Compact connected Lie groups acting on simply connected 4-manifolds

Hae Soo Oh

Vol. 109 (1983), No. 2, 425–435

DOI: 10.2140/pjm.1983.109.425

Abstract

Suppose a compact connected Lie group G acts effectively on a simply connected 4-manifold M. Then we show that G is one of the groups SO(5), SU(3)∕Z(G), SO(3) × SO(3), SO(4), SO(3) ×T¹, (SU(2) ×T¹)∕D, SU(2), SO(3), T², T¹, and that the representatives of the conjugacy classes of the principal isotropy groups for these groups on M are, respectively, SO(4), U(2), T², SO(3), S¹, S¹, SO(2) or e, SO(2) or D_2n, e, and e. We also show that in each of these cases M is a connected sum of copies of S⁴, S² × S², CP², and −CP² (except when G is T¹, see Theorem 2.6).

Mathematical Subject Classification 2000

Primary: 57S15

Secondary: 57S25

Milestones

Received: 11 May 1981

Published: 1 December 1983