|
Closed sets which are not  -critical
Author(s):
Cornel
Pintea
Journal:
Proc. Amer. Math. Soc.
133
(2005),
923-930.
MSC (2000):
Primary 55R05;
Secondary 55Q05, 55N10
Posted:
September 16, 2004
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this paper we first observe that the complement of a countable closed subset of an  -dimensional manifold  has large  -homology group. In the last section we use this information to prove that, under some topological conditions on the given manifold, certain families of fibers, in the total space of a fibration over  , are not critical sets for some special real or  -valued functions.
References:
-
- 1.
- Dodson, C.T.J., Parker, P.E., A User's Guide to Algebraic Topology, Kluwer Academic Publishers, 1997. MR 1430097 (97j:55001)
- 2.
- Doubrovine, B., Novikov, S., Fomenko, A., Géométrie contemporaine. Méthodes et applications, MIR, Moscow, vol. 2, 1982. MR 0728386 (84m:53002a)
- 3.
- Hu, S-T., Homotopy Theory, Academic Press, New York and London, 1959. MR 0106454 (21:5186)
- 4.
- Greenberg, M.J., Harper, J.R., Algebraic Topology. A first course, Addison-Wesley, 1981. MR 0643101 (83b:55001)
- 5.
- M. Grayson, C. Pugh, Critical Sets in
 -space, Institut des Hautes Études Scientifiques, No. 77(1993), 5-61. MR 1249169 (94k:57049) - 6.
- A. Norton, C. Pugh, Critical Sets in the Plane, Michigan Math. J. 38 (1991), 441-459. MR 1116500 (92f:57032)
- 7.
- Pintea, C., Differentiable Mappings with an Infinite Number of Critical Points, Proceedings of the A.M.S., vol. 128, Nr. 11, (2000), 3435-3444. MR 1670419 (2001b:57076)
- 8.
- Pintea, C., Some Pairs of Manifolds with Infinite Uncountable
 -Category, Topological Methods in Nonlinear Analysis, Vol. 21(2002), 101-113. MR 1980138 (2004d:57039) - 9.
- Spanier, E., Algebraic Topology, McGraw Hill, 1966. MR 0210112 (35:1007)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
55R05,
55Q05, 55N10
Retrieve articles in all Journals with MSC
(2000):
55R05,
55Q05, 55N10
Additional Information:
Cornel
Pintea
Affiliation:
Babes-Bolyai University, Faculty of Mathematics and Computer Sciences, 400084 M. Kogalniceanu 1, Cluj-Napoca, Romania
Email:
cpintea@math.ubbcluj.ro
DOI:
10.1090/S0002-9939-04-07584-7
PII:
S 0002-9939(04)07584-7
Keywords:
Closed countable sets,
homology and homotopy groups of fiber spaces,
critical points.
Received by editor(s):
November 12, 2002
Received by editor(s) in revised form:
November 9, 2003
Posted:
September 16, 2004
Additional Notes:
This research was partially supported by the European Research and Training Network \textit{Geometric Analysis}, Contract Number: HPRN-CT-1999-00118
Communicated by:
Paul Goerss
Copyright of article:
Copyright
2004,
American Mathematical Society
|
Comments: webmaster@ams.org