Int Σ(t)(i x ) ω (Ω (M), d h (t)) and (C (X), δh,o (t)) are complexes of G- representations and Int h (t) is G equivariant. 15

16 (Int h(t)) = (Int h(t)) ξ (Int h(t)) ξ : (Ω (M) ξ, d h(t)) (C (X, ) ξ, δ h (t)) with (C (X) ξ, δh,o (t)) finite dimensional cochain complex. 4)WHS-theorem Theorem 5. Given a Riemannian metric g admissible for (X, h) and ξ an irreducible representation there exists T ξ > 0 and a canonical decomposition (Ω (M) ξ, d h(t)) = (Ω ξ,sm(m), d h(t)) (Ω ξ,la(m), d h(t)) so that (Int h (t)) ξ : (Ω ξ,sm(m), d h(t)) (C (X) ξ, δh,o(t)) is an O(1/t) isometry provided t > T ξ. 16

17 PROOF OF THEOREMS 3,4) K := W Σ M c 2ɛ, K G H K x, K x = W x M c 2ɛ Z = ( h(σ )) c W + Σ M c 2ɛ, Z M c 2ɛ Whitney stratif. Maps G (K, M K x Z ) Maps G (K, M ) Maps H (K x, M K x Z ) Maps H (K x, M ) Maps (K x, M H K x Z H ) Maps (K x, M H ) 17

18 MANIFOLDS WITH CORNERS P: P is locally diffeomorphic R n + = {(x 1, x 2,..., x n ) x i 0}. P k points of P which in some chart have exactly k vanishing coordinates. P k is a smooth manifold of dimension (n k). P = P 1 P 2 P n is closed in P and a topological manifold. (P, P) is a topological manifold with boundary P. A product of manifolds with boundary is a manifold with corners. Remark 1. If P is a smooth manifold with corners, O, S smooth manifolds, p : P O and s : S O smooth maps so that p and s are transversal (p is transversal to s if its restriction to each k-boundary P k is transversal to s), then p 1 (s(s)) is a smooth submanifold with corners of P. 18

19 PROOF OF THEOREM 3 1) c N > > c i > c i 1 > > c 1, all critical values of h. ɛ > 0 small so that c i ɛ > c i 1 + ɛ Denote: M i :=h 1 (c i ), M ± i :=h 1 (c i ± ɛ i ) M(i) :=h 1 (c i 1, c i+1 ). P i trajectories (possible broken) from M i to M i P i compact smooth manifold with boundary. Let h(σ) = c r+1, h(σ ) = c r k 1 and take P := P r,r k := P r P r 1 P r k, O := r k i=r (M + i M i ), S := S Σ M r M r k+1 S+ Σ. Define p : P O s : S O. Show p and s are smooth and transversal. ˆT (Σ, Σ ) = p 1 (s(s)) 19

20 T (Σ, Σ 1 ) Σ1 T (Σ 1, Σ 2 ) u l T (Σ, Σ 1 ) l Σ1 u Σ1 T (Σ 1, Σ 2 ) u Σ Σ Σ 1 Σ 2 l Σ2 u T (Σ, Σ 1 ) Σ1 W Σ 1 p T (Σ, Σ 1 ) l Σ1 p Σ1 W Σ 1 u Σ Σ Σ 1 20