Hostname: page-component-5c6d5d7d68-wtssw Total loading time: 0 Render date: 2024-08-17T17:55:02.688Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Ganea strong category in proper homotopy

Published online by Cambridge University Press:  20 January 2009

R. Ayala  and
A. Quintero
R. Ayala
Affiliation:
Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, Apartado 1160, 41080-Sevilla, Spain
A. Quintero
Affiliation:
Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, Apartado 1160, 41080-Sevilla, Spain
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper contains some basic relations between Ganea strong category and Lusternik Schnirelmann category in proper homotopy theory. We focus our interest on the case of category 2 in order to show that ℚn is the unique open n-manifold with proper Lusternik-Schnirelmann category 2 (n ≠ 3).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 2 , June 1998 , pp. 247 - 263
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

REFERENCES

1.Ayala, R., Domínguez, E. and Quintero, A., A theoretical framework for proper homotopy theory, Math. Proc. Cambridge Philos. Soc. 107 (1990), 475–482.CrossRefGoogle Scholar
2.Ayala, R., Domínguez, E., Márquez, A. and Quintero, A., Lusternik-Schnirelmann invariants in proper homotopy, Pacific J. Math. 153 (1992), 201–215.CrossRefGoogle Scholar
3.Ayala, R., Domínguez, E. and Quintero, A., Hurewicz Theorem for homology at infinity, Quart. J. Math. Oxford (2) 44 (1993), 139–153.CrossRefGoogle Scholar
4.Ball, B. J. and Sher, R. B.. A theory of proper shape for locally compact metric spaces, Fund. Math. 86 (1974), 162–192.CrossRefGoogle Scholar
5.Baues, H. J., Algebraic Homotopy (Cambridge Studies in Advanced Maths., 15, Cambridge Univ. Press, 1989).CrossRefGoogle Scholar
6.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
7.Cornea, O., Cone-length and Lusternik-Schnirelmann category, Topology 33 (1993), 95–111.CrossRefGoogle Scholar
8.Edwards, D. A. and Hastings, H. M., ech and Steenrod homotopy theories with applications to Geometric Topology (Lecture Notes 542, Springer, 1976).CrossRefGoogle Scholar
9.Engelking, R. and Sieklucki, K., Topology. A geometric approach (Heldermann, 1992).Google Scholar
10.Fox, R. H., On the Lusternik-Schnirelmann category, Ann. of Math. 42 (1941), 333–370.CrossRefGoogle Scholar
11.Freedman, M. H., The topology of 4-manifolds, J. Differential Geom. 17 (1982), 375–453.CrossRefGoogle Scholar
12.Ganea, T., Lusternik-Schnirelmann category and strong category, Illiois J. Math. 11 (1967), 417–427.Google Scholar
13.Glaser, L., Geometrical Combinatorial Topology. Vol II (Van Nostrand, 1972).Google Scholar
14.Hess, K. and Lemaire, J. M., Generalizing a definition of Lusternik and Schnirelmann to model categories, J. Pure Appl. Algebra 91 (1994), 165–182.CrossRefGoogle Scholar
15.James, I. M., On category in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331–349.CrossRefGoogle Scholar
16.Kirby, R. C. and Siebenmann, L., Foundational essays on topological manifolds, smoothings, and triangulations (Annals of Math. Studies, 88, Princeton Univ. Press, 1977).CrossRefGoogle Scholar
17.Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory (Springer, 1977).Google Scholar
18.Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory (Wiley, 1966).Google Scholar
19.Mardesić, S. and Segal, J., Shape Theory (North-Holland, 1982).Google Scholar
20.Massey, W. S., Homology and Cohomology Theory (M. Dekker, 1978).Google Scholar
21.Mihalik, M., Semistability at the end of a group extension, Trans. Amer. Math. Soc. (1983), 307–321.CrossRefGoogle Scholar
22.Montejano, L., Lusternik-Schnirelmann category: A geometric approach (Banach Center Publications 18. PWN, 1986), 119–131.Google Scholar
23.Siebenmann, L., On detecting Euclidean Space homotopically among topological manifolds, Invent. Math. 6 (1968), 245–261.CrossRefGoogle Scholar
24.Takens, F., The Lusternik-Schnirelmann categories of a product space, Compositio Math. 22 (1970), 175–180.Google Scholar
25.Wall, C. T. C., Open 3-manifolds which are 1-connected at infinity, Quart. J. Math. Oxford (1) 16 (1965), 263–268.CrossRefGoogle Scholar
26.Whitehead, G. W., Elements of Homotopy Theory (GTM, 61. Springer, 1978).CrossRefGoogle Scholar