Spherical complexities and closed geodesics

Stephan Mescher(Mathematisches Institut, Universität Leipzig)5 February 2020

Lusternik-Schnirelmann categoryand critical points

The Lusternik-Schnirelmann category of a space

De�nition

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=1Uj}.

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 boundedfrom below and satisfy the Palais-Smale condition withrespect to a complete Finsler metric on M. Then

#Crit f ≥ cat(M).

• There are various generalisations, e.g. generalizedPalais-Smale conditions (Clapp-Puppe ’86), extensions to�xed points of self-maps (Rudyak-Schlenk ’03).

• Advantage over Morse inequalities: No nondegeneracycondition 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 anindex function.

3

Method of proof of the Lusternik-Schnirelmann theorem

f ∈ C1,1(M) bounded from below and satis�es PS conditionw.r.t. Finsler metric on M. Put fa := f−1((−∞,a]). Useproperties (1)-(4) and minimax methods to show:

• If [a,b] contains no critical value of f , then

catM(fb) = catM(fa).

• 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(fa) ≤ # (Crit f ∩ fa) ∀a ∈ R

and �nally the theorem. 4

Lusternik-Schnirelmann and closed geodesics

Let M be a closed manifold, F : TM→ [0,+∞) be a Finslermetric (e.g. F(x, v) =

√gx(v, v) for g Riemannian metric),

EF : ΛM := H1(S1,M)→ R, EF(γ) =

∫ 1

0F(γ(t), γ̇(t))2 dt.

Then EF is C1,1 and satis�es PS condition (Mercuri, ’77) with

Crit EF = {closed geodesics of F} ∪ {constant loops}.

Q: Can we use Lusternik-Schnirelmann theory to obtainlower 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 ≥ 0that #(Crit EF ∩ ΛMa) = +∞.

• catΛM({constant loops}) =?

• Critical points of EF come in S1-orbits, butcatΛM(S1 · γ) ∈ {1, 2} for each γ ∈ ΛM.

Idea: Replace catΛM : P(ΛM)→ N ∪ {+∞} by a di�erentindex function.

6

Spherical complexities

De�nition 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}.De�nition

• Let A ⊂ SnX. A sphere �lling on A is a continuous maps : 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 �llingssj : Uj → Bn+1X ∀j and A ⊂

r⋃j=1Uj}∈ 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 �niteCW 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.

De�ne a sphere �lling s : cn(X)→ Bn+1X by

s(cn(x)) = (Bn+1 → X, p 7→ 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)-actionson 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 andγ ∈ SnX and let Gγ denote its isotropy group. If Gγ is trivial orn = 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 de�ne s : G · γ → B2X by

s(A · γ) = A · (β ◦ pk),

where pk : B2 → B2, z 7→ 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 aG-invariant Hilbert manifold, f ∈ C1,1(M) be G-invariant. Let

ν(f , λ) := #{non-constant G-orbits in Crit f ∩ fλ}.

If

• f satis�es the Palais-Smale condition w.r.t. a completeFinsler metric onM,

• f is constant on cn(X),• G acts freely on Crit f ∩ fλ or n = 1,

thenSCn,X(fλ) ≤ ν(f , λ) + 1.

10

Consequences for closed geodesics

Corollary

Let M be a closed manifold, F : TM→ [0,+∞) be a Finslermetric and λ ∈ R. Let EF : H1(S1,M) ∩ S1M→ R be therestriction of the energy functional of F.

Let ν(F, λ) be the number of SO(2)-orbits of non-constantcontractible closed geodesics of F of energy ≤ λ. Then

ν(F, λ) ≥ SC1,M(EλF )− 1.

If F is reversible, e.g. induced by a Riemannian metric, thesame holds for the number of O(2)-orbits of contractibleclosed geodesics.

Remark The counting does not distinguish iterates of thesame prime closed geodesic. 11

Closed geodesics of Finsler metrics

De�nition A Finsler metric on a manifold M is a mapF : 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 de�nite 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. Aclosed geodesic of F is a critical point of

EF : ΛM := H1(S1,M)→ R, EF(γ) =∫ 10 F(γ(t), γ̇(t))2 dt.

Closed geodesics occur in SO(2)-orbits, if reversible, then inO(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 anon-constant closed geodesic.

De�nition γ1, γ2 : S1 → X are geometrically distinct ifγ1(S1) 6= γ2(S1). They are called positively distinct if they areeither 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 twopositively distinct ones

• Rademacher, 2009: every bumpy Finsler metric on Sn hastwo positively distinct ones

• etc., Long-Duan 2009 for S3, Wang 2019 for pinchedmetrics 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 6= 0 (e.g. CPn).Let F : TM→ [0,+∞) be a Finsler metric of reversibility λwhose �ag curvature satis�es

14

(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 EaF = E−1F ((−∞,a])

contains only prime closed geodesics and such thatSC1,M(EaF) ≥ 3.

14

Lower bounds for sphericalcomplexities

Lower bounds for spherical complexities

Aim Find "computable" lower bounds on SCn,X(A).

Method Put spherical complexities in a bigger frameworkand use results by A. S. Schwarz from a more general context.

15

Spherical complexities and sectional category

De�nition (A. Schwarz, ’62)

Let p : E→ B be a �bration. The sectional category or Schwarzgenus of p is given by

secat(p) = inf{n ∈ N

∣∣∣ ∃ n⋃j=1Uj = B open cover,

sj : UjC0→ E, p ◦ sj = inclUj ∀j

}.

