GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 1, 1997, 65-80

GEOMETRICAL DECOMPOSITION OF THE FREE LOOP

SPACE ON A MANIFOLD WITH FINITELY MANY CLOSED GEODESICS

THOMAS MORGENSTERN

Abstract. In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of a big relative CW-complex, representing the degenerate part, and a small cell having the dimension of the Morse index.

Introduction

In [1] Gromoll and Meyer succeeded in extending the Morse theory to isolated critical points and orbits. Applying these methods in [2] they were able to show that the sequence of Betti numbers with rational coefficients of the free (Sobolev) loop space ΛM is bounded for a closed manifold M with only finitely many bona fide closed geodesics. The free loop space ΛM is homotopy equivalent to a CW-complex. The intention of this paper is to show that not only these homological numbers are bounded but also the number of cells in all but finitely many dimensions is bounded for some homotopy equivalent CW-complex. It also gives a geometrical insight into why the Gromoll–Meyer theorem is true and motivates the use of special homotopies adapted to the free loop space.

If, on the other hand, the sequence of Betti numbers is unbounded then there have to be infinitely many prime non-trivial closed geodesics for every differentiable structure and Riemannian metric on M. This was shown to be true for all rationally nonmonogenic simply connected manifolds by Vigué-Poirrier and Sullivan [3]. This answers the existence question for a lot

65 of manifolds excluding, for example, spheres and projective spaces. There are also results for nonsimply connected manifolds but this case becomes more complicated the more complicated the fundamental group gets. The homological Gromoll–Meyer theorem is true for arbitrary field coefficients and one sees it to be true for integer coefficients as well, interpreting the Betti numbers as minimal number of generators. For simply connected ΛM our Theorem 1 thus follows from the abstract Proprosition [4, V. 8.3]. A nonsimply connected space is not determined by its homology, as the example of the Poincaré manifolds shows.

1991 Mathematics Subject Classification. 58E10.

Key words and phrases. Morse theory, closed geodesics, critical points, free loop space, Lusternik–Schnirelmann category.

1072-947X/97/0100-0065$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

Theorem 1 (Geometrical Gromoll–Meyer Theorem). If (M, g) is a closed differentiable Riemannian manifold, I(M) is the isometry group of M, G := I(M) ⊕ 𝕆(2), and if there are only finitely many G-orbits of geometrically distinct nontrivial closed geodesics, then there are numbers R and Q and a CW-complex which is homotopy equivalent to ΛM and has less than R cells of dimension greater than or equal to Q.

The extension of Morse theory was given by three lemmas. The first is a generalization of the Morse lemma [1, Lemma 1] introducing characteristic manifolds. The second step is a localization argument [1, Lemma 3] using what is now called Gromoll–Meyer pairs. Finally, the last and most important one is the Shifting theorem [1, Theorem], which can also be regarded as a kind of product formula. These lemmas proved to be very fruitful and a great deal of work nowadays relies on them. Using some classical results by Bott and observations about iterated characteristic manifolds the homological proof is then given in [2].

The homological proof uses excision strongly, which we don’t have at hand in homotopy theory. We will therefore introduce pairs of subsets very similar to the isolating neighborhoods of the Conley index. With the help of these and a telescope argument we are able to recover the homotopy type of the space under consideration. To do this we don’t take the step towards a local index, we don’t divide out the exit set. Consequently we have to find appropriate properties for these pairs enabling the step from local to global and conditions under which we can verify them. As in the original Gromoll–Meyer theory the product property turns out to be important and involved. We as well divide the proof into two parts, first setting up an appropriate general machinery and then using it in our special case.

1. h-Morse Theory

We are interested in the homotopy category of compactly generated pairs (X, A). From the homotopy type of filtration pairs (X_{n}, X_{n−1}) of a relative NDR filtration we can draw conclusions about the homotopy type of the pair (X, A) (see [5, Lemma 6] or [6, A.5.11]).

Definition 2. A relative h-CW-decomposition of the pair (X, A) is a relative NDR-filtration of (X, A) such that

[hCW]: for all n there is some relative CW-complex with (X_{n}, X_{n−1})≃ (Y_{n}, B_{n}) CW-complex.

A relative h-CW-complex (X, A) is a pair of compactly generated spaces admitting a relative h-CW-decomposition.

Lemma 3. If there is a relative h-CW-decomposition of (X, A) then the pair is homotopy equivalent to a relative CW-complex with one cell (of the same dimension) for every cell in (Y_{n}, B_{n}) [6, A.4.12 and A.4.15].

The step from local to global will use the following proposition:

Proposition 4 ([12, 7.5.7]). If X = X_{1} ∪ X_{2} is the union of closed subsets, the inclusion i_{2} : X_{1} ∩ X_{2} ↪ X_{2} is a closed cofibration and if (Y, B) is a closed cofibred pair, homotopy equivalent via r to (X_{2}, X_{1} ∩ X_{2}), then there is a map Φ such that (Φ, id_{X_{1}}) : (X_{1} ⊔_{r} Y, X_{1}) ≃ (X, X_{1}).

Corollary 5. If X = X_{1}∪X_{2} as in Proposition 4 and in addition, (Y, B) is a relative CW-complex then (X_{1} ⊔_{r} Y, X_{1}) is a relative CW-complex with one cell for each cell of (Y, B).

1.1. Local h-Decomposition Pairs. A strict Liapunov function for the topological flow φ on the Hausdorff space M is a continuous function f : M → ℝ, which is strictly decreasing along φ(x) for all x ∉ K^{φ}, the stationary points. As usual, the set f^{k} = M^{k} := {x ∈ M|f(x) ≤ k}. A closed subset T has the mean value property with respect to the flow if for any x ∈ T and two times t_{1}, t_{2} with t_{1} < t_{2}, φ(x, t_{1}) ∈ T and φ(x, t_{2}) ∈ T implies φ(x, t) ∈ T for all t_{1} < t < t_{2}. We define the following sets: the exit set T_{−} := {x ∈ T |∀t > 0 : φ(x, t) ∉ T }, the entrance set T_{+} := {x ∈ T |∀t > 0 : φ(x, −t) ∉ T }, and the vertical boundary set T_{0} := cl(∂T − (T_{−} ∪ T_{+})).

Lemma 6. Let T ⊆ M be a closed set with the mean value property with respect to the flow φ and let k_{T} ∈ ℝ such that f(T_{+}) ≥ k_{T} and f(T_{0} ∪ T_{−}) ≤ k_{T}. Then x ∈ T_{+} − (T_{0} ∪ T_{−}) if and only if there is t_{1} > 0 with φ(x, t_{1}) ∈ Int T; furthermore φ(x, t) ∈ Int T for all 0 < t ≤ t_{1}.

