Klein, John R. Poincaré submersions. (English) Zbl 1061.57024 Algebr. Geom. Topol. 5, 23-29 (2005). 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. Reviewer: Jonathan A. Hillman (Sydney) Cited in 1 Document MSC: 57P10 Poincaré duality spaces 55R99 Fiber spaces and bundles in algebraic topology Keywords:fibration; homotopy finite; Poincaré duality; submersion; umkehr PDFBibTeX XMLCite \textit{J. R. Klein}, Algebr. Geom. Topol. 5, 23--29 (2005; Zbl 1061.57024) Full Text: DOI arXiv EuDML EMIS References: [1] W Browder, J Levine, Fibering manifolds over a circle, Comment. Math. Helv. 40 (1966) 153 · Zbl 0134.42802 · doi:10.1007/BF02564368 [2] A J Casson, Fibrations over spheres, Topology 6 (1967) 489 · Zbl 0163.45302 · doi:10.1016/0040-9383(67)90006-7 [3] D H Gottlieb, Poincaré duality and fibrations, Proc. Amer. Math. Soc. 76 (1979) 148 · Zbl 0423.57009 · doi:10.2307/2042934 [4] J R Klein, The dualizing spectrum of a topological group, Math. Ann. 319 (2001) 421 · Zbl 0982.55004 · doi:10.1007/PL00004441 [5] J R Klein, Poincaré duality spaces, Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 135 · Zbl 0946.57027 [6] E H Spanier, Algebraic topology, McGraw-Hill Book Co. (1966) · Zbl 0145.43303 [7] C T C Wall, Poincaré complexes I, Ann. of Math. \((2)\) 86 (1967) 213 · Zbl 0153.25401 · doi:10.2307/1970688 [8] C T C Wall, Finiteness conditions for \(\mathrm{CW}\)-complexes, Ann. of Math. \((2)\) 81 (1965) 56 · Zbl 0152.21902 · doi:10.2307/1970382 [9] C T C Wall, Finiteness conditions for \(\mathrm{CW}\) complexes II, Proc. Roy. Soc. Ser. A 295 (1966) 129 · Zbl 0152.21902 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.