Spherical complexities and closed geodesics Stephan Mescher - PowerPoint PPT Presentation

Spherical complexities and closed geodesics Stephan Mescher (Mathematisches Institut, Universität Leipzig) 5 February 2020

Lusternik-Schnirelmann category and critical points

The Lusternik-Schnirelmann category of a space Definition For a topological space X and A ⊂ X put � � � cat X ( A ) := inf r ∈ N � ∃ U 1 , . . . , U r ⊂ X open, r � � s.t. U j ֒ → X nullhomotopic ∀ j and A ⊂ U j . j = 1 cat ( X ) := cat X ( X ) is the Lusternik-Schnirelmann category of X . • cat ( X ) is a homotopy invariant of X . • cat ( X ) is hard to compute explicitly. 1

Lusternik-Schnirelmann category and critical points Theorem (Lusternik-Schnirelmann ’34, Palais ’65) Let M be a Hilbert manifold and let f ∈ C 1 , 1 ( M ) be bounded from below and satisfy the Palais-Smale condition with respect to a complete Finsler metric on M. Then # Crit f ≥ cat ( M ) . • There are various generalisations, e.g. generalized Palais-Smale conditions (Clapp-Puppe ’86), extensions to fixed points of self-maps (Rudyak-Schlenk ’03). • Advantage over Morse inequalities: No nondegeneracy condition required. 2

Properties of cat X Proposition Let X be a normal ANR. Put ν ( A ) := cat X ( A ) . (1) (Monotonicity) A ⊂ B ⊂ X ν ( A ) ≤ ν ( B ) . ⇒ (2) (Subadditivity) ν ( A ∪ B ) ≤ ν ( A ) + ν ( B ) ∀ A , B ⊂ X. (3) (Continuity) Every A ⊂ X has an open neighborhood U with ν ( A ) = ν ( U ) . (4) (Deformation monotonicity) If Φ t : A → X, t ∈ [ 0 , 1 ] , is a deformation, then ν (Φ 1 ( A )) ≥ ν ( A ) . A map ν : P ( X ) → N ∪ { + ∞} satisfying (1)-(4) is called an index function . 3

Method of proof of the Lusternik-Schnirelmann theorem f ∈ C 1 , 1 ( M ) bounded from below and satisfies PS condition w.r.t. Finsler metric on M . Put f a := f − 1 (( −∞ , a ]) . Use properties (1)-(4) and minimax methods to show: • If [ a , b ] contains no critical value of f , then cat M ( f b ) = cat M ( f a ) . • If c is a critical value of f , then cat M ( f c ) ≤ cat M ( f c − ε ) + cat M ( Crit f ∩ f − 1 ( { c } )) . Combining these observations yields cat M ( f a ) ≤ # ( Crit f ∩ f a ) ∀ a ∈ R and finally the theorem. 4

Lusternik-Schnirelmann and closed geodesics Let M be a closed manifold, F : TM → [ 0 , + ∞ ) be a Finsler � metric (e.g. F ( x , v ) = g x ( v , v ) for g Riemannian metric), � 1 γ ( t )) 2 dt . E F : Λ M := H 1 ( S 1 , M ) → R , E F ( γ ) = F ( γ ( t ) , ˙ 0 Then E F is C 1 , 1 and satisfies PS condition (Mercuri, ’77) with Crit E F = { closed geodesics of F } ∪ { constant loops } . Q: Can we use Lusternik-Schnirelmann theory to obtain lower bounds on # { non-constant closed geodesics of F } ? 5

Problems with the LS-approach and closed geodesics There are problems: • Since { constant loops } ⊂ Crit E F , it holds for each a ≥ 0 that #( Crit E F ∩ Λ M a ) = + ∞ . • cat Λ M ( { constant loops } ) =? • Critical points of E F come in S 1 -orbits, but cat Λ M ( S 1 · γ ) ∈ { 1 , 2 } for each γ ∈ Λ M . Idea: Replace cat Λ M : P (Λ M ) → N ∪ { + ∞} by a different index function. 6

