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On the Mislin Genus of Symplectic Groups
Author(s) -
Ghienne Pierre
Publication year - 2003
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502014041
Subject(s) - mathematics , genus , symplectic geometry , homotopy , combinatorics , pure mathematics , conjecture , type (biology) , botany , biology , ecology
In this paper we give lower bounds for the Mislin genus of the symplectic groups Sp( m ). This result appears to be the exact analogue of Zabrodsky's theorem concerning the special unitary groups SU( n ). It is achieved by the determination of the stable genus of the quasi‐projective quaternionic spaces QH( 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 Sp(2) is well known to contain exactly two elements. We show that the genus of 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 Sp( m ) fibers over the sphere S 4 m −1 with fiber in the genus of Sp( m −1), and that any homotopy type in the genus of SU( n ) fibers over the sphere S 2 n −1 with fiber in the genus of 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. 2000 Mathematics Subject Classification 55P60 (primary), 55P15, 55R35 (secondary)