Get Algebraic Topology Rational Homotopy: Proceedings of a PDF

By David J. Anick (auth.), Yves Felix (eds.)

ISBN-10: 3540193405

ISBN-13: 9783540193401

This court cases quantity facilities on new advancements in rational homotopy and on their effect on algebra and algebraic topology. many of the papers are unique study papers facing rational homotopy and tame homotopy, cyclic homology, Moore conjectures at the exponents of the homotopy teams of a finite CW-c-complex and homology of loop areas. Of specific curiosity for experts are papers on building of the minimum version in tame idea and computation of the Lusternik-Schnirelmann classification by means of potential articles on Moore conjectures, on tame homotopy and at the homes of Poincaré sequence of loop spaces.

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Additional info for Algebraic Topology Rational Homotopy: Proceedings of a Conference held in Louvain-la-Neuve, Belgium, May 2–6, 1986

Example text

The proof can be found in Smale (1965). 12. GENERICITY OF THE MORSE – SMALE CONDITION Let f be a C h+1 real function, h ≥ 1, on the smooth Hilbert manifold N, endowed with a Riemannian metric g of class C h . Let −∞ < a < b ≤ +∞, and assume (B1) – (B4). The aim of this section is to show that it is possible to perturb the metric g obtaining a uniformly equivalent metric such that the associated negative gradient of f has the Morse – Smale property up to order h. We shall assume N to be infinite-dimensional and second countable (in particular, it is modeled on a separable Hilbert space).

ABBONDANDOLO AND P. MAJER We conclude that the homology of the Morse complex of X is isomorphic to the singular homology of ( M, A), Hk W∗ (X) Hk ( M, A) ∀k ∈ N. Finally, having fixed an orientation for each unstable manifold, we have the isomorphisms Θak : Ck (X)a Wk (X)a , Ck (X)a being the subgroup of Ck (X) generated by the rest points x with f (x) < a, and the limit of this direct system defines an isomorphism Θk : Ck (X) Wk (X). 13. 11) holds also without assuming (A8). 10. MORSE FUNCTIONS ON HILBERT MANIFOLDS A particular but important case is the following situation: f is a C 2 Morse function on a smooth Hilbert manifold N, endowed with a C 1 Riemannian metric g, −∞ < a < b ≤ +∞, M = {p ∈ N | f (p) < b}, M = {p ∈ N | a < f (p) < b}, and X = −∇ f , the negative gradient of f with respect to the metric g.

20. Let f be a C h+1 function, h ≥ 1, on the smooth second countable Hilbert manifold N ⊂ H, endowed with a Riemannian metric g of class C h . Let −∞ < a < b ≤ +∞, and assume (B2). Assume that the continuous function θ: N → [0, +∞[ satisfies (42), that its set of zeroes is the closure of an open set, and that it has the following property: if x, y are critical points in {a < f < b} with m(x) − m(y) ≤ h, such that W u (x) and W s (y) (with respect to −∇g f ) have a nontransverse intersection at p, then θ > 0 somewhere on the orbit of p.

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Algebraic Topology Rational Homotopy: Proceedings of a Conference held in Louvain-la-Neuve, Belgium, May 2–6, 1986 by David J. Anick (auth.), Yves Felix (eds.)


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