Torus group actions on simply connected manifolds

Soon-Kyu Kim, Dennis McGavran and Jingyal Pak

Vol. 53 (1974), No. 2, 435–444

DOI: 10.2140/pjm.1974.53.435

Abstract

In this paper it is shown that effective torus Tⁿ-actions on simply connected closed (n + 2)-manifolds Mⁿ⁺² for all n ≧ 1 exist, and a complete orbit structure is given. It turns out that all maximal isotropy subgroups must generate the whole group Tⁿ. The cross-sectioning theorem for the orbit map π : M → M^∗ = Mⁿ⁺²∕Tⁿ is given, and as its application an equivariant classification theorem is obtained.

It is also shown that free torus Tⁿ-actions on simply connected closed (n + 4)-manifolds for all n ≧ 1 exist.

Mathematical Subject Classification

Primary: 57E15

Milestones

Received: 5 April 1973

Revised: 15 February 1974

Published: 1 August 1974