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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
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##### References:
  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  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  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  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  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  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  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  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  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  Y.H. Im , Submanifold decompositions that induce approximate fibrations and approximation by bundle maps , Ph.D. Dissertation, University of Tennessee, Knoxville, 1991.  R. Lee , Semicharacteristic classes , Topology 12 (1973), 183-199. · Zbl 0264.57012 · doi:10.1016/0040-9383(73)90006-2  J. Milnor , Groups which act on Sn without fixed points , Amer. J. Math. 79 (1957), 623-630. · Zbl 0078.16304 · doi:10.2307/2372566  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  J.R. Munkres , Elements of Algebraic Topology , Addison Wesley Publ. Co., New York, 1984. · Zbl 0673.55001  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  E.H. Spanier , Algebraic Topology , McGraw-Hill, New York, 1966. · Zbl 0145.43303  G.A. Swarup , On embedded spheres in 3-manifolds , Math. Ann. 203 (1973), 89-102. · Zbl 0241.55017 · doi:10.1007/BF01431437 · eudml:162430  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  F. Waldhausen , Algebraic K-theory of generalized free products , Ann. of Math. 108 (1978), 135-256. · Zbl 0407.18009 · doi:10.2307/1971166
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