Spherical complexities and closed geodesics

Stephan Mescher(Universität Leipzig)19 September 2020

BIRS Online Workshop "Topological Complexity and Motion Planning"

L-S category and critical points

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).

Remark

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

1

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 of catM(·) 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.

2

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→ R, EF(γ) =

∫ 1

0F(γ(t), γ(t))2 dt.

Here, ΛM := H1(S1,M)'⊂ C0(S1,M) is a Hilbert manifold locally

modelled on H1(S1,RdimM) = W1,2(S1,RdimM).

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

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

Q: Can we use Lusternik-Schnirelmann theory to obtainlower bounds on#{geometrically distinct non-constant closed geodesics of F}?

3

Problems with the LS-approach and closed geodesics

There are problems:

• Since {constant loops} ⊂ Crit EF withEF(const. loop) = 0, it holds for each a ≥ 0 that#((Crit EF) ∩ EaF) = +∞.

• 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�erentfunction with similar properties.

There are several similar approaches to G-invariant functions,e.g. by Clapp-Puppe, Bartsch et al.

4

Spherical complexities

De�nition of spherical complexities (M., 2019)

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{r ∈ N

∣∣∣ ∃ r⋃j=1

Uj = B open cover, sj : UjC0→ E, p◦sj = inclUj ∀j

}.

Let X be a top. space, n ∈ N0, put Bn+1X := C0(Bn+1, X),SnX := {f ∈ C0(Sn, X) | f is nullhomotopic}.De�nition

The n-spherical complexity of X is given by

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

For A ⊂ SnX de�ne SCn,X(A) := secat(rn|r−1n (A) : r−1n (A)→ A).

5

Properties of spherical complexities

Remark SC0(X) = TC(X), the topological complexity of X.

In the following, let X be a metrizable ANR (e.g. a locally �niteCW complex).Proposition

Let cn : X → SnX, (cn(x))(p) = x for all p ∈ Sn, x ∈ X. ThenSCn,X(cn(X)) = 1.

Consider the left O(n+ 1)-actions on SnX and Bn+1X byreparametrization, 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.

6

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 ,a) := #{non-constant G-orbits in Crit f ∩ fa}.

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 ) ∩ fa or n = 1,

thenSCn,X(fa) ≤ ν(f ,a) + 1.

7

Consequences for closed geodesics

Corollary

Let M be a closed manifold, F : TM→ [0,+∞) be a Finslermetric and a ∈ R. Let EF : ΛM ∩ S1M→ R be the restriction ofthe energy functional of F.

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

ν(F,a) ≥ SC1,M(EaF)− 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. 8

Lower bounds for sphericalcomplexities

Sectional categories and cup length

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

For 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 6= 0}.

Theorem (A. Schwarz, ’62)

Let p : E→ B be a �bration. Then

secat(p) ≥ cl(ker [p∗ : H∗(B;R)→ H∗(E;R)]

)+ 1.

9

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, cn(X);R) might be eitherhard to compute or not that interesting. (E.g. the cup producton H∗(LS2, c1(S2); Q) vanishes.)

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

10

Sectional category and �berwise joins

Given �brations p : E→ B, p′ : E′ → B, let

p ∗ p′ : E ∗f E′ → B

denote the �berwise join of p and p′. The �ber over each b ∈ Bis (E ∗f E′)b = Eb ∗ E′b. (∗ = topological join)

Let p : E→ B be a �bration. De�ne �brations pr : Er → B,r ∈ N, recursively by

p1 = p, E1 = E, pr = p ∗ pr−1, Er = E ∗f Er−1.

Theorem (A. Schwarz, ’62)

secat(p) = inf{r ∈ N | ∃s : B C0→ Er with pr ◦ s = idB}.

11

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{r ∈ N0 | p∗ru = 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).

Thus, if k := cl(ker p∗) and u1, . . . ,uk ∈ ker p∗ withu1 ∪ · · · ∪ uk 6= 0, then

secat(p) ≥k∑j=1

wgtp(uj) + 1 ≥ cl(ker p∗) + 1.

12

Construction of classes of weight ≥ 1 for rn(γ) = γ|Sn

Want to �nd classes in ker [r∗n : H∗(SnX;R)→ H∗(Bn+1X;R)].

Lemma

Let X be a top. space, R be a commutative ring and

evn : SnX × Sn → X, (α,p) 7→ α(p).

For k ≥ n let

Zn : Hk(X;R)→ Hk−n(SnX;R), Zn(σ) = ev∗nσ/[Sn],

where ·/· denotes the slant product. Then Zn(σ) ∈ ker r∗n.

Remark If n = 1 and X is simply connected, thenZ1 : H∗(X;R)→ H∗−1(LX;R) is injective. (Jones ’87)

13

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

14

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 ∈ ker p∗, thenp∗2u ∈ imδ. Try to �nd αu ∈ Hk−1(Q;R) with

δ(αu) = p∗2u,

�nd conditions that imply αu = 0.

15

Construction of classes of weight ≥ 2, cont.

