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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: [email protected], [email protected]
P. R. GARCÍA–DÍAZ
Affiliation:
Departamento de Matemática Fundamental, Universidad de La Laguna, 38271 Islas Canarias, Spain. e-mail: [email protected], [email protected]
A. MURILLO MAS
Affiliation:
Departamento of Álgebra, Geometría y Topología, Universidad of Málaga, Ap. 59, 29080 Málaga, Spain. e-mail: [email protected]
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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

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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