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Criteria for components of a function space to be homotopy equivalent

Published online by Cambridge University Press:  01 July 2008

GREGORY LUPTON  and
SAMUEL BRUCE SMITH
GREGORY LUPTON
Affiliation:
Department of Mathematics, Cleveland State University, Cleveland OH 44115, U.S.A. e-mail: [email protected]
SAMUEL BRUCE SMITH
Affiliation:
Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131, U.S.A. e-mail: [email protected]
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Abstract

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 95 - 106
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Allaud, G.. On the classification of fiber spaces. Math. Z. 92 (1966), 110–125.CrossRefGoogle Scholar
[2]Crabb, M. C. and Sutherland, W. A.. Counting homotopy types of gauge groups. Proc. London Math. Soc. (3) 81, no. 3 (2000), 747–768.CrossRefGoogle Scholar
[3]Dold, A.. Partitions of unity in the theory of fibrations. Ann. of Math. (2) 78 (1963), 223–255.CrossRefGoogle Scholar
[4]Dold, A.. Halbexakte homotopiefunktoren. Lecture Notes in Mathematics, vol. 12 (Springer-Verlag, 1966).CrossRefGoogle Scholar
[5]Dugundji, J.. Topology (Allyn and Bacon Inc., 1978), Reprinting of the 1966 original, Allyn and Bacon Series in Advanced Mathematics.Google Scholar
[6]Fuchs, M.. Verallgemeinerte homotopie-homomorphismen und klassifizierende Räume. Math. Ann. 161 (1965), 197–230.CrossRefGoogle Scholar
[7]Golasinski, M. and Mukai, J.. Gottlieb groups of spheres, preprint.Google Scholar
[8]Gottlieb, D. H.. On fibre spaces and the evaluation map. Ann. of Math. (2) 87 (1968), 42–55.CrossRefGoogle Scholar
[9]Gottlieb, D. H.. Evaluation subgroups of homotopy groups. Amer. J. Math. 91 (1969), 729–756.CrossRefGoogle Scholar
[10]Hansen, V. L.. Equivalence of evaluation fibrations. Invent. Math. 23 (1974), 163–171.CrossRefGoogle Scholar
[11]Hansen, V. L.. The homotopy problem for the components in the space of maps on the n-sphere. Quart. J. Math. Oxford Ser. (2) 25 (1974), 313–321.CrossRefGoogle Scholar
[12]Hansen, V. L.. On spaces of maps of n-manifolds into the n-sphere. Trans. Amer. Math. Soc. 265 no. 1, (1981), 273–281.Google Scholar
[13]James, I. M.. General Topology and Homotopy Theory (Springer-Verlag, 1984).CrossRefGoogle Scholar
[14]Mimura, M., Lee, K.-Y. and Woo, M. H.. Gottlieb groups of homogeneous spaces, Topology Appl. 145, no. 1–3, (2004), 147–155.Google Scholar
[15]Kono, A.. A note on the homotopy type of certain gauge groups. Proc. Roy. Soc. Edinburgh Sect. A 117, no. 3–4, (1991), 295–297.CrossRefGoogle Scholar
[16]Kono, A. and Tsukuda, S.. A remark on the homotopy type of certain gauge groups. J. Math. Kyoto Univ. 36, no. 1, (1996), 115–121.Google Scholar
[17]Kono, A. and Tsukuda, S.. 4-manifolds X over BSU(2) and the corresponding homotopy types Map(X, BSU(2)). J. Pure Appl. Algebra 151, no. 3, (2000), 227–237.CrossRefGoogle Scholar
[18]McClendon, J. F.. On evaluation fibrations. Houston J. Math. 7, no. 3, (1981), 379–388.Google Scholar
[19]Milnor, J.. On spaces having the homotopy type of a CW complex. Trans. Amer. Math. Soc. 90 (1959), 272–280.Google Scholar
[20]Møller, J. M.. On spaces of maps between complex projective spaces. Proc. Amer. Math. Soc. 91, no. 3, (1984), 471–476.CrossRefGoogle Scholar
[21]Mislin, G., Hilton, P. and Roitberg, J.. Localization of Nilpotent Groups and Spaces (North-Holland Publishing Co., 1975), North-Holland Mathematics Studies, No. 15, Notas de Matemática, No. 55. [Notes on Mathematics, No. 55.]Google Scholar
[22]Siegel, J.. G-spaces, H-spaces and W-spaces. Pacific J. Math. 31 (1969), 209–214.CrossRefGoogle Scholar
[23]Stasheff, J.. A classification theorem for fibre spaces. Topology 2 (1963), 239–246.CrossRefGoogle Scholar
[24]Sutherland, W. A.. Path-components of function spaces. Quart. J. Math. Oxford Ser. (2) 34, no. 134, (1983), 223–233.CrossRefGoogle Scholar
[25]Sutherland, W. A.. Function spaces related to gauge groups. Proc. Roy. Soc. Edinburgh Sect. A 121, no. 1–2, (1992), 185–190.CrossRefGoogle Scholar
[26]Tsukuda, S.. Comparing the homotopy types of the components of Map(S4,BSU(2)). J. Pure Appl. Algebra 161, no. 1–2, (2001), 235–243.CrossRefGoogle Scholar
[27]Varadarajan, K.. Generalised Gottlieb groups. J. Indian Math. Soc. 33 (1969), 141–164.Google Scholar
[28]Whitehead, G. W.. On products in homotopy groups. Ann. of Math (2) 47 (1946), 460–475.CrossRefGoogle Scholar
[29]Whitehead, G. W.. Elements of homotopy theory. Graduate Texts in Mathematics, vol. 61 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[30]Yoon, Y. S.. On n-cyclic maps. J. Korean Math. Soc. 26, no. 1, (1989), 17–25.Google Scholar