Corollary 7. Let the assumptions of Lemma 6 be valid and ∂T ∩ K^{φ} = ∅. Denote by ∂_{f^{k}∪T} T the boundary of T in f^{k}∪T. Then ∂_{f^{k}∪T} T = T_{0}∪T_{−} .

Lemma 8. If for some x ∈ T with f(x) ≥ k there is t_{1} ≥ 0 with φ(x, t_{1}) ∈ T_{−} ∩ f^{−1}[k, ∞) and t_{2} ≥ 0 with f(φ(x, t_{2})) = k and φ(x, t_{2}) ∈ T, then t_{1} = t_{2}.

Lemma 9. If x ∈ M with f(x) ≥ k ≥ k_{T} and there is t_{1} ≥ 0 with φ(x, t_{1}) ∈ T_{+} ∩ f^{−1}[k, ∞) and t_{2} ≥ 0 with f(φ(x, t_{2})) = k and φ(x, t_{2}) ∈ cl(f^{−1}(k) − T), then t_{1} = t_{2}.

For a subset A ⊆ T we define T^{+}(A) := {x ∈ T |∃t ≥ 0 : φ(x, t) ∈ A}. A closed subset A has in T the retraction property from above (with respect to φ), if the function x ↦ t_{A}(x) := min_{φ(x,t)∈A}{t ≥ 0} is continuous on T^{+}(A).

Definition 10. Let M be a Hausdorff space, φ a topological flow, and K^{φ} its stationary points. Let f : M → ℝ be a strict Liapunov function for φ. A local h-decomposition pair of K ⊆ K^{φ} corresponding to φ and f is a pair (U, Ũ) together with a number k, the separation level, having the following properties:

[hm1]: U, Ũ ⊆ M are closed;

[hm2]: the top T := cl(U − Ũ) has the mean value property with respect to the flow φ and K^{φ} ∩ T = K and K ⊆ Int(T); [hm3]: T has the retraction property from above with respect to φ; [hm4]: f(T_{+}) ≥ k, f(Ũ) ≤ k and T ∩ Ũ = T_{0} ∪ T_{−}; [hm5]: the inclusion i : Ũ ↪ U is a cofibration.

Corollary 11. By [hm4] f^{k} − (U − Ũ) is closed, and by [hm5] f^{k} − (U − Ũ) ↪ f^{k} ∪ U is a cofibration [7, Satz 7.36].

Definition 12. A local h-decomposition pair has the following properties:

[hm6]: if there is a relative CW-complex (Y, B) homotopy equivalent

to (U, Ũ); [exc]: if T_{0} = ∅, κ := inf f(K) > k, T_{−} has the retraction property

from above in T and if for all x ∈ T there is either t ≥ 0 with φ(x, t) ∈ T_{−} or otherwise φ(x, t) → K as t → ∞.

1.2. h-Morse Decomposition.

Definition 13. Let f be a strict Liapunov function for φ and T a closed subset with the mean value property. φ has the property (D) for f on T if:

(i) for every open neighborhood O of the stationary points K^{φ} the level sets T ∩ f^{−1}(u) − O have the retraction property from above; (ii) for every x ∈ T there is t > 0 with φ(x, t) ∈ T_{−} or, otherwise, for the flow line φ(x) : (α(x), β(x)) → M either f ∘ φ(x) → −∞ or φ(x) → K^{φ} as t → β(x).

If f is differentiable and satisfies the Palais–Smale condition (C) on T then the negative gradient flow φ has the property (D) for f on T. If f is bounded from below on T then the flow line either converges to the critical points or ends in T_{−}.

Definition 14. Let M be a metrizable space and A, X closed subsets with A ⊆ X. Let φ be a topological flow on M and T := cl(X − A) have the mean value property. Let f : M → ℝ be a strict Liapunov function for φ, and φ have the property (D) for f on T. Let {K_{l}}_{l∈L} be a partition of the stationary points K^{φ} ∩ (X − A) such that for every l ∈ L there is a local h-decomposition pair (U_{l}, Ũ_{l}), k_{l} of K_{l} with U_{l} ⊆ X and the following properties:

[hm7]: There is a countable index set J = [0, …, j_{max}] or J = ℕ_{0} and a partition L = ⋃_{j∈J} L_{j} such that for all l, m ∈ L_{j} we have k_{l} = k_{m}. There is a nondecreasing function u : J → ℝ such that u_{j} := u(j) = k_{l} for all l ∈ L_{j}. If J = ℕ_{0} we have the convergence u_{j} → sup_{x∈X−A} f(x) as j → ∞. [hm8]: For i < j we have T_{l} ∩ U_{m} = ∅ if l ∈ L_{j} and m ∈ L_{i}. [hm9]: For all l, m ∈ L_{j} with l ≠ m we have U_{l} ∩ U_{m} = ∅ and,

moreover, ⋃_{l∈L_{j}} U_{l} is closed.

[hm10]: For all l ∈ L we have T_{l} ⊆ X − A. A has the retraction property from above in X, T_{−} ∪ T_{0} = ∂_{X}A, f(T_{0}) < u_{0} and

A^{u_{0}} ↪ X^{u_{0}} − ⋃_{l∈L}(U_{l} − Ũ_{l}) is deformation retract.

The relative h-Morse decomposition of (X, A) is a relative NDR filtration of (X, A) defined by setting X_{−1} := A, and for n = 2j

X_{n} := X_{n−1} ∪ (X^{u_{j}} − ⋃_{l∈L_{i}, i≥j}(U_{l} − Ũ_{l})) ,
X_{n+1} := X_{n} ∪ ⋃_{l∈L_{j}} U_{l} .

If J = [0, …, j_{max}] is finite, we always set X_{n} := X for n ≥ 2(j_{max} + 1).

Proposition 15. If all local h-decomposition pairs have the property [hm6], then a relative h-Morse decomposition is a relative h-CW-decom-position of (X, A).

Proof. Let n = 2j. Let r_{l} : (Y_{l}, B_{l}) → (U_{l}, Ũ_{l}) be homotopy equivalences to relative CW-complexes, and r_{n} := ∐_{l∈L_{j}} r_{l} : ∐_{l∈L_{j}}(Y_{l}, B_{l}) → ⋃_{l∈L_{j}}(U_{l}, Ũ_{l}). All ι_{j} : ⋃_{l∈L_{j}} Ũ_{l} ↪ ⋃_{l∈L_{j}} U_{l} are cofibrations, and by Proposition 4 (X_{n+1}, X_{n}) = (X_{n} ∪ ⋃_{l∈L_{j}} U_{l}, X_{n}) ≃ (X_{n} ⊔_{r_{n}} ∐_{l∈L_{j}} Y_{l}, X_{n}), a relative CW-complex with cells e_{γ}^{λ} for every cell e_{γ}^{λ} of a relative CW-complex ∐_{l∈L_{j}}(Y_{l}, B_{l}).

