Nilpotency of self homotopy equivalences with coefficients
[Nilpotence des auto-équivalences d’homotopie avec des coefficients]
Cuvilliez, Maxence1 ; Murillo, Aniceto2 ; Viruel, Antonio3
1 Universidad de Málaga Departamento de Álgebra, Geometría y Topología Ap. 59, 29080 Málaga (Spain)
2 Universidad de Málaga Departamento de Álgebra, Geometría y Topología Ap. 59, 29080 Málaga, SPAIN
3 Universidad de Málaga, Departamento de Álgebra, Geometría y Topología, Ap. 59, 29080 Málaga, SPAIN.
Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 351-364.

Nous étudions la nilpotence de certains groupes d’auto-équivalences d’homotopie. Notre objectif principal est d’étendre, aux groupes d’homotopy localisés et/ou aux groupes homotopie avec des coefficients, le principe général de Dror et A.  Zabrodsky par lequel un groupe d’auto-équivalences d’homotopie d’un complexe fini, qui agit de façon nilpotente sur les groupes homotopie, est lui-même nilpotent

In this paper we study the nilpotency of certain groups of self homotopy equivalences. Our main goal is to extend, to localized homotopy groups and/or homotopy groups with coefficients, the general principle of Dror and Zabrodsky by which a group of self homotopy equivalences of a finite complex which acts nilpotently on the homotopy groups is itself nilpotent.

DOI : 10.5802/aif.2604
Classification : 55P10
Keywords: Self homotopy equivalence
Mot clés : auto équivalence d’homotopie
Cuvilliez, Maxence 1 ; Murillo, Aniceto 2 ; Viruel, Antonio 3

1 Universidad de Málaga Departamento de Álgebra, Geometría y Topología Ap. 59, 29080 Málaga (Spain)
2 Universidad de Málaga Departamento de Álgebra, Geometría y Topología Ap. 59, 29080 Málaga, SPAIN
3 Universidad de Málaga, Departamento de Álgebra, Geometría y Topología, Ap. 59, 29080 Málaga, SPAIN. 
@article{AIF_2011__61_1_351_0,
     author = {Cuvilliez, Maxence and Murillo, Aniceto and Viruel, Antonio},
     title = {Nilpotency of self homotopy equivalences with coefficients},
     journal = {Annales de l'Institut Fourier},
     pages = {351--364},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {1},
     year = {2011},
     doi = {10.5802/aif.2604},
     zbl = {1221.55008},
     mrnumber = {2828133},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2604/}
}
TY  - JOUR
AU  - Cuvilliez, Maxence
AU  - Murillo, Aniceto
AU  - Viruel, Antonio
TI  - Nilpotency of self homotopy equivalences with coefficients
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 351
EP  - 364
VL  - 61
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2604/
DO  - 10.5802/aif.2604
LA  - en
ID  - AIF_2011__61_1_351_0
ER  - 
%0 Journal Article
%A Cuvilliez, Maxence
%A Murillo, Aniceto
%A Viruel, Antonio
%T Nilpotency of self homotopy equivalences with coefficients
%J Annales de l'Institut Fourier
%D 2011
%P 351-364
%V 61
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2604/
%R 10.5802/aif.2604
%G en
%F AIF_2011__61_1_351_0
Cuvilliez, Maxence; Murillo, Aniceto; Viruel, Antonio. Nilpotency of self homotopy equivalences with coefficients. Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 351-364. doi : 10.5802/aif.2604. http://www.numdam.org/articles/10.5802/aif.2604/

[1] Arkowitz, Martin Problems on Self-homotopy equivalences, Contemp. Math., Volume 274 (2001), pp. 309-315 | MR | Zbl

[2] Arkowitz, Martin; Lupton, Gregory; Murillo, Aniceto Subgroups of the group of self-homotopy equivalences, Contemp. Math., Volume 274 (2001), pp. 21-32 | MR | Zbl

[3] Baues, H.J. Obstruction Theory, Lectures Notes in Math., 628, Springer, 1977 | MR

[4] Dror, E.; Dwyer, W.; Kan, D. Self-homotopy equivalences of virtually nilpotent spaces, Comm. Math. Helv., Volume 56 (1981), pp. 599-614 | DOI | MR | Zbl

[5] Dror, E.; Zabrodsky, A. Unipotency and nilpotency in homotopy equivalences, Topology, Volume 18 (1979), pp. 187-197 | DOI | MR | Zbl

[6] Garvín, A.; Murillo, P. A.and Pavesic; Viruel, A. Nilpotency and localization of groups of fiber homotopy equivalences, Contemporary Math., Volume 274 (2001), pp. 145-157 | MR | Zbl

[7] Gitler, S. Operations with local coefficients, Amer. Journal of Math., Volume 82 (1963) no. 2, pp. 156-188 | DOI | MR | Zbl

[8] Gorenstein, D. Finite groups, Harper and Row, 1968 | MR | Zbl

[9] Hilton, P.; Mislin, G.; Roitberg, J. Localization of Nilpotent Groups and Spaces, Mathematics Studies, 15, North-Holland, 1975 | MR | Zbl

[10] Magnus, W.; Karrass, A.; Solitar, D. Combinatorial Group Theory, Pure and Applied mathematics, 13, Interscience Publishers, 1966 | Zbl

[11] Maruyama, K. Localization of a certain group of self-homotopy equivalences, Pacific Journal of Math., Volume 136 (1989), pp. 293-301 | MR | Zbl

[12] Maruyama, K.; Mimura, M. Nilpotent groups of the group of self-homotopy equivalences, Israel Journal of Math., Volume 72 (1990), pp. 313-319 | DOI | MR | Zbl

[13] Møller, J. Spaces of sections of Eilenberg-Mac Lane fibrations, Pacific Jour. of Math., Volume 130 (1987) no. 1, pp. 171-186 | MR | Zbl

[14] Møller, J. Self-homotopy equivalences of H * (-;ℤ/p)-local spaces, Koday Math. Jour., Volume 12 (1989), pp. 270-281 | DOI | MR | Zbl

[15] Rutter, J. Homotopy self–equivalences 1988–1999, Contemporary Math., Volume 274 (2001), pp. 1-12 | MR | Zbl

[16] Scheerer, H.; Tanré, D. Variation zum Konzept der Lusternik-Schnirelmann Kategorie, Math. Nachr., Volume 207 (1999), pp. 183-194 | MR | Zbl

[17] Siegel, J. k-invariants in local coefficients theory, Proc. Amer. Math. Soc., Volume 29 (1971), pp. 169-174 | MR | Zbl

[18] Sullivan, D. Infinitesimal computations in topology, I.H.E.S. Publ. Math., Volume 47 (1977), pp. 269-331 | Numdam | MR | Zbl

[19] Whitehead, G. Elements of Homotopy Theory, Graduate Texts in Math., 61, Springer, 1978 | MR | Zbl

[20] Wilkerson, C. Applications of minimal simplicial groups, Topology, Volume 15 (1976), pp. 115-130 | DOI | MR | Zbl

Cité par Sources :