Actions of torus Tⁿ on (n + 1)-manifolds Mⁿ⁺¹

Jingyal Pak

Vol. 44 (1973), No. 2, 671–674

DOI: 10.2140/pjm.1973.44.671

Abstract

Let ξ be a principaI T^l-bundle over a lens space L(p,q). It is shown here that the total space of ξ can be identified with L(k,q) × S₁¹ ×⋯ × S_l¹, for some k ≦ p. Let (Tⁿ,Mⁿ⁺¹) be an effective torus action on an orientable (n + 1)-dimensional manifold. An elementary examination of the parity of dimensions of the slice S_x at x ∈ M and of the orbit Tⁿ(x), shows that the circle subgroups are the only possible stability groups on Mⁿ⁺¹. From these two results and the cross-sectioning theorem we can conclude that Tⁿ⁺¹ and L(k,q) ×Tⁿ⁻² are the only possible types of compact closed orientable (n+l)-manifolds which allow Tⁿ actions.

Mathematical Subject Classification

Primary: 57E15

Milestones

Received: 18 October 1971

Published: 1 February 1973