We show that X_{n−1} ↪ X_{n} is deformation retract. By [hm10] X_{−1} ↪ X_{0} is deformation retract. If u_{j−1} = u_{j} we have X_{n} = X_{n−1}. Let u_{j−1} < u_{j}. For every x ∈ X_{n} − X_{n−1} there is 0 < t < +∞ with φ(x, t) ∈ A or f(φ(x, t)) ≤ u_{j−1} or φ(x, t) ∈ T_{l} for some l ∈ L.

From Lemmas 8 and 9 we see that t_{(A∪⋃_{k_{l}<u_{j}} T_{l})−f^{<u_{j−1}} )}(x) =

t_{f^{−1}(u_{j−1})∩X−Int(A∪⋃_{l∈L} T_{l})}(x) when both are defined; therefore t_{X_{n−1}} is continuous on X_{n}. We define the deformation retraction by h : X_{n}×I → X_{n} (x, s) ↦ φ(x, st_{X_{n−1}}(x)). The homotopy stays in X^{u_{j}} − ⋃_{l∈L_{i}, i≥j}(U_{l} − Ũ_{l}) because for every x ∈ X_{n} − X_{n−1} we have φ(x, t) ∉ ⋃_{l∈L_{i}, i≥j} U_{l} − Ũ_{l} for all t > 0.

1.3. Existence of Local h-Decomposition Pairs. Let G be a compact Lie group operating continuously on a differentiable manifold μ : G × M → M, and let μ_{g}, defined by μ_{g}(x) := μ(g, x), be differentiable for every g ∈ G. We say that G operates differentiably on c if the map μ(c)c : G → M, g ↦ g.c is differentiable. Let (B, g) be a Riemannian manifold, π : E → B a differentiable Rimannian vector bundle with a bundle metric ⟨ , ⟩_{p}. Let G operate differentiably on c ∈ B and B ≅ G/G_{c}; moreover, let G operate continuously fiber-preserving and orthogonal on π : E → B. Let Õ ⊆ E be an open neighborhood of the zero section 0_{G.c}. Let g̃ : Õ → Pos T (Õ) and ⟨ , ⟩_{ξ_{p}} be Riemannian metrics on Õ. There is a closed invariant neighborhood D ⊆ Õ of the zero section on which ⟨ , ⟩_{ξ_{p}} and g̃(ξ_{p}) are uniformly equivalent, i.e., a^{2}⟨ , ⟩_{ξ_{p}} ≤ g̃(ξ_{p}) ≤ b^{2}⟨ , ⟩_{ξ_{p}} for some 0 < a < 1 < b and all ξ_{p} ∈ D. Let f̃: Õ → ℝ be an invariant differentiable function. Let Õ_{p} := Õ ∩ π^{−1}(p), f̃_{p} := f̃|Õ_{b} and grad f̃_{p} ∈ E_{p} be the gradient of f̃_{p} formed by the Hilbert metric ⟨ , ⟩_{p} on E_{p}. Let grad f̃ be the gradient formed by ⟨ , ⟩_{ξ_{p}} and φ be the flow of − grad f̃. Assume that grad f̃ = grad f̃_{p} under the canonical identification of the vertical tangential bundle with the bundle itself. Let g̃rad f̃ be the gradient formed by g̃ and φ̃ the flow of − g̃rad f̃.

Lemma 16. If (U_{c}, Ũ_{c}), k is a G_{c}-invariant local h-decomposition pair of K_{c} corresponding to f̃_{c} and the flow φ_{c} of − grad f̃_{c} with U_{c} ⊆ O_{c}, then (U, Ũ) := (G.U_{c}, G.Ũ_{c}), k is a local h-decomposition pair of K := G.K_{c} corresponding to f̃ and φ. If (U_{c}, Ũ_{c}) has [exc] so does (U, Ũ).

Proof. The flow φ preserves fibres, hence [hm2] and [hm4] are fulfilled. ξ_{p} ↦ inf{t ≥ 0|φ(ξ_{p}, t) ∈ U} is invariant because the flow is. Choosing continuous bundle charts over V with the help of the G-operation one sees that [hm3] holds and also that V × Ũ_{c} ↪ V × U_{c} is a cofibration. That it is globally a cofibration can be seen by partitions of unity and [5], hence [hm5] holds.

Let the gradient grad e(ξ_{p}) of e(ξ_{p}) = ⟨ξ_{p}, ξ_{p}⟩_{p} formed by ⟨ , ⟩_{ξ_{p}} be bounded on D, and let f̃satisfy the Palais–Smale condition (C) with respect to g̃ on D. Let 0_{G.c} be an isolated critical submanifold of f̃ and f̃(0_{G.c}) = κ. Kf̃ = Kφ̃ = K^{φ}.

Lemma 17. For constants ε, ϱ > 0 with D_{ϱ}(0_{G.c}) ⊆ D and f̃^{−1}[κ − ε, κ + ε] ∩ D_{ϱ}(0_{G.c}) ∩ Kf̃ = 0_{G.c} there is a (classical) G-invariant Gromoll– Meyer pair (W, W_{−}), κ − δ of 0_{G.c} corresponding to φ, φ̃ and f̃ inside f̃^{−1}[κ − ε, κ + ε] ∩ D_{ϱ}(0_{G.c}). Every Gromoll–Meyer pair is a local h-decom-position pair.

Proof. Choose ϱ ≥ ϱ_{1} > ϱ_{0} > 0, set β^{2} := inf{g̃(ξ_{p})( g̃radf̃(ξ_{p}), g̃radf̃(ξ_{p})) | ξ_{p} ∈ D_{ϱ_{1}}(0_{G.c}) − B_{ϱ_{0}}(0_{G.c})} > 0 because condition (C) is fulfilled and set, finally, α^{2} := sup_{ξ_{p}∈D_{ϱ_{1}} (0_{B})}⟨grad e(ξ_{p}), grad e(ξ_{p})⟩_{ξ_{p}} < ∞. We define a function w(ξ_{p}) := ⟨ξ_{p}, ξ_{p}⟩_{p} + η(f̃(ξ_{p}) − κ) and

W := w^{ω} ∩ f̃^{−1}[κ − δ, κ + δ′],
W_{−} := W ∩ f̃^{−1}(κ − δ),

where η > \frac{α}{aβ}, ϱ_{0}^{2} + ηδ′ ≤ ω ≤ ϱ_{1}^{2} − ηδ and 0 < δ, δ′ < min\{ε, \frac{ϱ_{1}^{2}−ϱ_{0}^{2}}{2η} \}. (cf. [2, §2], [8, Theorem 5.3 and p. 74]). The level set f^{−1}(κ − δ) ∩ W is regular, hence one can deform a small neighborhood of W_{−} in W with the help of the flow φ inside of W onto W_{−} and with [7, Satz 3.13, Satz 3.9, Lemma 3.4, Satz 3.26] it follows that W_{−} ↪ W is a cofibration.

