Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T02:06:13.379Z Has data issue: false hasContentIssue false
  • English
  • Français

On cyclic maps

Published online by Cambridge University Press:  24 October 2008

V. P. Snaith
V. P. Snaith
Affiliation:
Emmanuel College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Throughout X will denote a connected, finite C.W. complex.

Let G be a subgroup of ∑n, the symmetric group, which acts transitively on the Cartesian product, Xn, of the space X. A map f:Xn→X is G-symmetric if it commutes with the action of G. If x0εX is a base point let i:X→Xn denote the inclusion, i(x) = (x,x0, …,x0). In ((6); (7); (11)–(13)) the following problem is posed: if X is an orientable topological n-manifold, what is the set of integers which may be obtained as the degree of(f.i) where f is a G-symmetric map? The degree of(f.i) is called the James number of f. If G = ∑n(G = Zn) a G-symmetric map will be called a symmetric map (a cyclic map). If X = Sn, the n-sphere, this problem has been studied in ((6)–(8), (11)–(13))

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 3 , May 1972 , pp. 449 - 456
Copyright
Copyright © Cambridge Philosophical Society 1972

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Anderson, D. W. and Hodgkin, L.The K-theory of Eilenberg-Maclane complexes. Topology 7 (1968), 317–329.Google Scholar
(2)Atiyah, M. F.Power operations in K-theory. Quart. J. Math. Oxford Ser. 2 17 (1966), 165–93.Google Scholar
(3)Atiyah, M. F. and Segal, G. Equivariant K-theory (Notes by D. B. A. Epstein and R. L. E. Schwarzenberger, University of Warwick notes (1966)).Google Scholar
(4)Eilenberg, S. and Moore, J.Homology and fibrations I, Comment. Math. Helv. 40 (1965/1966), 199–236.CrossRefGoogle Scholar
(5)Hodgkin, L.On the K-theory of Lie groups, Topology 6 (1967), 1–36.CrossRefGoogle Scholar
(6)James, I. M.Symmetric functions of several variables whose range and domain is a sphere. Bol. Soc. Mat. Mexicana 1 (1956), 85–88.Google Scholar
(7)James, I. M., Thomas, E., Toda, H. and Whitehead, G. W.On the symmetric square of the sphere. J. Math. Mech. 12 (1963), 771–776.Google Scholar
(8)Landweber, P. S.On symmetric maps between spheres and equivariant K-theory Topology 9 (1970), 55–61.Google Scholar
(9)Rothenberg, M. and Steenrod, N. E. The cohomology of classifying spaces of associative H-spaces (to appear).Google Scholar
(10)Snaith, V. P. On the K-theory of homogeneous spaces and conjugate bundles of Lie groups (to appear: Proc. London Math. Soc).Google Scholar
(11)Ucci, J. J.On the symmetric cube of the sphere, Trans. Amer. Math. Soc. 151 (1970), 527–549.Google Scholar
(12)Ucci, J. J. On cyclic and iterated cyclic products of spheres (to appear).Google Scholar
(13)Ucci, J. J.On symmetric maps of spheres. Invent Math. 5 (1968), 8–18.CrossRefGoogle Scholar