On the rank of a space
HTML articles powered by AMS MathViewer
- by Christopher Allday PDF
- Trans. Amer. Math. Soc. 166 (1972), 173-185 Request permission
Abstract:
The rank of a space is defined as the dimension of the highest dimensional torus which can act almost-freely on the space. (By an almost-free action is meant one for which all the isotropy subgroups are finite.) This definition is shown to extend the classical definition of the rank of a Lie group. A conjecture giving an upper bound for the rank of a space in terms of its rational homotopy is investigated.References
- Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 51508, DOI 10.2307/1969728
- Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
- Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221500
- Glen E. Bredon, Cohomological aspects of transformation groups, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 245–280. MR 0244990
- Haskell Cohen, A cohomological definition of dimension for locally compact Hausdorff spaces, Duke Math. J. 21 (1954), 209–224. MR 66637
- P. E. Conner, Retraction properties of the orbit space of a compact topological transformation group, Duke Math. J. 27 (1960), 341–357. MR 163987, DOI 10.1215/S0012-7094-60-02732-0 W.-Y. Hsiang, On generalizations of a theorem of A. Borel and their applications in the study of topological actions, University of California, Berkeley, 1969 (preprint).
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- Elon L. Lima, Commuting vector fields on $S^{3}$, Ann. of Math. (2) 81 (1965), 70–81. MR 171286, DOI 10.2307/1970383
- Mark Mahowald, On obstruction theory in orientable fiber bundles, Trans. Amer. Math. Soc. 110 (1964), 315–349. MR 157386, DOI 10.1090/S0002-9947-1964-0157386-8
- D. Montgomery and C. T. Yang, The existence of a slice, Ann. of Math. (2) 65 (1957), 108–116. MR 87036, DOI 10.2307/1969667
- Jean-Pierre Serre, Groupes d’homotopie et classes de groupes abéliens, Ann. of Math. (2) 58 (1953), 258–294 (French). MR 59548, DOI 10.2307/1969789
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Emery Thomas, Seminar on fiber spaces, Lecture Notes in Mathematics, vol. 13, Springer-Verlag, Berlin-New York, 1966. Lectures delivered in 1964 in Berkeley and 1965 in Zürich; Berkeley notes by J. F. McClendon. MR 0203733, DOI 10.1007/BFb0097864
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 166 (1972), 173-185
- MSC: Primary 55C10
- DOI: https://doi.org/10.1090/S0002-9947-1972-0292071-8
- MathSciNet review: 0292071