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A Homotopical Conner-Raymond Theorem and a Question of Gottlieb

Published online by Cambridge University Press:  20 November 2018

John Oprea
John Oprea*
Affiliation:
Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
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Abstract

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A homotopy theoretic version is given of the following result of Conner and Raymond: If the circle acts on a space so that the orbit map induces an injection in homology, then the space fibres over the circle with finite structure group. This homotopical analogue is related to recent results pertaining to the effect of the fundamental group's structure on the Euler characteristic. It is also used in the construction of a compact, simple 7-manifold with trivial Gottlieb group which, together with an infinite dimensional example of Ganea, answers a question of Gottlieb.

Keywords

55P9957S99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 2 , 01 June 1990 , pp. 219 - 229
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Conner, P. and Raymond, F., Injective operations of the toral groups, Top. 10 (1971), 283–296.Google Scholar
2. Eckmann, B., Nilpotent group action and euler characteristic, Springer Lecture Notes Math. vol. 1298 (1987), 120–123.Google Scholar
3. Ganea, T., Cyclic homotopies, 111. J. Math. 12 (1968), 1–4.Google Scholar
4. Gottlieb, D. H., A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840–856.Google Scholar
5. Gottlieb, D. H., Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729–755.Google Scholar
6. Gottlieb, D. H., On fibre spaces and the evaluation map, Ann. of Math. (2) 87 No. 1 (1968), 42–55. Google Scholar
7. Gottlieb, D. H. Splitting off tori and the evaluation subgroup, Israel Journal of Mathematics, Vol. 66, (1989), 216-222.Google Scholar
8. Hilton, P., Homotopy theory and duality, Gordon & Breach, New York 1965.Google Scholar
9. Hilton, P., Mislin, G. and Roitberg, J., Localization of nilpotent groups and spaces, North Holland, Amsterdan 1975.Google Scholar
10. Lang, G., Evaluation subgroups of factor spaces, Pac. J. Math. 42 No. 3 (1972), 701–709.Google Scholar
11. Lück, Wolfgang, The transfer maps induced in the algebraic K0- and K1-groups by a fibration II, J. Pure Appl. Algebra 45(1987), 143–169.Google Scholar
12. Oprea, J., Decompositions of localized fibres and cofibres, Canad. Bull. Math. vol. 31(4) (1988), 424–431.Google Scholar
13. Oprea, J., Decomposition theorems in rational homotopy theory, Proc. Amer. Math. Soc. 96 (1986), 505–512.Google Scholar
14. Oprea, J., The Samelson space of a fibration, Mich. Math. J. 34 (1987), 127–141.Google Scholar
15. Pak, J., On the fibered Jiang spaces, Comtemporary Math. 72(1988), 179–181.Google Scholar
16. Rosset, S., A vanishing theorem for euler characteristics, Math. Zeit. 185 (1984), 211–215.Google Scholar
17. Stallings, J., Centerless groups - an algebraic formulation of Gottlieb's theorem, Top. 4 (1965), 129–134.Google Scholar
18. Siegel, J., G-spaces, H-spaces and W-spaces, Pac. J. Math. 31 no. 1 (1969), 209–214.Google Scholar