Homotopie de l'espace des équivalences d'homotopie fibrées
Didierjean, Geneviève
Annales de l'Institut Fourier, Tome 35 (1985), p. 33-47 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

On construit une suite spectrale qui converge vers le bigradué associé à une filtration convenable des groupes d’homotopie du monoïde simplicial des équivalences d’homotopie fibrées d’un fibré de Kan dans lui-même. On obtient de nouveaux calculs de ces groupes. En particulier, on calcule le groupe des classes d’homotopie des équivalences d’homotopie d’un espace ayant trois groupes d’homotopie non nuls en dessous de sa dimension.

We construct a spectral sequence converging to the bigraded group associated with a suitable filtration of the homotopy groups of the simplicial monoid consisting of the fibre homotopy equivalences from a Kan fibration into itself. In particular, we calculate the homotopy group of self-equivalences of spaces with three non vanishing homotopy groups under their dimension.

@article{AIF_1985__35_3_33_0,
     author = {Didierjean, Genevi\`eve},
     title = {Homotopie de l'espace des \'equivalences d'homotopie fibr\'ees},
     journal = {Annales de l'Institut Fourier},
     volume = {35},
     year = {1985},
     pages = {33-47},
     doi = {10.5802/aif.1017},
     mrnumber = {87e:55008},
     zbl = {0563.55005},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1985__35_3_33_0}
}

Didierjean, Geneviève. Homotopie de l'espace des équivalences d'homotopie fibrées. Annales de l'Institut Fourier, Tome 35 (1985) pp. 33-47. doi : 10.5802/aif.1017. http://gdmltest.u-ga.fr/item/AIF_1985__35_3_33_0/

[1] M. Arkowitz et C.R. Curjel, Groups of homotopy classes, Lecture Notes in mathematics, Springer Verlag, 4 (1964). | Zbl 0131.20502

[2] W.D. Barcus et M.G. Barratt, On the homotopy classification of a fixed map, Trans. Amer. Math. Soc., 88 (1958), 57-74. | MR 20 #3540 | Zbl 0095.16801

[3] G. Didierjean, Groupes d'homotopie du monoïde des équivalences d'homotopie fibrées, C.R.A.S. Paris, t. 292 (1981), 555-558. | MR 82e:55012 | Zbl 0472.55009

[4] A. Legrand, Homotopie des espaces de sections, Lecture Notes in Mathematics, Springer Verlag, 941 (1981). | Zbl 0535.55001

[5] S. Mac Lane, Homology, Springer Verlag, (1963).

[6] J.P. May, Simplicial objects in algebraic topology, Van Nostrand, (1967). | Zbl 0165.26004

[7] S. Oka, On the group of self homotopy equivalences of H-spaces of low rank I. Memoirs of the Faculty of Science, Kyushu University, Série A. Vol. 35 (1981), 247-282, 307-323. | MR 83h:55006 | Zbl 0486.55003

[8] P. Olum, Self-equivalences of pseudo-projective planes, Topology, 4 (1965), 109-127. | MR 31 #2725 | Zbl 0138.18404

[9] J.W. Rutter, The group of homotopy self-equivalence classes of CW complexes, Math. Proc. Camb. Phil. Soc., 93 (1983), 275-293. | MR 84i:55002 | Zbl 0516.55008

[10] N. Sawashita, On the group of self-equivalences of the product of spheres, Hiroshima Math. J., 5 (1975), 69-86. | MR 51 #14035 | Zbl 0299.55013

[11] Schellenberg, The group of homotopy self-equivalences of some compact CW complexes, Math. Ann., 200 (1973), 253-266. | MR 48 #5062 | Zbl 0234.57005

[12] W. Shih, On the group ℰ(X) of homotopy equivalence maps, Bull. Amer. Math. Soc., 492 (1964), 361-365. | MR 28 #3423 | Zbl 0129.38803

[13] E.H. Spanier, Algebraic topology, Mc Graw-Hill, (1966). | MR 35 #1007 | Zbl 0145.43303

[14] K. Tsukiyama, Self homotopy-equivalences of a space with two non-vanishing homotopy group, Proc. of the Amer. Math. Soc., Vol. 79, n° 1 (1980), 134-138. | MR 81j:55012 | Zbl 0389.55014