Nonlinear smooth representations of compact Lie groups

Melvin Gordon Rothenberg and Jonathan David Sondow

Vol. 84 (1979), No. 2, 427–444

DOI: 10.2140/pjm.1979.84.427

Abstract

We study semi-free (= free off the fixed-point set) smooth actions of a compact Lie group G on disks and spheres with fixed-point set a disk or sphere, respectively. In dimensions ≧ 6 and codimension ≠2 we obtain a complete classification for such actions on disks and a partial classification for spheres, together with partial results in dimension 5 or codimension 2. We show that semi-free smooth actions of G on the n-disk Dⁿ, n ≧ 6 + dimG, with fixed-point set an (n-k)-disk, k≠2, are classified by two invariants:

1. a free orthogonal action of G on the (k-1)-sphere S^k−1 (the representation at the fixed points) and
2. an element of the Whitehead group Wh(π_o(G)).

Mathematical Subject Classification 2000

Primary: 57S15

Milestones

Received: 10 February 1978

Published: 1 October 1979