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Poincaré submersions. (English) Zbl 1061.57024

Let \(f:X\to P\) be a 2-connected map of connected Poincaré duality complexes with homotopy fibre \(F\). In Theorem A it is shown that if \(P\) is aspherical, \(F\) is a Poincaré duality complex if and only if the homotopy groups of \(X\) are finitely generated in each degree. In Theorem B it is shown that \(F\) is a Poincaré duality complex if and only if \(H_*(F;\mathbb{Z})\) is finitely generated, and that if moreover \(X\) is 1-connected the duality isomorphisms are given by cap product with a specified element. These results are easy applications of the mod-\(\mathcal C\) Hurewicz Theorem and the finiteness conditions of Wall, together with a homotopical realization of the umkehr homomorphism for the final result.

MSC:

57P10 Poincaré duality spaces
55R99 Fiber spaces and bundles in algebraic topology
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References:

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