Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T14:23:32.182Z Has data issue: false hasContentIssue false
  • English
  • Français

A Whitehead–Ganea approach for proper Lusternik–Schnirelmann category†

Published online by Cambridge University Press:  01 May 2007

J. M. GARCÍA–CALCINES ,
P. R. GARCÍA–DÍAZ  and
A. MURILLO MAS
J. M. GARCÍA–CALCINES
Affiliation:
Departamento de Matemática Fundamental, Universidad de La Laguna, 38271 Islas Canarias, Spain. e-mail: jmgarcal@ull.es, prgdiaz@ull.es
P. R. GARCÍA–DÍAZ
Affiliation:
Departamento de Matemática Fundamental, Universidad de La Laguna, 38271 Islas Canarias, Spain. e-mail: jmgarcal@ull.es, prgdiaz@ull.es
A. MURILLO MAS
Affiliation:
Departamento of Álgebra, Geometría y Topología, Universidad of Málaga, Ap. 59, 29080 Málaga, Spain. e-mail: aniceto@agt.cie.uma.es
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish Whitehead and Ganea characterizations for proper LS-category. We use the embedding of the proper category into the exterior category, and construct in the latter a suitable closed model structure of Strøm type. Then, from the axiomatic LS-category arising from the exterior homotopy category we can recover the corresponding proper LS invariants. Some applications are given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 439 - 457
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ayala, R., Domínguez, E. and Quintero, A..A theoretical framework for proper homotopy theory. Math. Proc. Camb. Phil. Soc. 107 (1990), 475–482.CrossRefGoogle Scholar
[2]Ayala, R., Domínguez, E., Márquez, A. and Quintero, A.. Lusternik–Schnirelmann invariants in proper homotopy theory. Pacific J. Math. 153 (1992), 201–215.CrossRefGoogle Scholar
[3]Ayala, R. and Quintero, A.. On the Ganea strong category in proper homotopy. Proc. Edinburgh Math. Soc. 41 (1998), 247–263.CrossRefGoogle Scholar
[4]Baues, H. J.. Algebraic Homotopy. Cambridge Stud. Adv. Math., 15 (Cambridge University Press, 1989).CrossRefGoogle Scholar
[5]Bousfield, A. K. and Kan, D. M.. Homotopy limits, completions and localizations. Lecture Notes in Mathematics 304 (1972).Google Scholar
[6]Cárdenas, M., Lasheras, F. F. and Quintero, A.. Minimal covers of open manifolds with half-spaces and the proper L-S category of product spaces. Bull. Belgian Math. Soc. 9 (2002), 419–431.Google Scholar
[7]Cárdenas, M., Lasheras, F. F., Muro, F. and Quintero, A.. Proper L-S category, fundamental pro-groups and 2-dimensional proper co-H-spaces. Topology Appl., in press. DOI: 10.1016/j.topol.2005.01.032.CrossRefGoogle Scholar
[8]Cárdenas, M., Muro, F. and Quintero, A.. The proper L-S category of Whitehead manifolds. Topology Appl., in press. DOI: 10.1016/j.topol.2005.01.031.CrossRefGoogle Scholar
[9]Clapp, M. and Puppe, D.. Invariants of the Lusternik–Schnirelmann type and the topology of critical sets. Trans. Amer. Math. Soc. 298 (1986), 603–620.CrossRefGoogle Scholar
[10]Cornea, O., Lupton, G., Oprea, J. and D. Tanré. Lusternik–Schnirelmann category. Math. Surveys Monogr. 103 (2003).CrossRefGoogle Scholar
[11]Doeraene, J. P.. Homotopy pullbacks, homotopy pushouts and joins. Bull. Belg. Math. Soc. 5 (1) (1998), 15-37.Google Scholar
[12]Doeraene, J. P.. L. S.-category in a model category. J. Pure Appl. Algebra 84 (1993), 215–261.CrossRefGoogle Scholar
[13]Doeraene, J. P. and Tanré, D.. Axiome du cube et Foncteurs de Quillen. Ann. Instit. Fourier 45 (4), (1995), 1061–1077.CrossRefGoogle Scholar
[14]Edwards, D. and Hastings, H.. Čech and Steenrod Homotopy Theories with Applications to Geometric Topology. Lecture Notes in Math. 542 (Springer, 1976).CrossRefGoogle Scholar
[15]Félix, Y. and Murillo, A.. A bound for the nilpotency of a group of self homotopy equivalences. Proc. Amer. Math. Soc. 126 (2) (1998), 625–627.CrossRefGoogle Scholar
[16]García–Calcines, J. M., García–Pinillos, M. and Hernández–Paricio, L. J.. A closed model category for proper homotopy and shape theories. Bull. Austral. Math. Soc. 57 (2) (1998) 221–242.CrossRefGoogle Scholar
[17]García–Calcines, J. M., García–Pinillos, M. and Hernández–Paricio, L. J.. Closed simplicial model structures for exterior and proper homotopy theory. Appl. Categ. Structures 12 (3) (2004), 225–243.CrossRefGoogle Scholar
[18]García–Calcines, J. M. and Hernández–Paricio, L. J.. Sequential homology. Topology Appl. 114 (2001), 201–225.Google Scholar
[19]James, I. M.. On category in the sense of Lusternik–Schnirelmann. Topology 17 (1978), 331–349.CrossRefGoogle Scholar
[20]Mather, M.. Pull-backs in Homotopy Theory. Canad. J. Math. 28 (2) (1976), 225–263.CrossRefGoogle Scholar
[21]Quillen, D.. Homotopical Algebra. Lecture Notes in Math. 43 (Springer, 1967).CrossRefGoogle Scholar
[22]Strøm, A.. The homotopy category is a homotopy category. Arch. Math. 23 (1972), 435–441.CrossRefGoogle Scholar