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
De nition For a topological space X and A ⊂ X put n
catX(A) := inf r ∈ N ∃U1,..., Ur ⊂ X open, r [ o s.t. Uj ,→ X nullhomotopic ∀j and A ⊂ Uj . j=1
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 xed 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 satis es 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 b a catM(f ) = catM(f ).
• If c is a critical value of f, then c c−ε −1 catM(f ) ≤ catM(f ) + catM(Crit f ∩ f ({c})).
Combining these observations yields a a catM(f ) ≤ #(Crit f ∩ f ) ∀a ∈ R and nally the theorem. 4 Lusternik-Schnirelmann and closed geodesics
Let M be a closed manifold, F : TM → [0, +∞) be a Finsler p metric (e.g. F(x, v) = gx(v, v) for g Riemannian metric),
Z 1 1 1 2 EF :ΛM := H (S , M) → R, EF(γ) = F(γ(t), γ˙ (t)) dt. 0
1,1 Then EF is C and satis es 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 a that #(Crit EF ∩ ΛM ) = +∞.
• catΛM({constant loops}) =? 1 • Critical points of EF come in S -orbits, but 1 catΛM(S · γ) ∈ {1, 2} for each γ ∈ ΛM.
Idea: Replace catΛM : P(ΛM) → N ∪ {+∞} by a dierent index function.
6 Spherical complexities De nition of spherical complexities (M., 2019)
0 n+1 Let X top. space, n ∈ N0, Bn+1X := C (B , X), 0 n SnX := {f ∈ C (S , X) | f is nullhomotopic}. De nition
• Let A ⊂ SnX.A sphere lling on A is a continuous map
s : A → Bn+1X with s(γ)|Sn = γ for all γ ∈ A.
• For A ⊂ SnX put n
SCn,X(A) := inf r ∈ N ∃U1,..., Ur ⊂ SnX open and sphere llings r [ o sj : Uj → Bn+1X ∀j and A ⊂ Uj ∈ N ∪ {∞}. j=1
Call SCn(X) := SCn,X(SnX) the n-spherical complexity of X.
7 Remark SC0(X) = TC(X), the topological complexity of X. Properties of spherical complexities (1)
In the following, let X be a metrizable ANR (e.g. a locally nite CW complex). Proposition
SCn,X : P(SnX) → N ∪ {+∞} is an index function on SnX. Proposition n Let cn : X → SnX, (cn(x))(p) = x for all p ∈ S , x ∈ X. Then
SCn,X(cn(X)) = 1. Proof.
De ne a sphere lling s : cn(X) → Bn+1X by
n+1 s(cn(x)) = (B → 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)-actions
on SnX and Bn+1X by reparametrization, i.e. −1 n (A · γ)(p) = γ(A p) ∀γ ∈ SnX, A ∈ O(n + 1), p ∈ S . 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.
0 n+1 C Proof If Gγ trivial, take β : B → X with β|Sn = γ, put
s : G · γ → Bn+1X, s(A · γ) = A · β ∀A ∈ G. ∼ k If n = 1 and G = Zk for k ∈ N, s.t. γ = α 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), 2 2 k where pk : B → B , z 7→ z . 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 satis es 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 λ SCn,X(f ) ≤ ν(f, λ) + 1. 10 Consequences for closed geodesics
Corollary Let M be a closed manifold, F : TM → [0, +∞) be a Finsler 1 1 metric and λ ∈ R. Let EF : H (S , 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
λ ν(F, λ) ≥ SC1,M(EF ) − 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
De nition A Finsler metric on a manifold M is a map C0 ∞ F : TM → [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, 2 • D Fx 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. A closed geodesic of F is a critical point of E :ΛM := H1(S1, M) → R E (γ) = R 1 F(γ(t), γ˙ (t))2 dt F , F 0 . 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.
1 De nition γ1, γ2 : S → X are geometrically distinct if 1 1 γ1(S ) 6= γ2(S ). 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 Sn metrics on , ... 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 λ whose ag curvature satis es
1 λ 2 1 < K ≤ 1, i.e. if F reversible: < K ≤ 1, 4 1 + λ 16
then F admits two positively distinct closed geodesics (geometrically distinct if F is reversible).
a −1 Idea of proof Find a > 0 such that EF = EF ((−∞, a]) contains only prime closed geodesics and such that a SC1,M(EF ) ≥ 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
De nition (A. Schwarz, ’62) Let p : E → B be a bration. The sectional category or Schwarz genus of p is given by
n n [ secat(p) = inf n ∈ N ∃ Uj = B open cover, j=1 C0 o sj : Uj → E, p ◦ sj = inclUj ∀j .
