Uniform and Lipschitz homotopy classes of maps
Author:
Sol Schwartzman
Journal:
Trans. Amer. Math. Soc. 354 (2002), 50395047
MSC (2000):
Primary 54E15, 55N10
Published electronically:
August 1, 2002
MathSciNet review:
1926848
Fulltext PDF Free Access
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Abstract: If is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line into an element of where is the space of uniformly continuous functions from to and is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to is surjective. If is the dimensional torus, it is bijective, while if is a compact orientable surface of genus , it is not injective. In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds to compact smooth manifolds With each such Lipschitz homotopy class we associate an element of where is the dimension of is the space of bounded continuous functions from the positive real axis to and is the set of all such that
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 H. Bohr, Überfast Periodischen Bewegungen, In his collected mathematical works, Vol. 2, Dansk. Mat. ForeningCopenhagen, C44 (1952). MR 15:276i
 2.
 N. Dunford and J. Schwartz, Linear Operators Part II, Spectral theory. Selfadjoint operators in Hilbert space, Interscience Publishers, (1963). MR 32:6181
 3.
 R. Moussu and F. Pelletier, Sur le Théorème de PoincaréBendixson, Annales Institute Fourier, Vol. 24 (1974), pp. 131148. MR 50:11266
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 J. F. Plante, A Generalization of the PoincaréBendixson Theorem for Foliations of Codimension One, Topology, Vol. 12 (1973) pp. 177181. MR 49:6253
 5.
 J. F. Plante, Foliations with Measure Preserving Holonomy, Annals of Math., Vol. 102 (1975), pp. 327361. MR 52:11947
 6.
 S. Schwartzman, Asymptotic Cycles, Annals of Math., Vol. 66 (1957) pp. 270284. MR 19:568i
 7.
 S. Schwartzman, Bohr Almost Periodic Maps into (,1) Spaces, Proceedings Amer. Math. Soc., Vol. 125 (1997), pp. 427431. MR 97d:58163
 8.
 H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, NJ (1957). MR 19:309c
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Additional Information
Sol Schwartzman
Affiliation:
Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881
DOI:
http://dx.doi.org/10.1090/S0002994702031070
PII:
S 00029947(02)031070
Received by editor(s):
April 1, 2000
Received by editor(s) in revised form:
May 22, 2002
Published electronically:
August 1, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