On a differentiable Riemannian manifold (M, g) let there be a differentiable function f : M → ℝ, its negative gradient − grad f formed with g and its flow φ. For a diffeomorphism ψ : Õ → ψ(Õ) ⊆ M set f̃:= ψ^{∗}f : Õ → ℝ and g̃ := ψ^{∗}g.

Corollary 18. If K ⊆ ψ(Õ) is a critical set and (U, Ũ), k a local hdecomposition pair corresponding to φ̃ and f̃, then the pair (ψ(U), ψ(Ũ)), k is a local h-decomposition pair of K corresponding to φ and f.

Proof. Flow lines entering ψ(U) in a finite time enter ψ(Õ) − ψ(U) beforehand. But there t_{ψ(T)}(x) is continuous, hence it is continuous everywhere. ψ is a homeomorphism, hence the claim.

Let H = H^{+} ⊕ H^{−} be a direct sum of Hilbert spaces, O an open neighborhood of 0 and f : (x, y) ↦ \frac{1}{2}(∥x∥^{2} − ∥y∥^{2}). Let D^{λ} ⊆ H^{−} be the closed unit disc with λ = dim H^{−}. Then the following is true:

Lemma 19. (W, W_{−}) ≃ (W_{−} ⊔_{α} D^{λ}, W_{−}) ≃ (D^{λ}, S^{λ−1}), where α : S^{λ−1} → W is given by ȳ ↦ (0, \sqrt{2δ}ȳ) [9, I §3].

Let H = ℝ^{ν} be finite-dimensional and f : O → ℝ be differentiable, 0 an isolated critical point, (U, Ũ), k a local h-decomposition pair of 0 corresponding to f and φ. Furthermore, let ϱ > 0 with B_{ϱ}(0) ⊆ U − Ũ. Then there is a differentiable function f̄: O → ℝ with only finitely many nondegenerate critical points and f|O − B_{ϱ}(0) = f̄|O − B_{ϱ}(0) [10, Lemma 8.6], [1, eq. (3)] or [8, Theorem 5.7]. (U, Ũ), k is also a local h-decomposition pair corresponding to f̄ and its negative gradient flow φ̄.

Corollary 20. If (U, Ũ), k has [exc] it has [hm6], i.e., there is a homotopy equivalent finite relative CW-complex (Y, Ũ).

Proof. Construct a relative h-Morse decomposition of (U, Ũ) with the help of f̄. Take open disjoint Morse charts for the finitely many critical points of f̄ in the interior of U − Ũ and the images of Gromoll–Meyer-pairs, using Corollary 18 and Lemma 19. The claim follows from Lemma 19, Proposition 15 and Lemma 3.

Proposition 21 ([11]). If (p, p̃) : (E, Ẽ) → B is a fibration pair of Hurewicz fibrations (including the property of it being a closed cofibred pair), if B is 0-connected and homotopy equivalent to a CW-complex P, and if the fibres (F, F̃) are homotopy equivalent to some relative CW-complex (K, L), then (E, Ẽ) is homotopy equivalent to a relative CW-complex having the same numbers of cells of the same dimension as the product of CW-complexes P × (K, L).

Proof. We can assume that B is already a CW-complex. (E, Ẽ) is a cofibred pair over B. Look first at B′ ∪_{φ} D^{n} ≅ B′ ∪ e^{n} = B′ ∪ χ(D^{n}). Set E′ := p^{−1}(B′) and Ẽ′ := E′ ∩ Ẽ. (E′ ∪ Ẽ) ∩ p^{−1}(e^{n}) = p̃^{−1}(e^{n}) ∪ p^{−1}(∂e^{n}) ↪ p^{−1}(e^{n}) is a closed cofibration (use [12, 7.3.9]). (χ^{∗}E, χ^{∗}Ẽ) is a cofibred pair, hence because D^{n} ≃ ∗ there is a fiber homotopy equivalence of pairs Φ : D^{n} × (F, F̃) → (χ^{∗}E, χ^{∗}Ẽ). Using a homotopy equivalence i : (K, L) → (F, F̃), set μ := p^{∗}χ ∘ Φ ∘ (id ×i)|D^{n} × A ∪ S^{n−1} × X and define

(K′, L′) := ((E′ ∪ Ẽ) ⊔_{μ} (D^{n} × X), E′ ∪ Ẽ) ,

which is a relative CW-complex with one cell for every cell in (D^{n}, S^{n−1}) × (K, L). From Corollary 5 we have (K′, L′) ≃ (E′ ∪ Ẽ ∪ p^{−1}(e^{n}), E′ ∪ Ẽ). If B has only countably many cells we can define by this process a relative h-Morse decomposition of (E, Ẽ) (otherwise proceed by skeletons) and use Lemma 3.

Corollary 22. If in Lemma 16 (U_{c}, Ũ_{c}) is homotopy equivalent to the relative CW-complex (K, L) and G/G_{c} homotopy equivalent to the finite CW-complex P, then (U, Ũ) := (G.U_{c}, G.Ũ_{c}) is homotopy equivalent to a relative CW-complex with the same number of cells as P × (K, L), hence it has the property [hm6].

1.4. Excision of Local h-Decomposition Pairs. Cf. [1, §2 (V_{1}, V_{1}^{−}) and equation (9)].

Proposition 23 (Excision and Change of Metric). If (U, Ũ), k is a local h-decomposition pair of the isolated critical orbit 0_{G.c} corresponding to φ and f̃ with the property [exc], if U ⊆ Õ and if f̃ has (C) on T_{U} with respect to ⟨ , ⟩_{ξ_{p}}, then for all ϱ, ε > 0 there is a homotopy equivalent local h-decomposition pair (V, Ṽ), k_{V} of 0_{G.c} corresponding to φ, φ̃ and f̃ with T_{V} ⊆ B_{ϱ}(0_{G.c}) ∩ f̃^{−1}[κ − ε, κ + ε] and again with [exc]. If (U, Ũ) and f̃ are G-invariant, then (V, Ṽ) can be chosen G-invariant as well and homotopy G-equivariant.

Let W ⊆ T be a closed subset with the mean value property, W_{+} have the retraction property from above, and K ⊆ Int(W). Let δ > 0 with f̃(∂W − W_{+}) ≤ κ − δ and f̃(T_{U−}) ≤ κ − δ. Choose δ ≥ δ̃ > 0, denote ε̃:= δ − δ̃, set k_{V} := κ − δ̃ and define the pair as follows:

S := T^{k_{V} −ε̃} ∪ W,
S̃ := T^{k_{V} −ε̃} ∪ W^{k_{V}} .

