Stabilizers and orbits of circle-valued smooth functions

2005/03/31 by Sergey Maksymenko, Maksymenko, Sergey
Mathematics · #14H40 #58K05 #Algebraic Topology (math.AT) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Topology (math.GT) #math.AT #math.DS #math.FA #math.GT #msc:14H40 #msc:58K05

paper · pdf · doi:10.48550/arxiv.math/0503734

11 pages, 1 figure. This paper is an extension of author's preprint http://xxx.lanl.gov/math.FA/0411612 to circle-valued functions

arxiv created 2005/03/31 · arxiv updated 2009/12/01

Abstract

Let M be a smooth compact manifold and P be either R¹ or S¹. There is a natural action of the groups Diff(M) and Diff(M) × Diff(P) on the space of smooth mappings C^∞(M,P). For f∈ C^∞(M,P) let S_f, S_MP, O_f, and O_MP be the stabilizers and orbits of f under these actions. Recently, the author proved that under mild conditions on f∈ C^∞(M,R¹) the corresponding stabilizers and orbits are homotopy equivalent: S_MR ∼ S_M and O_MR ∼ O_M. These results are extended here to the actions on C^∞(M,S¹). It is proved that under the similar conditions (that are rather typical) we have that S_MS∼ S_M and O_MS ∼ O_M × S¹.

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