Vol. 44, No. 2, 1973

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

Jingyal Pak

Vol. 44 (1973), No. 2, 671–674

Let ξ be a principaI Tl-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 × × Sl1, 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 Sx 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) ×Tn2 are the only possible types of compact closed orientable (n+l)-manifolds which allow Tn actions.

Mathematical Subject Classification
Primary: 57E15
Received: 18 October 1971
Published: 1 February 1973
Jingyal Pak