Tⁿ-actions on simply connected (n + 2)-manifolds

Dennis McGavran

Vol. 71 (1977), No. 2, 487–497

DOI: 10.2140/pjm.1977.71.487

Abstract

In this paper we show that, for each n ≧ 2, there is a unique, closed, compact, connected, simply connected (n + 2)-manifold, M_n+2, admitting an action of Tⁿ satisfying the following condition: there are exactly n T¹-stability groups T₁,⋯,T_n with each F(T_i,M_n+2) connected. In this case we have Tⁿ≅T₁ ×⋯ × T_n. Any other action (Tⁿ,Mⁿ⁺²), Mⁿ⁺² simply connected, can be obtained from an action (Tⁿ,M_n+2) by equivariantly replacing copies of D⁴ × Tⁿ⁻² with copies of S³ × D² × Tⁿ⁻³. As an application, we classify all actions of Tⁿ on simply connected (n + 2)-manifolds for n = 3,4.

Mathematical Subject Classification

Primary: 57E25, 57E25

Milestones

Received: 20 May 1976

Published: 1 August 1977