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Spaces of Maps into Eilenberg-Maclane Spaces

Published online by Cambridge University Press:  20 November 2018

Vagn Lundsgaard Hansen
Vagn Lundsgaard Hansen*
Affiliation:
The Technical University of Denmark, Lyngby, Denmark
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In this note we provide alternative and unified proofs for two theorems on the homotopy groups of spaces of (continuous) maps into Eilenberg-MacLane spaces. The first theorem is due to Thorn, and independently Federer, and deals with spaces of maps into Eilenberg- MacLane spaces of type (π, n) for n ≧ 1 with π abelian. The second theorem is due to Gottlieb and deals with spaces of maps into Eilenberg- MacLane spaces of type (π, 1) with π nonabelian. As a main tool we shall use the homotopy sequences for certain fibrations of spaces of maps.

2. Basic notation and some preliminary remarks. For any pair of connected CW-complexes X and Y with base points, we denote by M(X, Y), respectively F(X, Y), the space of free maps, respectively based maps, of X into Y.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 782 - 785
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Dyer, M., Two term conditions in * exact couples, Can. J. Math. 19 (1967), 1263–1288.Google Scholar
2. Fédérer, H., A study of function spaces by spectral sequences, Trans. Amer. Math. Soc. 82 (1956), 340–361.Google Scholar
3. Gottlieb, D. H., Covering transformations and universalfibrations, Illinois J. Math. 18 (1969), 432–437.Google Scholar
4. Gottlieb, D. H., A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840–856.Google Scholar
5. Griffiths, H. B., The fundamental group of a surface, and a theorem of Schreier, Acta Math. 110 (1963), 1–17.Google Scholar
6. Gromoll, D., Klingenberg, W. and Meyer, W., Riemannsche Géométrie im Grossen, Lecture Notes in Mathematics 55 (Springer, 1968).Google Scholar
7. Hansen, V. L., On the fundamental group of a mapping space. An example, Compositio Math. 28 (1974), 33–36.Google Scholar
8. Hansen, V. L., On a theorem of AVber on spaces of maps, J. Diff. Geom. 12 (1977), 565–566.Google Scholar
9. Spanier, E. H., Algebraic topology (McGraw-Hill, 1966).Google Scholar
10. Thorn, R., Uhomologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain (1956). Thone, Liège; Masson, Paris, 1957, 29–39.Google Scholar