# Adiabatic limits of closed orbits for some Newtonian systems by Malchiodi A. By Malchiodi A.

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If X is a 2 complex with X (1) = point, then X is a wedge of 2-spheres X = S2. S 2 → S(2 ) satisfies i) and ii), and is a localization. Assume the Theorem true for all complexes of dimension less than or equal to n − 1. Let X have dimension n. If f : A → A satisfies i), ii) and is a localization then f: A→ A does also. We have the Puppe sequence S n−1  f _  i f S(n−1 ) G X (n−1) c _  GX d G Sn f G X (n−1) G ... 1 X( ) is a local space. Let X( ) be the cofiber of f . Then define : X → X( ) by = n−1 ∪ c(i) : X n−1 ∪f C S n−1 → X(n−1 ) ∪f C S(n−1 .

1 that the natural map X ∪f C(A ) → (X ∪f CA) is an equivalence. We leave to the reader the task of deciding whether this is so in case Y is not simply connected. We note here that no extension of the localization functor to the entire homotopy category can preserve fibrations and cofibrations. For this consider the diagram S2 S1 G S1 double cover double cover  G R P2  natural inclusion R P∞ = K(Z/2, 1) The vertical sequence is a fibration and the horizontal sequence is a cofibration. e. does not contain 2) 40 we obtain S2 S1  G R P2 G S1 ∼ =  R P∞ ∼ =∗ If cofibrations were preserved R P2 should be a point.

This completes the proof for p > 2. If p = 2 certain modifications are required. In step 1, the exact sequence ∗ T 1 → U → Z2 − → Z/2 → 1 comes from “reduction modulo 4” and the equivalence (Z/4)∗ ∼ = Z/2 . 23 Algebraic Constructions The natural splitting is obtained by lifting Z/2 = {0, 1} to {±1} ⊆ Z2 . But the exponential map is only defined on the square of the maximal ideal, e 4Z2 − →U. ) = n − ϕ2 (n) n (2y) means is defined (and even) for all n but only approaches zero n! as required if y is also even.