×

zbMATH — the first resource for mathematics

Submanifold decompositions that induce approximate fibrations. (English) Zbl 0684.57009
A closed n-manifold N is called a codimension-k fibrator if for every usc decomposition G of an \((n+k)\)-manifold M such that each \(g\in G\) is shape equivalent to N and the decomposition space M/G is finite dimensional, then the projection \(\pi\) : \(M\to M/G\) is an approximate fibration. The main problem studied in this paper is the identification of codimension-k fibrators. As a sample, we state the following results proved in this paper:
(1) A closed orientable n-manifold N is a codimension-1 fibrator provided one of the following is satisfied: (a) \(Z\pi_ 1(N)\) is Noetherian; or (b) \(\pi_ 1(N)\) is Hopfian and N is aspherical.
(2) A closed n-manifold N is a codimension-2 fibrator provided one of the following is satisfied: (a) N is the real projective n-space \((n>1)\); (b) N is a surface, \(n=2\), and the Euler characteristic of N is nonzero; or (c) each element of \(\pi_ 1(N)\) has order two. Several examples, counterexamples, and relevant questions are also given.
Reviewer: S.Singh

MSC:
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
55R65 Generalizations of fiber spaces and bundles in algebraic topology
57N25 Shapes (aspects of topological manifolds)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
54B15 Quotient spaces, decompositions in general topology
57N10 Topology of general \(3\)-manifolds (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Coram, D.S.; Duvall, P.F., Approximate fibrations, Rocky mountain J. math., 7, 275-288, (1977) · Zbl 0367.55019
[2] Coram, D.S.; Duvall, P.F., Approximate fibrations and a movability condition for maps, Pacific J. math., 72, 41-56, (1977) · Zbl 0368.55016
[3] Coram, D.S.; Duvall, P.F., Mappings from S^3 to S2 whose point inverses have the shape of a circle, General topology appl., 10, 239-246, (1979) · Zbl 0417.54014
[4] Daverman, R.J., Decompositions into codimension one submanifolds, Compositio math., 55, 185-207, (1985) · Zbl 0593.57005
[5] Daverman, R.J., Decompositions into codimension two submanifolds: the nonorientable case, Topology appl., 24, 71-81, (1986) · Zbl 0605.57007
[6] Daverman, R.J.; Husch, L.S., Decompositions and approximate fibrations, Michigan math. J., 31, 197-214, (1984) · Zbl 0584.57011
[7] Daverman, R.J.; Walsh, J.J., Decompositions into codimension two spheres and approximate fibrations, Topology appl., 19, 103-121, (1985) · Zbl 0589.57012
[8] Daverman, R.J.; Walsh, J.J., Decompositions into codimension two manifolds, Trans. amer. math. soc., 288, 273-291, (1985) · Zbl 0568.57013
[9] Daverman, R.J.; Walsh, J.J., Decompositions into submanifolds that yield generalized manifolds, Topology appl., 26, 143-162, (1987) · Zbl 0617.57009
[10] Epstein, D.B.A., The degree of a map, (), 369-383 · Zbl 0148.43103
[11] Frederick, K., The Hopfian property for a class of fundamental groups, Comm. pure appl. math., 16, 1-8, (1963) · Zbl 0115.40503
[12] Hempel, J., Residual finiteness of surface groups, (), 323 · Zbl 0231.55002
[13] Hopf, H., Zur algebra der abbildungen von mannigfaltigkeiten, J. reine angew. math., 163, 71-88, (1930) · JFM 56.0501.03
[14] Husch, L.S., Approximating approximate fibrations by fibrations, Canad. J. math., 29, 897-913, (1977) · Zbl 0366.55006
[15] Jaco, W., Surfaces embedded in M^2xs1, Canad. J. math., 22, 553-568, (1970) · Zbl 0205.53501
[16] Orlik, P., Seifert manifolds, () · Zbl 0263.57001
[17] Scott, G.P., The geometries of 3-manifolds, Bull. London math. soc., 15, 401-487, (1983) · Zbl 0561.57001
[18] Tollefson, J.L., On 3-manifolds that cover themselves, Michigan math. J., 16, 103-109, (1969) · Zbl 0176.53402
[19] Wolf, J.A., Spaces of constant curvature, (1967), McGraw-Hill New York · Zbl 0162.53304
[20] Zieschang, H.; Vogt, E.; Coldeway, H.-D., Surfaces and planar discontinuous groups, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.