Back to our setting: Here p = rn : Bn+1X → SnX, γ 7→ γ|Sn , andthe pullback is

Q = {(γ1, γ2) ∈ (Bn+1X)2 | γ1|Sn = γ2|Sn} ≈ C0(Sn+1, X),

hence for E2 := Bn+1X ∗f Bn+1X the Mayer-Vietoris sequencehas the form

· · · → Hk−1(C0(Sn+1, X);R)δ→ Hk(E2;R)→ ⊕2i=1H

k(Bn+1X;R)→ . . .

Q: If u ∈ Hk(X;R), Zn(u) 6= 0 ∈ Hk−n(SnX;R), what is

αZn(u) ∈ Hk−n−1(C0(Sn+1, X);R) ?

16

Construction of classes of weight ≥ 2, cont.

Lemma

Let u ∈ Hk(X; Q), k ≥ n+ 1. Then

αZn(u) = e∗n+1u/[Sn+1],

where en+1 : C0(Sn+1, X)× Sn+1 → X, en+1(γ,p) = γ(p).

Use lemma and Mayer-Vietoris sequence to show:

Theorem

Let u ∈ Hk(X; Q) with Zn(u) 6= 0. If f ∗u = 0 for allf : Sn+1 × P C0→ X, where P is any closed oriented manifold withdimP = k− n− 1, then

wgt(Zn(u)) ≥ 2.

17

Consequences for topological complexity (the case n = 0)

Theorem (Grant-M. ’18, M. ’19)

Let M be a closed or. manifold with dimM ≥ 3. If there existsu ∈ H2(M; Q) with f ∗u = 0 for all f ∈ C0(T2,M), then TC(M) ≥ 6.

Theorem (M. ’19)

Let M be an even-dim. closed or. manifold. If TC(M) ≤ 4, then Mis dominated by a manifold P× S1, i.e. there exists a degree-1map P× S1 → M. Here, P is a closed or. manifold withdimP = dimM− 1.

Proof.

Assume M is not dominated by ... Let n := dimM,u 6= 0 ∈ Hn(M; Q), u := 1× u− u× 1 ∈ Hn(M×M; Q). Thenwgt(u) ≥ 2 by the assumptions. Since n is even,u2 = −2u× u 6= 0, hence wgt(u2) ≥ 4, so TC(M) ≥ 5. 18

Results on closed geodesics

Overview

Theorem (Lusternik-Fet, ’51, for Riemannian manifolds)

Every Finsler metric on a closed manifold admits anon-constant closed geodesic.

De�nition Two closed geodesics γ1, γ2 : S1 → X are positivelydistinct if either γ1(S1) 6= γ2(S1) or ∃A ∈ O(2) \ SO(2) withγ1 = A · γ2.

• 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, ...

19

New results using spherical complexities

Theorem (M., 2020)

Let M be a closed oriented manifold, F : TM→ [0,+∞) be aFinsler metric of reversibility λ and �ag curvature K. Let `F > 0be the length of the shortest non-const. closed geodesic of F.

a) If M = S2d, d ≥ 2, 0 < K ≤ 1 and F ≤ 1+λλ

√g1, then Fadmits two pos. distinct closed geodesics of length < 2`F .(g1 = round metric of constant curvature 1)

b) If M = CPn or M = HPn, n ≥ 3, 0 < K ≤ 1 and F ≤ 1+λλ

√g1,then ∃ two pos. distinct closed geodesics of length < 2`F .

c) If M = S2d+1, d ∈ N, λ2

(1+λ)2 < K ≤ 1 and F ≤ (k+1)(1+λ)mλ

√g1,then F admits d 2mk e pos. distinct closed geodesics oflength < (k+ 1)`F.

20

Method of proof of results for closed geodesics

Let ιa : EaF ↪→ ΛM be the inclusion. For parts a) and b):

• If γ is a closed geodesic of length `F , then its iteratessatisfy EF(γk) ≥ EF(γ2) = 4`2F ∀k ≥ 2. Thus, if a < 4`2F andν(F,a) ≥ 2, then EaF contains two distinct closedgeodesics.

• If u ∈ H∗(ΛM;R) satis�es ι∗au 6= 0, then ν(F,a) ≥ wgt(u).Thus, it su�ces to �nd such u with wgt(u) ≥ 2 andι∗au 6= 0, where a < 4`2F.

• For classes of �xed degree, positive curvature boundsprovide such energy bounds.

21

Perspectives and possible applications

• Equivariant versions, use richer ring structure inH∗S1(LM,M; Q)

• Applicable in greater generality to periodic Reeb orbitson contact manifolds? (generalizing closed geodesics onT1M)

• Higher-dimensional applications, i.e. for SCn,M if n > 1?• Any ideas? Contact me!

22

Spherical complexities and closed geodesics

Thank you for your attention!

talk based on:

S. Mescher, Spherical complexities, with applications to closedgeodesics, arXiv:1911.03948, to appear in Algebr. Geom. Topol.

S. Mescher, Existence results for closed geodesics viaspherical complexities, to appear in Calc. Var. 59, 2020.

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

mescher@math.uni-leipzig.de