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By de Rham G., Maumary S., Kervaire M.A.

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9. HOW TO REMOVE THE ASSUMPTION (A8) If we drop assumption (A8), there need not exist a function ρ satisfying (34), and it becomes more diﬃcult to associate a cellular filtration to X. Nevertheless, we can make the graded group C∗ (X) into a chain complex by taking a direct limit of the Morse complexes on sublevels { f < a}, for a ↑ sup f . On these domains indeed, there are finitely many rest points and condition (A7) guarantees condition (A8). 12. If the supremum of f on M is attained, by (A2) and (A6) X has finitely many rest points, so (A8) is implied by (A7).

Indeed, by the mean value theorem there is sn ∈]0, tn [ such that D f φ(sn , pn ) X φ(sn , pn ) = f (pn ) − f φ(tn , pn ) , tn and by (29), qn = φ(sn , pn ) is a (PS) sequence. Actually, the above observation could be used to give a weaker formulation of the (PS) condition, which does not require f to be diﬀerentiable, and could be used to study flows in the continuous category. 2. THE MORSE – SMALE CONDITION We recall that two closed linear subspaces V1 , V2 of a Banach space E are said transverse if V1 +V2 = E and V1 ∩V2 is complemented in E.

Then α is homotopic to the map β: ∂Dk → Mk−1 , ζ → φ b(ζ), α(ζ) , 36 A. ABBONDANDOLO AND P. MAJER so θ∗x (ωk ) = i∗ α∗ (σk−1 ) = i∗ β∗ (σk−1 ). Denote by γi : (Dk−1 , ∂Dk−1 ) → (Mk−1 , Mk−2 ) the composition of i ◦ β with an orientation preserving homeomorphism (Dk−1 , ∂Dk−1 ) → Bρ (ζi ), ∂Bρ (ζi ) . 12 shows that: h θ∗x (ωk ) = i∗ β∗ (σk−1 ) = γi ∗ (ωk−1 ). (40) i=1 Fix some i ∈ {1, . . , of β(ζi ), converges for t → +∞, and let Wi be the connected component of W u (x) ∩ W s (y) consisting of such an orbit.