Claim 24. (S, S̃) ≃ (T, T_{−}) rel T_{−}.

Proof. t_{S}(x) := min{t ≥ 0|φ(x, t) ∈ S} is continuous on T according to Lemma 9. Define i : (T, T_{−}) → (S, S̃) by x ↦ φ(x, t_{S}(x)) and

r : (S, S̃) → (T, T_{−})

x ↦ \{ \begin{matrix}φ(x, t_{T_{−}}(x)) & for & f(x) ≤ k_{V} \\ φ (x, \frac{2(κ−f(x))−δ̃}{δ̃}t_{T_{−}}(x)) & for & f(x) ∈ [k_{V} , κ − δ̃/2] \\ x & for & f(x) ≥ κ − δ̃/2 \end{matrix} .

All maps are the identity on T_{−}.

Corollary 25. The pair (V, Ṽ) := (S ∪ Ũ, S̃ ∪ Ũ), k_{V} is a local hdecomposition pair with (U, Ũ) ≃ (V, Ṽ) and T_{V} = cl(S − S̃) ⊆ W.

Proof of Proposition 23. Choose an invariant Gromoll–Meyer pair (W, W_{−}) of 0_{G.c} with W ⊆ B_{ϱ′}(0_{G.c}) ∩ f^{−1}[κ − ε′, κ + ε′], where ε′ := min{ε, κ − k_{U}} and ϱ > ϱ′ > 0 is such that B_{ϱ′}(0_{G.c}) ∩ f^{−1}[κ − ε′, κ + ε′] ⊆ U − Ũ. Choose δ̃ > 0 with ε′ > δ̃ and set k_{V} := κ − δ̃. Then use Corollary 25.

1.5. Product Proposition. Let there be given an equivariant orthogonal bundle sum E ≅ E^{+} ⊕ E^{−} ⊕ E^{0}, ξ_{p} = (ξ_{p}^{+}, ξ_{p}^{−}, ξ_{p}^{0}), and an equivariant characteristic manifold

W_{char} : (D_{ϱ_{2}}^{0}(0_{G.c}), 0_{G.c}) → (Õ, 0_{G.c})
ξ_{p}^{0} ↦ (h(ξ_{p}^{0}), ξ_{p}^{0})

as in the generalized equivariant Morse Lemma. Then there are ϱ > 0 such that D_{ϱ}(W_{char}(D_{ϱ_{2}}(0_{G.c}))) ⊆ Õ, some numbers ϱ_{1}, ϱ_{3} > 0 and a continuous equivariant fiber preserving homeomorphism, a generalized equivariant Morse chart,

Φ : (D_{ϱ_{3}}^{±}(0_{G.c}) × D_{ϱ_{1}}^{0}(0_{G.c}), 0_{G.c}) ⟶ D_{ϱ}(W_{char}(D_{ϱ_{2}}(0_{G.c})))
(ξ_{p}^{±}, ξ_{p}^{0}) ↦ η(ξ_{p}^{±}, ξ_{p}^{0}) + ξ_{p}^{0} + h(ξ_{p}^{0}) ,

which extends W_{char} (which means η(0_{p}^{±}, ξ_{p}^{0}) = 0_{p}), with η(ξ_{p}^{±}, ξ_{p}^{0}) ∈ E^{+} ⊕ E^{−}, and

f̃_{p} ∘ Φ(ξ_{p}) = \frac{1}{2}(∥ξ_{p}^{+}∥_{p}^{2} − ∥ξ_{p}^{−}∥_{p}^{2}) + f̃_{p}(h(ξ_{p}^{0}), ξ_{p}^{0}) =: f̃_{p}^{1}(ξ_{p}^{+}, ξ_{p}^{−}) + f̃_{p}^{0}(ξ_{p}^{0}) .

If f̃ is sufficiently differentiable, then Φ is a local diffeomorphism [13, Prop. 4.2.1], [10, Theorem 8.3], [8, Corollary 7.1].

To clarify here the dark points in [2] and some other places we should make the following point clearer. The Whitney sum representation of E and Morse-charts as above exist under conditions on the spectrum of the second derivative d^{2}f̃(0_{p}). The subbundles are G-invariant if the map induced by every element g ∈ G is differentiable and isometric. If this is the case and E_{p}^{0} is finite-dimensional, then E^{0} is a finite-dimensional manifold on which G acts differentiably [14, V.1. Corollary]. In our case the map is fiberwise linear and orthogonal. Choose some linear connection C^{±} on E^{±} := E^{+} ⊕ E^{−}.

Lemma 26. If E^{0} is finite dimensional and the action of G is fiber preserving and orthogonal, then there are a canonical linear connection C^{0} : T (B) ⊕ E^{0} → T (E^{0}) and a canonical Riemannian metric ⟨ , ⟩_{ξ_{p}} on E. The gradient formed by this metric grad(f̃∘ Φ) is in the vertical tangent space T^{v}(E).

Proof. Represent X_{p} ∈ T_{p}(B) by a differentiable curve γ : (−ε, ε) → G and for ξ_{p}^{0} ∈ E_{p}^{0} set C_{p}^{0}(X_{p}, ξ_{p}^{0}) = (γ(t).ξ_{p}^{0})′(0). This map is linear in the second variable because G operates linearly on fibers and is continuous in both variables because G operates differentiably. Let C := C^{±} ⊕ C^{0} be the product connection. The canonical horizontal tangent space is given by T_{ξ_{p}}^{h}(E) := C_{p}(T_{p}(B), ξ_{p}) and the canonical metric by [15, Def. 1.9.12, 1.11.9].

Therefore grad(f̃∘Φ) = grad(f̃∘Φ)_{p}. It also follows that 0_{p} is an isolated critical point of f̃_{p} iff 0_{G.c} is an isolated critical manifold of f̃.

Lemma 27. Given constants 0 < ϱ_{0} < ϱ_{1}, there is an equivariant diffeomorphism Φ̃ : D_{ϱ_{3}}^{±}(0_{G.c}) × D_{ϱ_{2}}^{0}(0_{G.c}) → D_{ϱ}(W_{char}(D_{ϱ_{2}}(0_{G.c}))) extending W_{char} such that Φ̃|D_{ϱ_{3}} × D_{ϱ_{0}} = Φ|D_{ϱ_{3}} × D_{ϱ_{0}}.

