On the homotopy type of Diff (M n ) and connected problems
Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 3-17.

Cet article est un rapport contenant des résultats sur le type d’homotopie du groupe des difféormorphismes Diff (M n ) d’une variété différentiable compacte M n (munie de la topologie C ) et sur la comparaison homotopique de cet espace avec le groupe des homéomorphismes de la variété M n (munie de la topologie C o ). Comme applications, on obtient des renseignements nouveaux sur les groupes d’homotopie de Diff (D n ,D n ), Top n et Top n /O n et sur le nombre des composantes connexes de l’espace des pseudo-isotopies topologiques et combinatoires.

Les résultats sont énoncés dans les sections 1 et 2 et les idées géométriques sont expliquées dans la section 3.

This paper reports on some results concerning:

a) The homotopy type of the group of diffeomorphisms Diff (M n ) of a differentiable compact manifold M n (with C -topology).

b) the result of the homotopy comparison of this space with the group of all homeomorphisms Homeo M n (with C o -topology). As a biproduct, one gets new facts about the homotopy groups of Diff (D n ,D n ), Top n , Top n /O n and about the number of connected components of the space of topological and combinatorial pseudoisotopies.

The results are contained in Section 1 and Section 2 and the geometric ideas in Section 3.

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Burghelea, Dan. On the homotopy type of ${\rm Diff}(M^n)$ and connected problems. Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 3-17. doi : 10.5802/aif.453. https://aif.centre-mersenne.org/articles/10.5802/aif.453/

[1] P. Antonelli, D. Burghelea, P. J. Kahn, The non finite homotopy type of some Diff... Topology V. 11, 1-49 (1972). | Zbl

[2] P. Antonelli, D. Burghelea, P. J. Kahn, The concordance homotopy groups. Lectures Notes in Math., Springer Verlag, Vol. 215. | Zbl

[3] D. Burghelea et N. Kuiper, Hilbert manifolds. Ann. of Math. V. 90 (1969), 379-417. | MR | Zbl

[4] J. Cerf, La stratification naturelle des espaces des fonctions différentiables réelles et le théorème de pseudoïsotopie. Publications Math. I.H.E.S., Vol. 39. | Numdam | Zbl

[5] J. Cerf, Invariants des paires d'espaces, Applications à la topologie différentielle, C.I.M.E. (1961) (Urbino). | Zbl

[6] A. Haefliger et C.T.C. Wall, Piecewise linear bundles in the stable range, Topology Vol. 4 (1965) p. 209. | MR | Zbl

[7] A. Hatcher, This Issue.

[8] D. Henderson, Infinite dimensional manifolds are open sets of Hilbert space. Topology V. 9 (1970) 25-33. | MR | Zbl

[9] R. Lashof, The immersion approach to triangulation and smoothing. Proceedings of symposia in pure mathematics. Vol. XXII, 131-164. | MR | Zbl

[10] C. Morlet, Isotopie et pseudoïsotopie. C.R. Acad. Sc., A t. 266 (1968) 559-560 and «Cours Pecot», Collège de France (1969). | MR | Zbl

[11] C. Rourke et B. Sanderson, Δ-sets (to appear).

[12] H. Toda, p-primary components of homotopy groups IV. Memoirs of College of Science Univ. Kyoto V. XXXVII p. 297-332. | Zbl

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