Commuting Vector Fields on S 3
1965; Princeton University; Volume: 81; Issue: 1 Linguagem: Inglês
10.2307/1970383
ISSN1939-8980
Autores Tópico(s)Geometric and Algebraic Topology
ResumoThe same type of argument used to prove Theorem A, together with a simple induction procedure, will show that every continuous action of the additive group Rm (m_ 1) on S2 (and hence on P2) has a fixed point. Thus, any finite set Xi, * * *, Xm of pairwise commuting vector fields on S2 (or P2) has a common singularity. Of course, there are differentiable actions of Rm without fixed points on the torus T2 and on the Klein bottle K2, for every m ? 1. The natural action of R2 on T2 has only one orbit, which has dimension 2. Every continuous nontrivial action of R2 on K2 must have at least one 1-dimensional orbit. There is one differentiable action of R2 on K2 with exactly two orbits or dimensions 2 and 1, respectively. It seems plausible that every continuous action of R2 on a compact 2-manifold, other than T2 or K2, must have a fixed point, but, this question is open at present. Added in proof. Meanwhile, I have been able to prove this conjecture. See the Research Announcement in the Bull. Amer. Math. Soc. of May 1963, pp. 366-368. I wish to thank Robert Ellis for interesting conversations.
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