×

zbMATH — the first resource for mathematics

Hyperhopfian groups and approximate fibrations. (English) Zbl 0788.57012
An approximate fibration \(p: M\to B\) is a proper mapping having the usual lifting property of a fibration, but now up to a given open covering of \(B\). The advantage of this notion is that on one hand there exists an exact homotopy sequence but on the other hand there are more such approximate fibrations available. Suppose \(M\) is an \((n+k)\)-manifold and \(p\) is induced by a decomposition \(\{p^{-1}(b)\}\) of \(M\) such that each \(p^{-1}(b)\mid b\in B\) is shape equivalent to a single \(n\)-manifold \(N\), then \(N\) is called a codimension \(k\)-fibrator. There are Hopfian manifolds \(N\) (every \(g: N\to N\) of degree 1 is a homotopy equivalence) and, analogously Hopfian groups \(G\) (every epic \(g: G\to G\) is an isomorphism) resp. hyper-Hopfian groups (\(G\) finitely presented, every \(g: G\to G\), \(g(G)\triangleleft G\), \(G/g(G)\) cyclic, is an isomorphism). The main results of the present paper are:
(1) All closed Hopfian manifolds with hyper-Hopfian fundamental group are codimension 2-fibrators.
(2) Every closed Hopfian \(n\)-manifold \(N\) with \(\pi_ 1(N)\) Hopfian and \(\chi(N)\neq 0\) is a codimension 2-fibrator.
(3) Some related results for \(n=4\).
Moreover the paper contains some results on hyper-Hopfian groups.
One of the basic tools for establishing these results is a theorem providing sufficient conditions about \(\pi_ 1\) ensuring that \(N\) is Hopfian. This assertion is derived from results of Hausmann, Hempel and Swarup.

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
55N25 Homology with local coefficients, equivariant cohomology
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] D.S. Coram and P.F. Duvall , Approximate fibrations , Rocky Mtn. J. Math. 7 (1977), 275-288. · Zbl 0367.55019 · doi:10.1216/RMJ-1977-7-2-275
[2] D.S. Coram and P.F. Duvall , Approximate fibrations and a movability condition for maps , Pacific J. Math. 72 (1977), 41-56. · Zbl 0368.55016 · doi:10.2140/pjm.1977.72.41
[3] R.J. Daverman , Submanifold decompositions that induce approximate fibrations , Topology Appl. 33 (1989), 173-184. · Zbl 0684.57009 · doi:10.1016/S0166-8641(89)80006-9
[4] R.J. Daverman , Manifolds with finite first homology as codimension 2 fibrators , Proc. Amer. Math. Soc. 113 (1991), 471-477. · Zbl 0727.55009 · doi:10.2307/2048533
[5] R.J. Daverman , 3-Manifolds with geometric structures and approximate fibrations , Indiana Univ. Math. J. 40 (1991), 1451-1469. · Zbl 0739.57007 · doi:10.1512/iumj.1991.40.40065
[6] R.J. Daverman and J.J. Walsh , Decompositions into codimension two manifolds , Trans. Amer. Math. Soc. 288 (1985), 273-291. · Zbl 0568.57013 · doi:10.2307/2000440
[7] K.W. Gruenberg , Residual properties of infinite solvable groups , Proc. London Math. Soc. (3) 7 (1957), 29-62. · Zbl 0077.02901 · doi:10.1112/plms/s3-7.1.29
[8] J.C. Hausmann , Geometric Hopfian and non-Hopfian situations , in Geometry and Topology (C. McCrory and T. Shiflin, eds.) Lecture Notes in Pure Appl. Math., Marcel Dekker, Inc., New York, 1987, 157-166. · Zbl 0607.57015
[9] J. Hempel , Residual finiteness for 3-manifolds , in Combinatorial Group Theory and Topology (S. M. Gersten and J. R. Stallings, eds.), Annals of Math. Studies, No. 111, Princeton Univ. Press, Princeton, NJ, 1987, 379-396. · Zbl 0772.57002
[10] Y.H. Im , Submanifold decompositions that induce approximate fibrations and approximation by bundle maps , Ph.D. Dissertation, University of Tennessee, Knoxville, 1991.
[11] R. Lee , Semicharacteristic classes , Topology 12 (1973), 183-199. · Zbl 0264.57012 · doi:10.1016/0040-9383(73)90006-2
[12] J. Milnor , Groups which act on Sn without fixed points , Amer. J. Math. 79 (1957), 623-630. · Zbl 0078.16304 · doi:10.2307/2372566
[13] J. Milnor , Infinite cyclic coverings , in Conference on the Topology of Manifolds (J. G. Hocking, ed.), Prindle Weber & Schmidt, Inc., Boston, 1968, 115-133. · Zbl 0179.52302
[14] J.R. Munkres , Elements of Algebraic Topology , Addison Wesley Publ. Co., New York, 1984. · Zbl 0673.55001
[15] P. Scott and T. Wall , Topological methods in group theory , in Homological Group Theory (C. T. C. Wall, ed.), Cambridge Univ. Press, Cambridge, 1979, 137-203. · Zbl 0423.20023
[16] E.H. Spanier , Algebraic Topology , McGraw-Hill, New York, 1966. · Zbl 0145.43303
[17] G.A. Swarup , On embedded spheres in 3-manifolds , Math. Ann. 203 (1973), 89-102. · Zbl 0241.55017 · doi:10.1007/BF01431437 · eudml:162430
[18] G.A. Swarup , On a theorem of C. B. Thomas , J. London Math. Soc. (2) 8 (1974), 13-21. · Zbl 0281.57003 · doi:10.1112/jlms/s2-8.1.13
[19] F. Waldhausen , Algebraic K-theory of generalized free products , Ann. of Math. 108 (1978), 135-256. · Zbl 0407.18009 · doi:10.2307/1971166
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.