Spherical complexities and closed geodesics Stephan Mescher (Universität Leipzig) September BIRS Online Workshop "Topological Complexity and Motion Planning"
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).
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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.
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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.
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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.
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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.
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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)