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A theoretical framework for proper homotopy theory

Published online by Cambridge University Press:  24 October 2008

R. Ayala ,
A. Quintero  and
E. Dominguez
R. Ayala
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain
A. Quintero
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain
E. Dominguez
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
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Abstract

Following the techniques of ordinary homotopy theory, a theoretical treatment of proper homotopy theory, including the known proper homotopy groups, is provided within Baues's theory of cofibration categories.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 3 , May 1990 , pp. 475 - 482
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Arkhangelskii, A. V. and Ponomarev, V. I.. Fundamentals of General Topology (D. Reidel Publishing Co., 1984).Google Scholar
[2]Ayala, R., Dominguez, E. and Quintero, A.. Calculations of p-cylindrical homotopy groups. Proc. Edinburgh Math. Soc. (To appear.)Google Scholar
[3]Ayala, R., Dominguez, E. and Quintero, A.. About p-cylindrical homotopy groups. (Preprint.)Google Scholar
[4]Baues, H. J.. Algebraic Homotopy. Cambridge Studies in Advanced Math. no. 15 (Cambridge University Press, 1988).Google Scholar
[5]Bourbaki, N.. General Topology, vol. 1 (Hermann, 1966).Google Scholar
[6]Brin, M. G. 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
[7]Brown, E. M.. Proper homotopy theory in simplicial complexes. In Topology Conference Virginia Polytechnic Inst. 1973, Lecture Notes in Math. vol. 375 (Springer-Verlag, 1974), pp. 41–46.Google Scholar
[8]Brown, E. M. and Tucker, T. W.. On proper homotopy theory for noncompact 3-manifolds. Trans. Amer. Math. Soc. 188 (1974), 105–126.CrossRefGoogle Scholar
[9]Čerin, Z.. On various relative proper homotopy groups. Tsukuba J. Math. 4 (1980), 177–202.Google Scholar
[10]Cordier, J. M. and Porter, T.. Homotopy limits and homotopy coherence: a report on joint work. Notes on Lectures given at Perugia, Sept.–Oct. 1984 (Università di Perugia).Google Scholar
[11]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
[12]Dominguez, E. and Hernández, L. J.. Remarks about proper ends. (Preprint.)Google Scholar
[13]Dugundji, J.. Topology (Allyn & Bacon, 1966).Google Scholar
[14]Edwards, D. A. and Hastings, H. M.. Čech and Steenrod Homotopy Theories with Applications to Geometric Topology. Lecture Notes in Math. vol. 542 (Springer-Verlag, 1976).CrossRefGoogle Scholar
[15]Farrell, F. T., Taylor, L. R. and Wagoner, J. B.. The Whitehead Theorem in the proper category. Compositio Math. 27 (1973), 1–23.Google Scholar
[16]Freudenthal, H.. Über die Enden topologischer Räume und Gruppen. Math. Z. 33 (1931), 692–71.CrossRefGoogle Scholar
[17]Gabriel, P. and Zisman, M.. Calculus of Fractions and Homotopy Theory. Ergeb. Math. Grenzgeb. vol. 35 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[18]Hernández, L. J.. A note on proper homology and homotopy groups. (Preprint.)Google Scholar
[19]Hernández, L. J. and Porter, T.. Global analogues of the Brown–Grossman proper homotopy groups of an end. Math. Proc. Cambridge Philos. Soc. 104 (1988), 483–496.CrossRefGoogle Scholar
[20]Hernández, 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
[21]James, I. M.. General Topology and Homotopy Theory (Springer-Verlag, 1984).CrossRefGoogle Scholar
[22]MacLane, S.. Categories for the Working Mathematician. Graduate Texts in Math. vol. 5 (Springer-Verlag, 1971).Google Scholar
[23]Mihalik, M.. Semistability at the end of a group extension. Trans. Amer. Math. Soc. 277 (1983), 307–321.CrossRefGoogle Scholar
[24]Porter, T.. Čech and Steenrod homotopy and the Quigley exact couple in strong shape and proper homotopy theory. J. Pure Appl. Algebra 24 (1983), 303–312.CrossRefGoogle Scholar
[25]Porter, T.. Homotopy groups for strong shape and proper homotopy. In Convegno di Topologia, Suppl. Rendiconti Circ. Mat. Palermo. Serie II, no. 4 (1984), pp. 101–111.Google Scholar
[26]Porter, T.. Proper Homotopy, Prohomotopy and Coherence. Sem. Mat. Garcia Galdeano, Ser. II, Sec. 3, no. 10 (Universidad de Zaragoza).Google Scholar
[27]Quigley, J. B.. An exact sequence from the n-th to the (n − 1)-st fundamental groups. Fund. Math. 77 (1973), 195–210.CrossRefGoogle Scholar
[28]Quillen, D.. Homotopical Algebra. Lecture Notes in Math. vol. 43 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[29]Siebenmann, L.. The obstruction to finding a boundary for an open manifold of dimension ≥ 5. Ph.D. thesis, Princeton University (1965).Google Scholar
[30]Siebenmann, L.. On detecting Euclidean space homotopically among topological manifolds. Invent. Math. 6 (1968), 245–261.CrossRefGoogle Scholar
[31]Strøm, A.. Note on cofibrations. I. Math. Scand. 19 (1966), 11–14.CrossRefGoogle Scholar
[32]Strøm, A.. Note on cofibrations. II. Math. Scand. 22 (1968), 130–142.CrossRefGoogle Scholar