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 catX(A) := inf

• r ∈ N
• ∃U1, . . . , Ur ⊂ X open,

s.t. Uj ֒ → X nullhomotopic ∀j and A ⊂

r

• j=1

Uj

• .

cat(X) := catX(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 ∈ C1,1(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 catX

Proposition Let X be a normal ANR. Put ν(A) := catX(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 Φt : A → X, t ∈ [0, 1], is a deformation, then ν(Φ1(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 ∈ C1,1(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

catM(f b) = catM(f a).

• If c is a critical value of f, then

catM(f c) ≤ catM(f c−ε) + catM(Crit f ∩ f −1({c})). Combining these observations yields catM(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) =

• gx(v, v) for g Riemannian metric),

EF : ΛM := H1(S1, M) → R, EF(γ) = 1 F(γ(t), ˙ γ(t))2 dt. Then EF is C1,1 and satisfies PS condition (Mercuri, ’77) with Crit EF = {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 EF, it holds for each a ≥ 0

that #(Crit EF ∩ ΛMa) = +∞.

• catΛM({constant loops}) =?
• Critical points of EF come in S1-orbits, but

catΛM(S1 · γ) ∈ {1, 2} for each γ ∈ ΛM. Idea: Replace catΛM : P(ΛM) → N ∪ {+∞} by a different index function.

6

Spherical complexities

Definition of spherical complexities (M., 2019)

Let X top. space, n ∈ N0, Bn+1X := C0(Bn+1, X), SnX := {f ∈ C0(Sn, X) | f is nullhomotopic}. Definition

• Let A ⊂ SnX. A sphere filling on A is a continuous map

s : A → Bn+1X with s(γ)|Sn = γ for all γ ∈ A.

• For A ⊂ SnX put

SCn,X(A) := inf

• r ∈ N
• ∃U1, . . . , Ur ⊂ SnX open and sphere fillings

sj : Uj → Bn+1X ∀j and A ⊂

r

• j=1

Uj

• ∈ N ∪ {∞}.

Call SCn(X) := SCn,X(SnX) the n-spherical complexity of X. Remark SC0(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 SCn,X : P(SnX) → N ∪ {+∞} is an index function on SnX. Proposition Let cn : X → SnX, (cn(x))(p) = x for all p ∈ Sn, x ∈ X. Then SCn,X(cn(X)) = 1. Proof. Define a sphere filling s : cn(X) → Bn+1X by s(cn(x)) = (Bn+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

• n SnX and Bn+1X by reparametrization, i.e.

(A · γ)(p) = γ(A−1p) ∀γ ∈ SnX, A ∈ O(n + 1), p ∈ Sn. Proposition Let G ⊂ O(n + 1) be a closed subgroup and γ ∈ SnX and let Gγ denote its isotropy group. If Gγ is trivial or n = 1, then SCn,X(G · γ) = 1. Proof If Gγ trivial, take β : Bn+1 C0 → X with β|Sn = γ, put s : G · γ → Bn+1X, s(A · γ) = A · β ∀A ∈ G. If n = 1 and G ∼ = Zk for k ∈ N, s.t. γ = αk for some α ∈ S1X, take β ∈ B2X with β|S1 = α and define s : G · γ → B2X by s(A · γ) = A · (β ◦ pk), where pk : B2 → B2, z → zk. Extend to open nbhd. of G · γ.

9

A Lusternik-Schnirelmann-type theorem for SCn

Theorem (M., 2019) Let G ⊂ O(n + 1) be a closed subgroup, M ⊂ SnX be a G-invariant Hilbert manifold, f ∈ C1,1(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 cn(X),
• G acts freely on Crit f ∩ f λ or n = 1,

then SCn,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 EF : H1(S1, M) ∩ S1M → 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) ≥ SC1,M(Eλ

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 C0 → [0, +∞), such that F|TM\{zero-section} is C∞ and

• Fx(λv) = λFx(v) ∀(x, v) ∈ TM, λ ≥ 0,
• Fx(v) = 0

⇔ v = 0 ∈ TxM,

• D2Fx is positive definite for all x ∈ M.

The reversibility of F is λ := sup{Fx(−v) | Fx(v) = 1}. F is reversible if λ = 1, i.e. if Fx(−v) = Fx(v) for all (x, v) ∈ TM. A closed geodesic of F is a critical point of EF : ΛM := H1(S1, M) → R, EF(γ) = 1

0 F(γ(t), ˙

γ(t))2 dt. 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 γ1, γ2 : S1 → X are geometrically distinct if γ1(S1) = γ2(S1). They are called positively distinct if they are either geom. distinct or ∃A ∈ O(2) \ SO(2) with γ1 = A · γ2. Existence results for closed geodesics:

• Bangert-Long, 2007: every Finsler metric on S2 has two

positively distinct ones

• Rademacher, 2009: every bumpy Finsler metric on Sn has

two positively distinct ones

• etc., Long-Duan 2009 for S3, Wang 2019 for pinched

metrics on Sn, ...