Spherical complexities

Definition of spherical complexities (M., 2019) Let X top. space, n ∈ N 0 , B n + 1 X := C 0 ( B n + 1 , X ) , S n X := { f ∈ C 0 ( S n , X ) | f is nullhomotopic } . Definition • Let A ⊂ S n X . A sphere filling on A is a continuous map s : A → B n + 1 X with s ( γ ) | S n = γ for all γ ∈ A . • For A ⊂ S n X put � � � SC n , X ( A ) := inf r ∈ N � ∃ U 1 , . . . , U r ⊂ S n X open and sphere fillings r � � s j : U j → B n + 1 X ∀ j and A ⊂ U j ∈ N ∪ {∞} . j = 1 Call SC n ( X ) := SC n , X ( S n X ) the n-spherical complexity of X . Remark SC 0 ( X ) = TC ( X ) , the topological complexity of X . 7

Properties of spherical complexities (1) In the following, let X be a metrizable ANR (e.g. a locally finite CW complex). Proposition SC n , X : P ( S n X ) → N ∪ { + ∞} is an index function on S n X. Proposition Let c n : X → S n X, ( c n ( x ))( p ) = x for all p ∈ S n , x ∈ X. Then SC n , X ( c n ( X )) = 1 . Proof. Define a sphere filling s : c n ( X ) → B n + 1 X by s ( c n ( x )) = ( B n + 1 → X , p �→ x ) ∀ x ∈ X , extend continuously to an open neighborhood. 8

Properties of spherical complexities (2) Let X be a metrizable ANR. Consider the left O ( n + 1 ) -actions on S n X and B n + 1 X by reparametrization, i.e. ( A · γ )( p ) = γ ( A − 1 p ) ∀ γ ∈ S n X , A ∈ O ( n + 1 ) , p ∈ S n . Proposition Let G ⊂ O ( n + 1 ) be a closed subgroup and γ ∈ S n X and let G γ denote its isotropy group. If G γ is trivial or n = 1, then SC n , X ( G · γ ) = 1 . If G γ trivial, take β : B n + 1 C 0 Proof → X with β | S n = γ , put s : G · γ → B n + 1 X , s ( A · γ ) = A · β ∀ A ∈ G . = Z k for k ∈ N , s.t. γ = α k for some α ∈ S 1 X , If n = 1 and G ∼ take β ∈ B 2 X with β | S 1 = α and define s : G · γ → B 2 X by s ( A · γ ) = A · ( β ◦ p k ) , where p k : B 2 → B 2 , z �→ z k . Extend to open nbhd. of G · γ . 9

A Lusternik-Schnirelmann-type theorem for SC n Theorem (M., 2019) Let G ⊂ O ( n + 1 ) be a closed subgroup, M ⊂ S n X be a G-invariant Hilbert manifold, f ∈ C 1 , 1 ( M ) be G-invariant. Let ν ( f , λ ) := # { non-constant G-orbits in Crit f ∩ f λ } . If • f satisfies the Palais-Smale condition w.r.t. a complete Finsler metric on M , • f is constant on c n ( X ) , • G acts freely on Crit f ∩ f λ or n = 1 , then SC n , X ( f λ ) ≤ ν ( f , λ ) + 1 . 10

Consequences for closed geodesics Corollary Let M be a closed manifold, F : TM → [ 0 , + ∞ ) be a Finsler metric and λ ∈ R . Let E F : H 1 ( S 1 , M ) ∩ S 1 M → R be the restriction of the energy functional of F. Let ν ( F , λ ) be the number of SO ( 2 ) -orbits of non-constant contractible closed geodesics of F of energy ≤ λ . Then ν ( F , λ ) ≥ SC 1 , M ( E λ F ) − 1 . If F is reversible, e.g. induced by a Riemannian metric, the same holds for the number of O ( 2 ) -orbits of contractible closed geodesics. Remark The counting does not distinguish iterates of the same prime closed geodesic. 11

