Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Uniform bounds for isoperimetric problems


Authors: Jerrold Siegel and Frank Williams
Journal: Proc. Amer. Math. Soc. 107 (1989), 459-464
MSC: Primary 55P99; Secondary 58E05
DOI: https://doi.org/10.1090/S0002-9939-1989-0984815-2
MathSciNet review: 984815
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we generalize our previous joint work with Allan Calder on the width of homotopies by considering an arbitrary finite polyhedral pair $ \left( {W,V} \right)$ rather than $ \left( {I,\left\{ {0,1} \right\}} \right)$. We show that given appropriate topological conditions on a Riemannian manifold $ M$, with respect to $ \left( {W.V} \right)$, there are bounds, $ {B_q}\left( {a,\left( {W,V} \right),M} \right)$, such that if $ F:K \times W \to M$ is a map with $ {\text{Lip}}\left( {F\left\vert {\left( {K \times V} \right)} \right.} \right) < a$, then $ F$ can be deformed $ {\text{rel}}\left( {K \times V} \right)$ to $ F'$ with $ {\text{Lip}}\left( {F'} \right) < {B_q}\left( {a,\left( {W,V} \right),M} \right) + \varepsilon $, where $ \varepsilon > 0$ is arbitrary and $ \dim \left( K \right) = q$.


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

  • [1] A. Calder and J. Siegel, On the width of homotopies, Topology 19 (1980), 209-220. MR 579572 (82c:55013)
  • [2] -, Homotopies of bounded width are almost Lipschitz, Topology Appl. 14 (1982), 117-129. MR 667658 (83m:55021)
  • [3] A. Calder, J. Siegel, and F. Williams, The width of homotopies into spheres, Topology 21 (1982), 281-290. MR 649759 (84a:55011)
  • [4] M. Gromov, Homotopical effects of dilation, J. Differential Geom. 13 (1978), 303-310. MR 551562 (82d:58017)
  • [5] P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Stud., no. 15, Amsterdam, 1975. MR 0478146 (57:17635)
  • [6] A. Lundell and S. Weingram, The topology of CW-complexes, Van Nostrand Reinhold, New York, 1969. MR 0238319 (38:6595)
  • [7] R. Olivier, Über die Dehnung von Spharenabbildungen, Invent. Math. 1 (1966), 380-390. MR 0203651 (34:3500)
  • [8] J. Roitberg, Dilatation phenomena in the homotopy groups of spheres, Adv. in Math. 15 (1975), 198-206. MR 0375302 (51:11498)
  • [9] J. Siegel, and F. Williams, Numerical invariants of homotopies into spheres, Pacific J. Math. 110 (1984), 417-428. MR 726499 (85i:55013)
  • [10] -, Variational invariants of Riemannian manifolds, Trans. Amer. Math. Soc. 294 (1984), 417-428. MR 743739 (85i:55014)
  • [11] -, Uniform bounds for equivariant homotopies, Topology Appl. (to appear). MR 1003302 (90h:55019)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55P99, 58E05

Retrieve articles in all journals with MSC: 55P99, 58E05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0984815-2
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society