Abstract
Let G=S1 or G=S3, and let p : z → X be a bundle with a fibre preserving action of G. Let q : V → Y be a vector space bundle with a fibre preserving action of G. Let f : Z→V be a fibre preserving map. The paper studies the size of the subset Af made up of the orbits over which the average of f is zero. The size of Af depends on the cohomology index of the action on Z and on the type of the action on V which can be described in terms of a Euler number. The result can be viewed as an extension of a continuous version of the Borsuk-Ulam theorem.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Conner, P. E. and Floyd, E. E.: Fixed point free involutions and equivariant maps. Bull. Amer. Math. Soc. 66 (1960), 416–441.
Fadell, E. R. and Rabinowitz, P. H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45 (1978), 139–174.
Fadell, E. R., Husseini, S. and Rabinowitz, P. H.: Borsuk-Ulam theorems for arbitrary S1 actions and applications. Trans. Amer. Math. Soc. 275 (1982), 345–360.
Gottlieb, D. H.: Fibre bundles and the Euler characteristic. J. Differential Geometry 10 (1975), 39–48.
Jaworowski, J.: A continuous version of the Borsuk-Ulam theorem. Proc. Amer. Math. Soc. 82 (1981), 112–114.
Jaworowski, J.: Fibre preserving maps of sphere bundles into vector space bundles. Proc. of the Fixed Point Theory Workshop, Sherbrooke, 1980; Lecture Notes in Mathematics, vol. 886, Springer-Verlag, 1981.
Jaworowski, J.: The set of balanced orbits of maps of S1 and S3 actions. To be published in Proc. Amer. Math. Soc.
Liulevicius, A.: Borsuk-Ulam theorems for spherical space forms. Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982). Contemp. Math. 19 (1983), 189–192.
Nakaoka, M.: Equivariant point theorems for fibre-preserving maps. Preprint.
Yang, C. T.: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson, I. Ann. of Math. 70 (1954), 262–282.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag
About this paper
Cite this paper
Jaworowski, J. (1986). Balanced orbits for fibre preserving maps of S1 and S3 actions. In: Jackowski, S., Pawałowski, K. (eds) Transformation Groups Poznań 1985. Lecture Notes in Mathematics, vol 1217. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072820
Download citation
DOI: https://doi.org/10.1007/BFb0072820
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16824-9
Online ISBN: 978-3-540-47097-7
eBook Packages: Springer Book Archive