13

New result using spherical complexities

Theorem (M., 2019) Let M be a closed oriented 2n-dimensional manifold, n ≥ 3. Assume that ∃x ∈ Hk(M; Q), 2 ≤ k < n, with x2 = 0 (e.g. CPn). Let F : TM → [0, +∞) be a Finsler metric of reversibility λ whose flag curvature satisfies 1 4

• λ

1 + λ 2 < K ≤ 1,

• i.e. if F reversible:

1 16 < K ≤ 1,

• then F admits two positively distinct closed geodesics

(geometrically distinct if F is reversible). Idea of proof Find a > 0 such that Ea

F = E−1 F ((−∞, a])

contains only prime closed geodesics and such that SC1,M(Ea

F) ≥ 3. 14

Lower bounds for spherical complexities

Lower bounds for spherical complexities

Aim Find "computable" lower bounds on SCn,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 secat(p) = inf

• n ∈ N
• ∃

n

• j=1

Uj = B open cover, sj : Uj

C0

→ E, p ◦ sj = inclUj ∀j

• .

In our setting: SCn(X) = secat

• rn : Bn+1X → SnX, γ → γ|Sn
• ,

SCn,X(A) ≥ secat

• rn|r−1

n (A) : r−1

n (A) → A

• ∀A ⊂ SnX.

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 | ∃u1, . . . , ur ∈ I∩ ˜ H∗(X; R) s.t. u1∪· · ·∪ur = 0}. Theorem (Lusternik-Schnirelmann ’34) cat(X) ≥ cl(H∗(X; R)) + 1. Theorem (A. Schwarz, ’62) Let p : E → B be a fibration. Then secat(p) ≥ cl

• ker [p∗ : H∗(B; R) → H∗(E; R)]
• + 1.

17

Consequences for spherical complexities

The previous theorem, some work and the long exact cohomology sequence of (SnX, cn(X)) yield: Theorem Let A ⊂ SnX and let ι : (A, ∅) ֒ → (SnX, cn(X)) be the inclusion of

• pairs. Then

SCn,X(A) ≥ cl

• im [ι∗ : H∗(SnX, cn(X); R) → H∗(A; R)]
• + 1.

Problem The cup product on H∗(SnX, X; R) might be either hard to compute or not that interesting. (E.g. the cup product

• n H∗(LS2, S2; Q) vanishes.)

Idea Improve cup length bounds by associating N-valued weights to cohomology classes.

18

Sectional category and fiberwise joins

Given fibrations f1 : E1 → B, f2 : E2 → B, let f1 ∗ f2 : E1 ∗f E2 → B denote the fiberwise join of p1 and p2. The fiber over each b ∈ B is (E1 ∗f E2)b = (E1)b ∗ (E2)b. Let p : E → B be a fibration. Define fibrations pn : En → B, n ∈ N, recursively by p1 = p, E1 = E, pn = p ∗ pn−1, En = E ∗f En−1. Theorem (A. Schwarz, ’62) secat(p) = inf{n ∈ N | ∃s : B C0 → En with pn ◦ s = idB.}

19

Sectional category weights

Let p : E → B be a fibration, R be a commutative ring. Definition (Farber-Grant 2007; Fadell-Husseini ’92, Rudyak ’99) Let u ∈ H∗(B; R), u = 0. The weight of u with respect to p is given by wgtp(u) := sup{n ∈ N0 | p∗

nu = 0}.

Properties: Let u, v ∈ H∗(B; R) with u = 0, v = 0.

• If wgtp(u) ≥ k, then secat(p) ≥ k + 1.
• wgtp(u ∪ v) ≥ wgtp(u) + wgtp(v).
• If f : X C0

→ B with f ∗u = 0, then wgtf ∗p(f ∗u) ≥ wgtp(u). Thus, can use weights to improve cup length bounds: if k := cl(ker p∗) and u1, . . . , uk ∈ ker p∗ with u1 ∪ · · · ∪ uk = 0, then secat(p) ≥

k

• j=1

wgtp(uj) + 1 ≥ cl(ker p∗) + 1.

20

Consequences for the proof of the main result

Let (M, F) be a Finsler manifold, a > 0 and let ιa : Ea

F ֒

→ ΛM be the inclusion. From the above properties we obtain: if u ∈ H∗(ΛM, c1(M); R) satisfies ι∗

au = 0, then

ν(F, a) ≥ wgt(u) := wgtr1(u). Thus, it suffices to find such u with wgt(u) ≥ 2 for sufficiently small a.

21

Construction of classes of weight ≥ 1

Lemma Let X be a simply connected top. space and R be a commutative ring. Let LX = C0(S1, X) and ev : LX × S1 → X, (α, t) → α(t). Then for k ≥ 2, Z : Hk(X; R) → Hk−1(LX; R), Z(σ) = ev∗σ/[S1] is injective and wgt(Z(σ)) ≥ 1 for all σ = 0 ∈ H∗(X; R). (Here, ·/· denotes the slant product.)

