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On the Mislin Genus of Symplectic Groups

Published online by Cambridge University Press:  25 June 2003

PIERRE Ghienne
Affiliation:
Département de Mathématiques, UMR 8524, Université de Lille 1, 59655 Villeneuve d'Ascq Cedex, France. E-mail: Pierre.Ghienne@agat.univ-lille1.fr Present address: Faculté Jean Perrin, Rue Jean Souvraz, SP18, 62307 Lens Cedex, France. E-mail: ghienne@euler.univ-artois.fr
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Abstract

In this paper we give lower bounds for the Mislin genus of the symplectic groups $\mathrm{Sp}(m)$. This result appears to be the exact analogue of Zabrodsky's theorem concerning the special unitary groups $\mathrm{SU}(n)$ . It is achieved by the determination of the stable genus of the quasi-projective quaternionic spaces $Q\mathbb{H}(m)$ , following the approach of McGibbon. It leads to a symplectic version of Zabrodsky's conjecture, saying that these lower bounds are in fact the exact cardinality of the genus sets. The genus of $\mathrm{Sp}(2)$ is well known to contain exactly two elements. We show that the genus of $\mathrm{Sp}(3)$ has exactly 32 elements and see that the conjecture is true in these two cases.

Independently, we also show that any homotopy type in the genus of $\mathrm{Sp}(m)$ fibers over the sphere $S^{4m-1}$ with fiber in the genus of $\mathrm{Sp}(m-1)$ , and that any homotopy type in the genus of $\mathrm{SU}(n)$ fibers over the sphere $S^{2n-1}$ with fiber in the genus of $\mathrm{SU}(n-1)$ . Moreover, these fibrations are principal with respect to some appropriate loop structures on the fibers. These constructions permit us to produce particular spaces realizing the lower bounds obtained.

Type
Research Article
Copyright
2003 London Mathematical Society

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