Algebraic realization problems for low dimensional G-manifolds

Abstract

The classification of equivariant vector bundles over a circle and algebraic realization problems for low dimensional manifolds are studied. For a compact Lie group G the fiber H-module of a G-vector bundle is characterized, where H is the trivializer of the action on the base space. We prove that the isomorphism class of a complex G-vector bundle over a circle is determined by the fibers over one or two points when regarded as complex representations of the isotropy subgroups at the points. The extension problem of representations is used to check whether a given G-vector bundle over a circle is trivial or not. For real G-vector bundles over a circle we also give isomorphism theorems according to the type of the fiber H-module. In a corollary of the isomorphism theorems it is proved that over a circle the stable isomorphism classes in equivariant K-theory are equivalent to the isomorphism classes in equivariant vector bundle theory. In addition, one of our main theorems for complex G-vector bundles over a circle provides information on how many indecomposable complex G-vector bundles have the same fiber H-module and how many these bundles are trivial. We also discuss a complete description of complex and real G-line bundles over a circle. Classification of G-vector bundles over a circle for abelian groups, dihedral groups, and generalized quaternion groups are also given. By an algebraic G-model we mean a real algebraic G-set X such that every real G-vector bundle over X is isomorphic to a strongly algebraic G-vector bundle. A nonsingular algebraic G-model is presented for one-dimensional closed smooth G-manifolds and two-dimensional closed smooth G-manifolds with effective action by applying the classification results of real G-vector bundles over a circle where G is a compact abelian Lie group. Using this result we also prove that every closed orientable smooth G-manifold of dimension three is equivariantly diffeomorphic to a nonsingular real algebraic G-set...