In our setting: SCn(X) = secat rn : Bn+1X → SnX, γ 7→ γ|Sn ,
−1 n,X(A) ≥ rn| −1 : r (A) → A ∀A ⊂ SnX. SC secat rn (A) n 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 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 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 on H∗(LS2, S2; Q) vanishes.) Idea Improve cup length bounds by associating N-valued weights 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 each b ∈ 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)
C0 secat(p) = inf{n ∈ N | ∃s : B → 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 is ∗ given by wgtp(u) := sup{n ∈ N0 | pnu = 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). C0 ∗ ∗ • If f : X → B with f u 6= 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 6= 0, then k X (p) ≥ (u ) + ≥ ( p∗) + . secat wgtp j 1 cl ker 1 20 j=1 Consequences for the proof of the main result
a Let (M, F) be a Finsler manifold, a > 0 and let ιa : EF ,→ ΛM be the inclusion. From the above properties we obtain: ∗ ∗ if u ∈ H (ΛM, c1(M); R) satis es ιau 6= 0, then
ν(F, a) ≥ (u) := (u). wgt wgtr1
Thus, it su ces to nd such u with wgt(u) ≥ 2 for su ciently 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) 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 ∈ He∗(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 as a homotopy pushout of a pullback (double mapping cylinder): pullback homotopy pushout
f2 f2 Q / E Q / E
p f1 f1 p p E / B E / E ∗f E p2 p !, B
23 Construction of classes of weight ≥ 2, cont.
As a homotopy pushout of a pullback, it has a Mayer-Vietoris sequence:
δ ... / Hk−1(Q; R) / Hk(E ∗ E; R) / ⊕2 Hk(E; R) / ... f i=1 O p∗⊕p∗ 7 ∗ p2 Hk(B; R)
k ∗ ∗ Want to nd u ∈ H (B; R) with p2u = 0. If u lies in ker p , then ∗ k−1 ∗ p2u ∈ imδ. Try to nd αu ∈ H (Q; R) with δ(αu) = p2u, nd conditions 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 the pullback is 2 Q = {(γ1, γ2) ∈ (B2X) |r1(γ1) = r1(γ2)} 0 2 2 0 2 = {(γ1, γ2) ∈ (C (B , X)) | γ1|S1 = γ2|S1 } ≈ C (S , X),
hence for E2 := B2X ∗f B2X the Mayer-Vietoris sequence has the form k−1 0 2 δ k k k · · · → H (C (S , X); Q) → H (E2; Q) → H (B2X; Q)⊕H (B2X; Q) → ...
Lemma
k ∗ Let u 6= 0 ∈ H (X; Q), k ≥ 2. Then p2(Z(u)) = δ(au), where k−2 0 2 au ∈ H (C (S , X); Q) is given by
∗ 2 0 2 2 25 au = e2u/[S ], e2 : C (S , X) × S → X, e2(γ, p) := γ(p). Construction of classes of weight ≥ 2, cont.
Theorem
C0 Let u 6= 0 ∈ Hk(X; Q), k ≥ 3. If f ∗u = 0 for all f : S2 × P → X, where P is any closed oriented (k − 2)-manifold, then wgt(Z(u)) ≥ 2. Proof. ∗ 2 By the previous lemma, wgt(Z(u)) ≥ 2 if e2u/[S ] = 0. 0 2 But since H∗(C (S , X); Q) is representable, this will hold if for C0 all closed or. (k − 2)-manifolds P and g : P → C0(S2, X) it holds that
∗ 2 ∗ 2 e2u/[S ], g∗[P] = 0 ⇔ < f u, [S × P] >= 0,
where f(p, x) = e (g(x), p). The claim immediately 2 26 follows. Sketch of proof of the main result
Reminder By assumption, x ∈ Hk(M; Q) with 2k < dim M and x2 6= 0.
2 M even-dim., F positive curv. ⇒ π1(M) = 0 ⇒ Z(x ) 6= 0. ! C0 If f ∗(x2) = 0 for all f : S2 × P → 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.
2 ∗ Remains to show for u := Z(x ) that ιau 6= 0 for some a > 0 a for which EF contains only prime geodesics.
27 Sketch of proof of the main result, cont.
Reminder 1 ( λ )2 < δ ≤ K ≤ By assumption, 4 1+λ 1.
Let γ0 be a non-constant closed geodesic of F of shortest length `0. 1+λ • Since K ≤ 1, `0 > π 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) ∗ π2 that ιau 6= 0 for a := δ , hence ν(F, a) ≥ wgt(u) = 2. δ > 1 ( λ )2 ` • One derives from 4 1+λ and the bound for 0 that 2 EF(γ0) > a. a ⇒ EF contains two positively distinct closed geodesics. (geometrically distinct, if F reversible).
28 Perspectives and possible applications
• Equivariant versions, use richer ring structure in ∗ HS1 (LM, M; Q) (−→ work in progress) • Apply the methods to more general ows 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)