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Global analogues of the Brown–Grossman proper homotopy groups of an end

Published online by Cambridge University Press:  24 October 2008

L. J. Hernandez  and
T. Porter
L. J. Hernandez
Affiliation:
Departamento de Matematicas, Facultad de Ciencias, 50009 Zaragosa, Spain
T. Porter
Affiliation:
Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd, LL57 2UW, Wales
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Abstract

The Brown–Grossman proper homotopy groups at an end of a locally compact Hausdorff space, X, were introduced by E. M. Brown in 1974. In this note we define and study ‘global’ versions of these groups and compare these global groups with the groups ‘at infinity’. We also obtain several interlocking exact sequences relating these groups with the Hurewicz and Steenrod homotopy groups of X.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 3 , November 1988 , pp. 483 - 496
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Brin, M. and Thickstun, T. L.. On the proper Steenrod homotopy groups and proper embeddings of planes into 3-manifolds. Trans. Amer. Math. Soc. 289 (1985), 737–755.CrossRefGoogle Scholar
[2]Brown, E. M.. On the proper homotopy type of simplicial complexes. In Topology Conference, Lecture Notes in Math. vol. 375 (Springer-Verlag, 1974). pp. 41–46.CrossRefGoogle Scholar
[3]Čerin, Z.. On various relative proper homotopy groups. Tsukuba J. Math. 4 (1980), 177–202.Google Scholar
[4]Cordier, J.-M.. Sur la notion de diagramme homotopiquement cohèrent. Cahiers Topologie Géom. Différentielle Catégoriques 23 (1982), 93–112.Google Scholar
[5]Cordier, J.-M. and Porter, T.. Homotopy limits and homotopy coherence. Lectures given at Perugia, Sept.–Oct. 1984.Google Scholar
[6]Cordier, J.-M. and Porter, T.. Vogt's theorem on categories of homotopy coherent diagrams. Math. Proc. Cambridge Philos. Soc. 100 (1986), 65–90.CrossRefGoogle Scholar
[7]Cordier, J.-M. and Porter, T.. Maps between homotopy coherent diagrams. Top Appl. 28 (1988), 255–275.CrossRefGoogle Scholar
[8]Edwards, D. A. and Hastings, H.. Čech and Steenrod Homotopy with Applications to Geometric Topology. Lecture Notes in Math. vol. 542 (Springer-Verlag, 1976).CrossRefGoogle Scholar
[9]Grossman, J. M.. Homotopy groups of prospaces. Illinois J. Math. 20 (1976), 622–625.CrossRefGoogle Scholar
[10]Hernandez, L. J.. A note on proper homology and homotopy groups. (Preprint.)Google Scholar
[11]Hernandez, L. J. and Porter, T.. Proper pointed maps from ℝn+1 to a σ-compact space. Math. Proc. Cambridge Philos. Soc. 103 (1988), 457–462.CrossRefGoogle Scholar
[12]Porter, T.. Čech and Steenrod homotopy and the Quigley exact couple in strong shape and proper homotopy theory. J. Pure Appl. Algebra 26 (1982), 325–330.CrossRefGoogle Scholar
[13]Porter, T.. Homotopy groups for strong shape and proper homotopy. Rend. Circ. Mat. Palermo (2)4(1984), 101–113.Google Scholar
[14]Porter, T.. Proper homotopy, prohomotopy and coherence. Publ. Sem. Mat. García de Galdeano Ser. II 10 (1987).Google Scholar
[15]Quigley, J. B.. An exact sequence from the nth to the (n – 1)st fundamental groups. Fund. Math. 77 (1973), 195–210.CrossRefGoogle Scholar
[16]Quillen, D.. Homotopical Algebra. Lecture Notes in Math. vol. 43 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[17]Vogt, R. M.. Homotopy limits and colimits. Math. Z. 134 (1973), 11–52.CrossRefGoogle Scholar