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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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