22

Construction of classes of weight ≥ 2

(Generalization of methods from Grant-M. 2018) If p : E → B is a fibration, then p2 : E ∗f E → B is constructed as a homotopy pushout of a pullback (double mapping cylinder): pullback homotopy pushout Q

f2

• f1
• E

p

• E

p

B

Q

f2

• f1
• E
• p
• E
• p
• E ∗f E

p2

B

23

Construction of classes of weight ≥ 2, cont.

As a homotopy pushout of a pullback, it has a Mayer-Vietoris sequence: . . .

Hk−1(Q; R)

δ

Hk(E ∗f E; R) ⊕2

i=1Hk(E; R)

. . .

Hk(B; R)

p∗

2

• p∗⊕p∗
• Want to find u ∈ Hk(B; R) with p∗

2u = 0. If u lies in ker p∗, then

p∗

2u ∈ imδ. Try to find αu ∈ Hk−1(Q; R) with δ(αu) = p∗ 2u, find

conditions that imply αu = 0.

24

Construction of classes of weight ≥ 2, cont.

Back to our setting: Here p = r1 : B2X → S1X, γ → γ|S1, and the pullback is Q = {(γ1, γ2) ∈ (B2X)2 |r1(γ1) = r1(γ2)} = {(γ1, γ2) ∈ (C0(B2, X))2 | γ1|S1 = γ2|S1} ≈ C0(S2, X), hence for E2 := B2X ∗f B2X the Mayer-Vietoris sequence has the form · · · → Hk−1(C0(S2, X); Q)

δ

→ Hk(E2; Q) → Hk(B2X; Q)⊕Hk(B2X; Q) → . . . Lemma Let u = 0 ∈ Hk(X; Q), k ≥ 2. Then p∗

2(Z(u)) = δ(au), where

au ∈ Hk−2(C0(S2, X); Q) is given by au = e∗

2u/[S2],

e2 : C0(S2, X) × S2 → X, e2(γ, p) := γ(p).

25

Construction of classes of weight ≥ 2, cont.

Theorem Let u = 0 ∈ Hk(X; Q), k ≥ 3. If f ∗u = 0 for all f : S2 × P C0 → X, where P is any closed oriented (k − 2)-manifold, then wgt(Z(u)) ≥ 2. Proof. By the previous lemma, wgt(Z(u)) ≥ 2 if e∗

2u/[S2] = 0.

But since H∗(C0(S2, X); Q) is representable, this will hold if for all closed or. (k − 2)-manifolds P and g : P C0 → C0(S2, X) it holds that

• e∗

2u/[S2], g∗[P]

• = 0

⇔ < f ∗u, [S2 × P] >= 0, where f(p, x) = e2(g(x), p). The claim immediately follows.

26

Sketch of proof of the main result

Reminder By assumption, x ∈ Hk(M; Q) with 2k < dim M and x2 = 0. M even-dim., F positive curv. ⇒ π1(M) = 0 ⇒ Z(x2) = 0. If f ∗(x2) ! = 0 for all f : S2 × P C0 → M, dim P = 2k − 2, then wgt(Z(x2)) ≥ 2. But this is clear since f ∗(x2) = (f ∗x)2 ∈ H2k(S2 × P; Q) ∼ = H2(S2; Q) ⊗ H2k−2(P; Q), which can not contain nontrivial squares in degree 2k. Remains to show for u := Z(x2) that ι∗

au = 0 for some a > 0

for which Ea

F contains only prime geodesics. 27

Sketch of proof of the main result, cont.

Reminder By assumption, 1

4( λ 1+λ)2 < δ ≤ K ≤ 1.

Let γ0 be a non-constant closed geodesic of F of shortest length ℓ0.

• Since K ≤ 1, ℓ0 > π 1+λ

λ

by an injectivity radius estimate.

• Since K ≥ δ and deg u < dim M − 1, it follows by

comparison arguments (Ballmann-Thorbergsson-Ziller ’82 for Riemannian metrics, Rademacher 2004 for Finsler) that ι∗

au = 0 for a := π2 δ , hence ν(F, a) ≥ wgt(u) = 2.

• One derives from δ > 1

4( λ 1+λ)2 and the bound for ℓ0 that

EF(γ2

0) > a.

⇒ Ea

F contains two positively distinct closed geodesics.

(geometrically distinct, if F reversible).

28

Perspectives and possible applications

• Equivariant versions, use richer ring structure in

H∗

S1(LM, M; Q) (−

→ work in progress)

• Apply the methods to more general flows having

Lyapunov functions.

• Applicable in greater generality to Reeb orbits on contact

manifolds? (generalizing closed geodesics on T1M)

• Higher-dimensional applications, i.e. for SCn,M if n > 1?

29

Spherical complexities and closed geodesics

Thank you for your attention!

talk based on:

• S. Mescher, Spherical complexities, with applications to

closed geodesics, arXiv:1911.03948 (slides at http://www.math.uni-leipzig.de/∼mescher)