In our setting:

SCn(X) = secat(rn : Bn+1X → SnX, γ 7→ γ|Sn

),

SCn,X(A) ≥ secat(rn|r−1

n (A) : r−1n (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∩H̃∗(X;R) s.t. u1∪· · ·∪ur 6= 0}.Theorem (Lusternik-Schnirelmann ’34)

cat(X) ≥ cl(H∗(X;R)) + 1.

Theorem (A. Schwarz, ’62)

Let p : E→ B be a �bration. 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 exactcohomology sequence of (SnX, cn(X)) yield:Theorem

Let A ⊂ SnX and let ι : (A,∅) ↪→ (SnX, cn(X)) be the inclusion ofpairs. 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 eitherhard to compute or not that interesting. (E.g. the cup producton H∗(LS2, S2; Q) vanishes.)

Idea Improve cup length bounds by associating N-valuedweights to cohomology classes. 18

Sectional category and �berwise joins

Given �brations f1 : E1 → B, f2 : E2 → B, let

f1 ∗ f2 : E1 ∗f E2 → B

denote the �berwise join of p1 and p2. The �ber over eachb ∈ B is (E1 ∗f E2)b = (E1)b ∗ (E2)b.

Let p : E→ B be a �bration. De�ne �brations 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 �bration, R be a commutative ring.De�nition (Farber-Grant 2007; Fadell-Husseini ’92, Rudyak ’99)

Let u ∈ H∗(B;R), u 6= 0. The weight of u with respect to p isgiven by wgtp(u) := sup{n ∈ N0 | p∗nu = 0}.

Properties: Let u, v ∈ H∗(B;R) with u 6= 0, v 6= 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 6= 0, then wgtf∗p(f ∗u) ≥ wgtp(u).

Thus, can use weights to improve cup length bounds: ifk := cl(ker p∗) and u1, . . . ,uk ∈ ker p∗ with u1 ∪ · · · ∪ uk 6= 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 : EaF ↪→ ΛM bethe inclusion. From the above properties we obtain:

if u ∈ H∗(ΛM, c1(M);R) satis�es ι∗au 6= 0, then

ν(F,a) ≥ wgt(u) := wgtr1(u).

Thus, it su�ces to �nd such u with wgt(u) ≥ 2 for su�cientlysmall a.

21

Construction of classes of weight ≥ 1

Lemma

Let X be a simply connected top. space and R be acommutative ring. Let LX = C0(S1, X) and

ev : LX × S1 → X, (α, t) 7→ 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 6= 0 ∈ H̃∗(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 �bration, then p2 : E ∗f E→ B is constructed asa homotopy pushout of a pullback (double mapping cylinder):

pullback homotopy pushout

Q f2 //

f1��

Ep��

E p // B

Q f2 //

f1��

E

�� p

��

E //

p,,

E ∗f Ep2

!!B

23

Construction of classes of weight ≥ 2, cont.

As a homotopy pushout of a pullback, it has a Mayer-Vietorissequence:

. . . // Hk−1(Q;R)δ // Hk(E ∗f E;R) // ⊕2i=1H

k(E;R) // . . .

Hk(B;R)

p∗2

OOp∗⊕p∗

77

Want to �nd u ∈ Hk(B;R) with p∗2u = 0. If u lies in ker p∗, thenp∗2u ∈ imδ. Try to �nd αu ∈ Hk−1(Q;R) with δ(αu) = p∗2u, �ndconditions that imply αu = 0.

24

Construction of classes of weight ≥ 2, cont.

Back to our setting: Here p = r1 : B2X → S1X, γ 7→ γ|S1 , and thepullback 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 hasthe form

· · · → Hk−1(C0(S2, X); Q)δ→ Hk(E2; Q)→ Hk(B2X; Q)⊕Hk(B2X; Q)→ . . .

Lemma

Let u 6= 0 ∈ Hk(X; Q), k ≥ 2. Then p∗2(Z(u)) = δ(au), whereau ∈ 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 6= 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 forall closed or. (k− 2)-manifolds P and g : P C0→ C0(S2, X) it holdsthat ⟨

e∗2u/[S2],g∗[P]⟩

= 0 ⇔ < f ∗u, [S2 × P] >= 0,

where f (p, x) = e2(g(x),p). The claim immediatelyfollows.

26

Sketch of proof of the main result

Reminder By assumption, x ∈ Hk(M; Q) with 2k < dimM andx2 6= 0.

M even-dim., F positive curv. ⇒ π1(M) = 0⇒ Z(x2) 6= 0.

If f ∗(x2) != 0 for all f : S2 × P C0→ M, dimP = 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 6= 0 for some a > 0for which EaF contains only prime geodesics.

27

Sketch of proof of the main result, cont.

Reminder By assumption, 14( λ1+λ)2 < δ ≤ K ≤ 1.

Let γ0 be a non-constant closed geodesic of F of shortestlength `0.

• Since K ≤ 1, `0 > π 1+λλ by an injectivity radius estimate.• Since K ≥ δ and deg u < dimM− 1, it follows bycomparison arguments (Ballmann-Thorbergsson-Ziller ’82for Riemannian metrics, Rademacher 2004 for Finsler)that ι∗au 6= 0 for a := π2

δ , hence ν(F,a) ≥ wgt(u) = 2.• One derives from δ > 1

4( λ1+λ)2 and the bound for `0 that

EF(γ20) > a.

⇒ EaF contains two positively distinct closed geodesics.

(geometrically distinct, if F reversible).28

Perspectives and possible applications

• Equivariant versions, use richer ring structure inH∗S1(LM,M; Q) (−→ work in progress)

• Apply the methods to more general �ows havingLyapunov functions.

• Applicable in greater generality to Reeb orbits on contactmanifolds? (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 toclosed geodesics, arXiv:1911.03948

(slides at http://www.math.uni-leipzig.de/∼mescher)