Proof. Choose χ ∈ C^{∞}([0, ϱ_{2}], ℝ_{0}^{+}) with χ|[0, ϱ_{0}] ≡ 1 and χ|[ϱ_{1}, ϱ_{2}] ≡ 0,

define Φ̃ : (ξ_{p}^{±}, ξ_{p}^{0}) ↦ ξ_{p}^{0} + h(ξ_{p}^{0}) + η(ξ_{p}^{±}, χ(∥ξ_{p}^{0}∥) · ξ_{p}^{0} + (1 − χ(∥ξ_{p}^{0}∥))ϱ_{1} · \frac{ξ_{p}^{0}}{∥ξ_{p}^{0}∥}). If ∥ξ_{p}^{0}∥ is invariant, so is χ(∥ξ_{p}^{0}∥).

Let H^{1} × H^{0} be the product of Hilbert spaces, O^{1} × O^{0} an open neighborhood of 0, and f : O^{1} × O^{0} → ℝ a differentiable function. Let there be D^{1} × D^{0} ⊆ O^{1} × O^{0}, a closed neighborhood of 0, on which f = f^{1} + f^{0} : (x, z) ↦ f^{1}(x) + f^{0}(z). Denote by φ^{1} (respectively φ^{0}) the partial negative gradient flows of f^{1} (respectively of f^{0}) and by φ that of f.

Proposition 28 (Product Proposition). If H^{0} is finite dimensional, if we are given a local h-decomposition pair (U^{0}, Ũ^{0}), k_{U^{0}} of 0^{0} corresponding to φ^{0} and f^{0} which is compact and has the property [exc], and if 0^{1} ∈ D^{1} is an isolated critical point of f^{1} which satisfies the Palais–Smale condition (C) on D^{1}, then there are a local h-decomposition pair (U^{1}, Ũ^{1}), k_{1} of 0^{1} corresponding to f^{1} and φ^{1}, a local h-decomposition pair (U, Ũ), k of 0 corresponding to φ and f and a homotopy equivalence to the product

(U, Ũ) ≃ (U^{1}, Ũ^{1}) × (U^{0}, Ũ^{0}) .

(U, Ũ), k again has the property [exc]. All pairs and homotopies can be chosen to be equivariant.

We prove a technical lemma first.

Let (V^{1}, Ṽ^{1}), k_{1} and (V^{0}, Ṽ^{0}), k_{0} be local h-decomposition pairs of K^{1}, K^{0} corresponding to φ^{1}, φ^{0} and f^{1}, f^{0} respectively with V^{1} ⊆ O^{1} and V^{0} ⊆ O^{0}. We use the following notations: T^{1} := T_{V^{1}} and T^{0} := T_{V^{0}}. Let T^{1} ⊆ Int D^{1} and T^{0} ⊆ Int D^{0}; then (T^{1} × T^{0}) has the mean value property, and K = K^{1} × K^{0} are the critical points of f in T^{1} × T^{0}. Let f(T^{i}) ⊆ [k_{i} − ε_{i}, k_{i} + ε′_{i}] and f(K^{i}) ≥ κ_{i} for i = 0, 1; then we have f(K) ≥ κ_{1} +κ_{0} and f(∂(T^{1} ×T^{0})−(T^{1} ×T^{0})_{+}) ≤ max{k_{1} +k_{0} +ε′_{0}, k_{0} +k_{1} +ε′_{1}} = k_{1} + k_{0} + max{ε′_{1}, ε′_{0}}.

Lemma 29. If k_{1} + k_{0} + max{ε′_{1}, ε′_{0}} < κ_{1} + κ_{0}, if both pairs have the property [exc], if there exists a number k ∈ [k_{1} + k_{0} + max{ε′_{1}, ε′_{0}}, κ_{1} + κ_{0}), such that f((V^{1} − D^{1}) × V^{0} ∪ V^{1} × (V^{0} − D^{0})) ≤ k and if condition (C) is satisfied by f^{0} on T^{0} and f^{1} on T^{1}, then there exists (equivariant) local h-decomposition pair (U, Ũ), k of K corresponding to φ and f with the property [exc], and there is (equivariant) homotopy equivalence to the product

(U, Ũ) ≃ (V^{1}, Ṽ^{1}) × (V^{0}, Ṽ^{0}) .

Proof. It immediately follows that f(Ṽ^{1} × V^{0} ∪ V^{1} × Ṽ^{0}) ≤ k. Define

U := V^{1} × V^{0},
Ũ := (V^{1} × V^{0})^{k} ,

and [hm1], [hm2], [hm4], [hm5] and [exc] follow easily. [hm3] is true, because t_{T_{1}×T_{0}}(x, z) = max{t_{T_{1}}(x), t_{T_{0}}(z)} and T^{1} × T^{0} ⊆ Int(D^{1} × D^{0}). It only remains to show the existence of homotopy equivalence. f satisfies (C) on T^{1}×T^{0} and (T_{1}×T_{0})_{−} has the retraction property from above in T^{1}×T^{0}; hence we can define a deformation retraction of pairs r : (T^{1} × T^{0}, (T^{1} × T^{0})^{k}) → (T^{1} × T^{0}, (T^{1} × T^{0})_{−}) as in Claim 24 and Proposition 23.

Proof of Proposition 28. By hypothesis B_{ϱ_{0}}(0^{0}) ⊆ D^{0} for some ϱ_{0} > 0; hence by Proposition 23 there is a compact local h-decomposition pair (V^{0}, Ṽ^{0}), k_{0} = κ_{0}−δ_{0} with [exc], T_{V^{0}} ⊆ B_{ϱ_{0}}(0^{0})∩(f^{0})^{−1}[k_{0} := κ_{0}−δ_{0}, κ_{0}+ δ′_{0}], where δ_{0}, δ′_{0} > 0 and δ′_{0} can later be chosen arbitrarily small. There is ϱ > 0 with f(B_{ϱ}({0^{1}}×V^{0})−Int D^{1}×B_{ϱ_{0}}(0^{0})) ≤ κ_{1}+κ_{0}−δ_{0}/2 and B_{ϱ}(0^{1}) ⊆ D^{1}. Choose a Gromoll–Meyer pair (W^{1}, W_{−}^{1}) of 0^{1} for f^{1} with W^{1} ⊆ B_{ϱ}(0^{1}), f^{1}(W^{1}) ⊆ [k_{1} := κ_{1}−δ_{1}, κ_{1}+δ′_{1}], W_{−}^{1} = W^{1}∩(f^{1})^{−1}(κ_{1}−δ_{1}), where δ_{1}, δ′_{1} > 0 and δ′_{1} < δ_{0}. ε′_{i} := δ_{1} + δ′_{i} and therefore k_{1} + k_{0} + max{ε′_{1}, ε′_{0}} = κ_{1} −δ_{1} +κ_{0} −δ_{0} +max{δ_{1} +δ′_{1}, δ_{0} +δ′_{0}} = κ_{1} +κ_{0} −min{δ_{0} −δ′_{1}, δ_{1} −δ′_{0}}. We see, that if we shrink δ′_{0} such that δ′_{0} < δ_{1}, then the sum becomes smaller than κ_{1} + κ_{0}. Choose κ_{1} + κ_{0} > k > κ_{1} + κ_{0} − min{δ_{0}/2, δ_{1} − δ′_{0}, δ_{0} − δ′_{1}} and use Lemma 29.

