Classification of $SO(3)$-Actions on Five Manifolds

Abstract

P. Orlik and F. Raymond showed that some invariants classify smooth 3-manifolds with smooth ^-action, up to equivariant diffeomorphism (preserving the orientation of the orbit space if it is orientable) [6]. And R. W. Richardson JR. studied SO (3) -actions on S [7]. Also, K. A. Hudson classified smooth SO (3) -actions on connected, simply connected, closed 5-manifolds admitting at least one orbit of dimension three [2]. In this paper, we discuss the equivariant classification of smooth SO (3) -actions on closed, connected, oriented, smooth 5-manifolds such that the orbit space is an orientable surface. We call oriented 50(3)manifolds M and N are equivalent if there is an equivariant homeomorphism between M and N which induces an orientation preserving homeomorphism of the orbit spaces M* to N*. Since there exist various types as the principal orbit, we classify SO (3) -manifolds about each type. It is well known that every subgroups of SO (3) are conjugate to one of the following [4], [5]. 50(2), 0(2), Zn, dihedral group Dn= {x, ; x =y»= (xyY = \], tetrahedral group T={x,, ; x= (xyY = y*= 1}, octahedral group O=[x, y x= (xyY=y = 1}, and icosahedral group 1= [x, ; x —