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Mod p associative H-spaces of given rank

Published online by Cambridge University Press:  24 October 2008

J. R. Hubbuck
J. R. Hubbuck
Affiliation:
University of Aberdeen
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Extract

1. Few mod 2 finite simply-connected spaces are known to support an H-structure. These examples arise from products of Lie groups and seven spheres. It has been shown in (12) and (14) that the space of a mod 2 finite H-space of given rank can belong to only a finite number of different homotopy types. The situation at odd primes is very different; many examples of mod p finite H-spaces are known, such as p-localized odd-dimensional spheres. Therefore there are infinitely many homotopy types of prescribed rank. If one insists that the mod pH-space has a strictly associative multiplication, or equivalently has the homotopy type of a topological group, the picture is less clear. Examples have been constructed which do not arise from Lie groups and spheres but there are strong restrictions on just what can occur by these techniques, see (5). There are only finitely many known examples in each rank. We establish that there can in fact only be a finite number of possible homotopy types for the spaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 1 , July 1980 , pp. 153 - 160
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Atiyah, M. F.K-Theory (Benjamin, 1967).Google Scholar
(2)Atiyah, M. F. and Hirzebruch, F.Vector bundles and homogeneous spaces. Proc. Symp. Amer. Math. Soc. 3 (1961), 7–38.CrossRefGoogle Scholar
(3)Atiyah, M. F. and Tall, D. O.Group representations, λ-rings and the J-homomorphism. Topology 8 (1969), 253–297.Google Scholar
(4)Adams, J. F.J(X) - II. Topology 3 (1965), 137–171.CrossRefGoogle Scholar
(5)Adams, J. F. and Wilkerson, C.Finite H-spaces and algebras over the Steenrod algebra. Ann. of Math., II. Ser. (to appear.)Google Scholar
(6)Body, R. A. and Douglas, R. R.Homotopy types within a rational homotopy type. Topology 13 (1974), 209–214.Google Scholar
(7)Browder, W.The cohomology of covering spaces of H-spaces. Bull. Amer. Math. Soc. 65 (1959), 140–141.CrossRefGoogle Scholar
(8)Browder, W.Differential Hopf algebras. Trans. Amer. Math. Soc. 107 (1963), 153–176.Google Scholar
(9)Curjel, C. R. and Douglas, R. R.On H-spaces of finite dimension. Topology 10 (1971), 385–390.CrossRefGoogle Scholar
(10)Hilton, P., Mislin, G. and Roitberg, J.Localization ofnilpotent groups and spaces (North-Holland, 1975).Google Scholar
(11)Hodgkin, L.On the K-theory of Lie groups. Topology 6 (1967), 1–36.CrossRefGoogle Scholar
(12)Hubbuck, J. R.H-spaces of given rank. Geometric Applications of Homotopy Theory I. Springer-Verlag. Lecture Notes in Maths. 657 (1978), 271–281.CrossRefGoogle Scholar
(13)Lin, J. P.Torsion in H-spaces II. Annals of Math. 107 (1978), 41–88.CrossRefGoogle Scholar
(14)Nunn, J. D. M.The homotopy types of finite H-spaces. Topology 18 (1979), 17–28.Google Scholar
(15)Rothenberg, M. and Steenrod, N. E.The cohomology of classifying spaces of H-spaces. Bull. Amer. Math. Soc. 71 (1965), 872–875.CrossRefGoogle Scholar
(16)Stasheff, J. D.Homotopy associativity of H-spaces: I and II. Trans. Amer. Math. Soc. 108 (1963), 275–292, 293–312.Google Scholar
(17)Sullivan, D.Geometric Topology, part I (M.I.T. Notes, 1970).Google Scholar