2. The Geometric Gromoll–Meyer Theorem

Let (M, g) be a closed differentiable Riemannian manifold and ΛM := H^{1}(S^{1}, M) the space of free closed Sobolev loops. If c is a differentiable closed loop then T_{c}(ΛM) ≅ H^{1}(S^{1}, c^{∗}T (M)), where H^{1}(S^{1}, c^{∗}T (M)) is the space of H^{1}-vector fields along c on which 𝕆(2) acts orthogonally. On differentiable curves 𝕆(2) acts differentiably. Even though ΛM is not a differentiable 𝕆(2)-manifold, there is a kind of equivariant exponential map ẽxp : T (ΛM) ⊇ Õ → ΛM fibrewise induced by the exponential map exp : T M ⊇ O → M of the manifold [15, 2.3.12]. We define fibrewise an energy integral Ẽ on T ΛM by ξ_{c} ↦ \frac{1}{2} ∫_{0}^{1} ∥∇_{1}ξ_{c}(t)∥_{c(t)}^{2}dt. Ẽ coincides on Õ with the pull back ẽxp^{∗}E. Denote by φ the negative gradient flow of E on ΛM.

2.1. Local h-Decomposition Pairs for Iterated Geodesics. Denote by I(M) the isometry group of M and set G := I(M)⊕𝕆(2). If c is a closed geodesic, then the G-orbit G.c is a differentiable submanifold [16, I.5.4]. If m_{c} is the order of c, choose, on T_{c}ΛM, the G_{c}-equivariant metric

⟨ξ_{c}, ξ′_{c}⟩_{c,m_{c}} := ⟨ξ_{c}, ξ′_{c}⟩_{0} + \frac{1}{m_{c}^{2}} ⟨∇_{1}ξ_{c}, ∇_{1}ξ′_{c}⟩_{0}

[2, §3 eqn. (14)], [13, §4.2] and denote its norm by ∥ ∥_{1,m_{c}}. The iteration map m_{∗} : T ΛM|G.c ⟶ T ΛM|G.c^{m} is then an isometric embedding. Identify the normal bundle π_{c} : N(G.c) → G.c with the orthogonal complement of T G.c. For arbitrary m we choose the domain of the exponential map Õ ∩ N(G.c^{m}) as tubular neighborhoods (Õ(G.c^{m}), π_{c^{m}}, ẽxp); then m_{∗}(Õ(G.c)) ↪ Õ(G.c^{m}) is a linear bundle map.

Lemma 30 ([2, Lemma 5], [13, Lemma 4.2.5]). Let c be a (iterated) nontrivial closed geodesic. If ν(c) = ν(c^{m}), then m_{∗}|N_{c}^{0} is an isomorphism onto N_{c^{m}}^{0}. If W_{char} is a characteristic manifold for Ẽ_{c}, then m_{∗} ∘ W_{char} ∘ m_{∗}^{−1} is a characteristic manifold for Ẽ_{c^{m}}.

Corollary 31 ([2, Theorem 3]). If 0_{c} ∈ (W_{c}^{0}, W_{c−}^{0}) ⊆ Õ_{c}^{0} is a Gromoll–Meyer pair for the degenerate part Ẽ_{c}^{0}, then m_{∗}(W_{c}^{0}, W_{c−}^{0}) is a homeomorphic compact Gromoll–Meyer pair for Ẽ_{c^{m}}^{0} (formed by the iterated characteristic manifold) and the chosen Riemannian metric.

Proof. We have ∥ξ_{c}^{m}∥_{1,mm_{c}}^{2} + \frac{1}{m^{2}} ηẼ_{c^{m}}^{0}(ξ_{c}^{m}) = ∥ξ_{c}∥_{1,m_{c}}^{2} +ηẼ_{c}^{0}(ξ_{c}) and, moreover, ξ_{c}^{m} + \frac{1}{m^{2}} η grad_{mm_{c}} Ẽ_{c^{m}}^{0}(ξ_{c}^{m}) = ξ_{c}^{m} + ηm_{∗} grad_{m_{c}} Ẽ_{c}^{0}(ξ_{c}).

Lemma 32. There is an arbitrarily small invariant Gromoll–Meyer pair (W_{c^{m}}^{1}, W_{c^{m}−}^{1}) of 0_{c^{m}}^{±} corresponding to the nondegenerate part Ẽ_{c^{m}}^{1}, an invariant local h-decomposition pair (U_{c^{m}}, Ũ_{c^{m}}) ≃ (W_{c^{m}}^{1}, W_{c^{m}−}^{1}) × (W_{c}^{0}, W_{c−}^{0}) of 0_{c^{m}} and an invariant local h-decomposition pair (V, Ṽ), k_{V} of the zero section 0_{G.c^{m}} with (V, Ṽ) ≃ G×_{G_{c^{m}}} (U_{c^{m}}, Ũ_{c^{m}}). Furthermore, there exists a homeomorphic invariant local h-decomposition pair (V′, Ṽ′), k_{V} of the orbit G.c^{m}. Given ϱ, ε > 0, there is a homotopy equivalent local h-decomposition pair (U, Ũ), k_{U} of the orbit with T_{U} ⊆ B_{ϱ}(G.c^{m}) ∩ [m^{2}κ − ε, m^{2}κ + ε] corresponding to E and φ.

Proof. By Lemma 27 there is an equivariant extended Morse chart Φ̃ : (D_{ϱ_{3}}^{±}(0_{G.c^{m}}) × D_{ϱ_{2}}^{0}(0_{G.c^{m}}), 0_{G.c^{m}}) → (Õ, 0_{G.c^{m}}) extending m_{∗} ∘ W_{char} ∘ m_{∗}^{−1}. Denote by Ẽ_{c^{m}}^{1} := E ∘ ẽxp ∘ Φ̃|D_{ϱ_{3}}^{±}(0_{c^{m}}). The function Ẽ_{c^{m}}^{1}(ξ^{±}) = \frac{1}{2}(∥ξ_{c}^{+}∥_{m_{c}}^{2} − ∥ξ_{c}^{−}∥_{m_{c}}^{2}) satisfies (C) with respect to ⟨ , ⟩_{m_{c}} on every closed neighborhood D_{c^{m}}^{±} ⊆ D_{ϱ_{3}}^{±}(0_{c^{m}}).

