Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Uniform and Lipschitz homotopy classes of maps


Author: Sol Schwartzman
Journal: Trans. Amer. Math. Soc. 354 (2002), 5039-5047
MSC (2000): Primary 54E15, 55N10
Published electronically: August 1, 2002
MathSciNet review: 1926848
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $X$ is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line $R$ into $X$ an element of $H_{1} (X, U/U_{0}),$ where $U$ is the space of uniformly continuous functions from $R$ to $R$ and $U_{0}$ is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to $H_{1}(X,U/U_{0})$ is surjective. If $X$ is the $n$-dimensional torus, it is bijective, while if $X$ is a compact orientable surface of genus $>1$, it is not injective.

In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds $P$ to compact smooth manifolds $X.$ With each such Lipschitz homotopy class we associate an element of $H_{n} (X, B^+/B_{0}^+),$ where $n$ is the dimension of $P,$ $B$ is the space of bounded continuous functions from the positive real axis to $R,$ and $B_{0}^+$ is the set of all $f\in B^+$ such that $\lim_{t \rightarrow \infty} f(t) = 0.$


References [Enhancements On Off] (What's this?)

  • 1. H. Bohr, Überfast Periodischen Bewegungen, In his collected mathematical works, Vol. 2, Dansk. Mat. Forening-Copenhagen, 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 Poincaré-Bendixson, Annales Institute Fourier, Vol. 24 (1974), pp. 131-148. MR 50:11266
  • 4. J. F. Plante, A Generalization of the Poincaré-Bendixson Theorem for Foliations of Codimension One, Topology, Vol. 12 (1973) pp. 177-181. MR 49:6253
  • 5. J. F. Plante, Foliations with Measure Preserving Holonomy, Annals of Math., Vol. 102 (1975), pp. 327-361. MR 52:11947
  • 6. S. Schwartzman, Asymptotic Cycles, Annals of Math., Vol. 66 (1957) pp. 270-284. MR 19:568i
  • 7. S. Schwartzman, Bohr Almost Periodic Maps into $K$ ($\pi$,1) Spaces, Proceedings Amer. Math. Soc., Vol. 125 (1997), pp. 427-431. MR 97d:58163
  • 8. H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, NJ (1957). MR 19:309c

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 54E15, 55N10

Retrieve articles in all journals with MSC (2000): 54E15, 55N10


Additional Information

Sol Schwartzman
Affiliation: Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03107-0
PII: S 0002-9947(02)03107-0
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