On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)