From Product Proposition 28 with D^{1} = D_{c^{m}}^{±} and D^{0} = D_{ϱ_{0}}^{0}(0_{c^{m}}) it follows that there are a Gromoll–Meyer pair (W_{c^{m}}^{1}, W_{c^{m}−}^{1}) of 0_{c^{m}}^{±} with W_{c^{m}}^{1} ⊆ B_{ϱ_{3}}^{±}(0_{c^{m}}) corresponding to Ẽ_{c^{m}}^{1} and ⟨ , ⟩_{mm_{c}}^{±} and a local h-decomposition pair (U_{c^{m}}, Ũ_{c^{m}}), k of 0_{c^{m}} with U_{c^{m}} ⊆ B_{ϱ_{3}}^{±}(0_{c^{m}}) × B_{ϱ_{2}}^{0}(0_{c^{m}}) corresponding to (Ẽ ∘ Φ̃)_{c^{m}} and ⟨ , ⟩_{mm_{c}}, with (U_{c^{m}}, Ũ_{c^{m}}) ≃ (W_{c^{m}}^{1}, W̃_{c^{m}−}^{1}) × (W_{c}^{0}, W_{c−}^{0}).

From Lemma 16 it follows that (U, Ũ) := (G.U_{c^{m}}, G.Ũ_{c^{m}}), k is a local h-decomposition pair of 0_{G.c^{m}} corresponding to Ẽ and the canonical metric ⟨ , ⟩_{ξ_{p}} of Lemma 26 formed by ⟨ , ⟩_{m_{c}}. From Proposition 23 (change of metric) it follows that there is a local h-decomposition pair (V, Ṽ), k_{V} corresponding to Ẽ ∘ Φ̃ and to the pull back metric g̃. (V′, Ṽ′) := (ẽxp ∘ Φ̃(V), ẽxp∘Φ̃(Ṽ)), k_{V} is a local h-decomposition pair of G.c^{m} corresponding to φ and E with properties [exc] by Corollary 18. The assertion of the lemma follows from Proposition 23.

Lemma 33. If λ := λ(c^{m}) is the index of the iterated geodesic and (W_{c}^{0}, W_{c−}^{0}) ≃ (K, L) is homotopy equivalent to some relative CW-complex and G.c^{m} ≃ P to some CW-complex, then (U, Ũ) is homotopy equivalent to a relative CW-complex (Y, B) having as many cells of the same dimension as P × (D^{λ}, S^{λ−1}) × (K, L).

Proof. From Lemma 19 it follows that (W_{c}^{1}, W_{c−}^{1}) ≃ (D^{λ}, S^{λ−1}). The claim follows from Corollary 22.

Note that the iteration map m_{∗} induces a homeomorphism between the orbits G.c and G.c^{m} because it is equivariant.

2.2. h-Morse Decomposition of ΛM.

Proof of Theorem 1. Let there be n prime distinct nontrivial closed geodesics {c_{n}}_{n∈{1,... ,n_{max}}} on M and denote κ_{n} := E(c_{n}). For every prime geodesic choose a partition

{l_{in}m_{ijn}} of ℕ^{∗}, {l_{in}}_{i∈{1,... ,r_{n}}},

(m_{ijn})_{i∈{1,... ,r_{n}},j∈ℕ^{∗}} with ν(c_{n}^{l_{i}m_{ij}}) = ν(c_{n}^{l_{i}}) as in [2, Lemma 2] or [13, Proposition 4.2.6]. For every n and i choose a characteristic manifold W_{char}^{in} : D^{0}(G.c_{n}^{l_{in}}) → Õ(G.c_{n}^{l_{in}}) and a Gromoll–Meyer pair (W_{in}^{0}, W_{in−}^{0}) ⊆ D^{0}(0_{c_{n}^{l_{in}}}) for 0_{c_{n}^{l_{in}}} corresponding to Ẽ_{c_{n}^{l_{in}}}^{0} with Ẽ_{c_{n}^{l_{in}}}^{0} (W_{in}^{0}) ≥ κ_{in} − δ_{in}, and

0 < 3δ_{in} < κ_{in} := l_{in}^{2}κ_{n}.

There are only finitely many critical orbits in E^{u} for all u and W_{in}^{0} is compact. From Lemma 32 with m = l_{in}m_{ijn} it follows that there is a local h-decomposition pair (V_{nm}, Ṽ_{nm}), k′_{nm} of 0_{G.c_{n}^{m}} containing only this critical orbit and having the property [exc] and (V_{nm}, Ṽ_{nm}) ∩ N_{c_{n}^{m}} ≃ (W_{nm}^{1}, W_{nm−}^{1}) × (W_{in}^{0}, W_{in−}^{0}) for some Gromoll–Meyer pair (W_{nm}^{1}, W_{nm−}^{1}). The pair can be chosen such that Ẽ(V_{nm}) ≥ m_{ijn}^{2}(κ_{in} − 2δ_{in}). Set (V′_{nm}, Ṽ′_{nm}) := ẽxp(V_{nm}, Ṽ_{nm}), k′_{nm}. Because E^{u} intersects only finitely many V′_{nm} and G.c_{n}^{m} is compact there are homotopy equivalent local hdecomposition pairs (U_{nm}, Ũ_{nm}), k_{nm} of the orbits G.c_{n}^{m} with k_{nm} ≥ k′_{nm} and tops T_{nm} disjoint from all other U_{n′m′}. Set K_{0} := Λ^{0}M ≅ M and choose k_{0} < min_{n, i}{κ_{in} − 2δ_{in}} which is not critical. Then k_{0} < k_{nm}, K_{0} ↪ Λ^{k_{0}}M is a deformation retract and (U_{0}, ∅) := (Λ^{k_{0}}M, ∅), k_{0} is a local h-decomposition pair of K_{0}. The set {K_{0}, K_{nm} := G.c_{n}^{m}}_{n, m} is a partition of the critical points of E. Choose J = ℕ, enumerate {k_{0}, k_{nm}} in nondecreasing order and obtain a (absolute) h-Morse decomposition of ΛM.

Choosing fixed relative CW-complexes (K_{in}, L_{in}) ≃ (W_{in}^{0}, W_{in−}^{0}) and P_{n} ≃ G.c_{n} and (Y_{nm}, B_{nm}) := P_{n} × (D^{λ(c_{n}^{m})}, S^{λ(c_{n}^{m})−1}) × (K_{in}, L_{in}) ≃ (U_{nm}, Ũ_{nm}), the claim follows from combinatorial considerations as in [2, Corollary 2, Theorem 4] and [13, Lemma 4.2.5].

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(Received 7.04.1995)

Author’s address: Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288, 69122 Heidelberg Germany