Closed geodesics of Finsler metrics Definition A Finsler metric on a manifold M is a map F : TM C 0 → [ 0 , + ∞ ) , such that F | TM \{ zero-section } is C ∞ and • F x ( λ v ) = λ F x ( v ) ∀ ( x , v ) ∈ TM , λ ≥ 0, • F x ( v ) = 0 v = 0 ∈ T x M , ⇔ • D 2 F x is positive definite for all x ∈ M . The reversibility of F is λ := sup { F x ( − v ) | F x ( v ) = 1 } . F is reversible if λ = 1, i.e. if F x ( − v ) = F x ( v ) for all ( x , v ) ∈ TM . A closed geodesic of F is a critical point of � 1 γ ( t )) 2 dt . E F : Λ M := H 1 ( S 1 , M ) → R , E F ( γ ) = 0 F ( γ ( t ) , ˙ Closed geodesics occur in SO ( 2 ) -orbits, if reversible, then in O ( 2 ) -orbits. Iterates are again closed geodesics. 12

Results on closed geodesics Theorem (Lusternik-Fet, ’51, for Riemannian manifolds) Every Finsler metric on a closed manifold admits a non-constant closed geodesic. Definition γ 1 , γ 2 : S 1 → X are geometrically distinct if γ 1 ( S 1 ) � = γ 2 ( S 1 ) . They are called positively distinct if they are either geom. distinct or ∃ A ∈ O ( 2 ) \ SO ( 2 ) with γ 1 = A · γ 2 . Existence results for closed geodesics: • Bangert-Long, 2007: every Finsler metric on S 2 has two positively distinct ones • Rademacher, 2009: every bumpy Finsler metric on S n has two positively distinct ones • etc., Long-Duan 2009 for S 3 , Wang 2019 for pinched metrics on S n , ... 13

New result using spherical complexities Theorem (M., 2019) Let M be a closed oriented 2 n-dimensional manifold, n ≥ 3 . Assume that ∃ x ∈ H k ( M ; Q ) , 2 ≤ k < n, with x 2 � = 0 (e.g. C P n ) . Let F : TM → [ 0 , + ∞ ) be a Finsler metric of reversibility λ whose flag curvature satisfies � � 2 � � 1 1 λ < K ≤ 1 , i.e. if F reversible: 16 < K ≤ 1 , 4 1 + λ then F admits two positively distinct closed geodesics (geometrically distinct if F is reversible). F = E − 1 Find a > 0 such that E a Idea of proof F (( −∞ , a ]) contains only prime closed geodesics and such that SC 1 , M ( E a F ) ≥ 3. 14

Lower bounds for spherical complexities

Lower bounds for spherical complexities Aim Find "computable" lower bounds on SC n , X ( A ) . Method Put spherical complexities in a bigger framework and use results by A. S. Schwarz from a more general context. 15

Spherical complexities and sectional category Definition (A. Schwarz, ’62) Let p : E → B be a fibration. The sectional category or Schwarz genus of p is given by n � � � � secat ( p ) = inf n ∈ N U j = B open cover , � ∃ j = 1 � C 0 s j : U j → E , p ◦ s j = incl U j ∀ j . In our setting: � � SC n ( X ) = secat r n : B n + 1 X → S n X , γ �→ γ | S n , � � n ( A ) : r − 1 SC n , X ( A ) ≥ secat r n | r − 1 n ( A ) → A ∀ A ⊂ S n X . 16

Sectional categories and cup length Given X top. space, R commutative ring, I ⊂ H ∗ ( X ; R ) an ideal, let cl ( I ) := sup { r ∈ N | ∃ u 1 , . . . , u r ∈ I ∩ ˜ H ∗ ( X ; R ) s.t. u 1 ∪· · ·∪ u r � = 0 } . Theorem (Lusternik-Schnirelmann ’34) cat ( X ) ≥ cl ( H ∗ ( X ; R )) + 1 . Theorem (A. Schwarz, ’62) Let p : E → B be a fibration. Then � � ker [ p ∗ : H ∗ ( B ; R ) → H ∗ ( E ; R )] secat ( p ) ≥ cl + 1 . 17