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Let M be a connected closed smooth manifold of dimension at least 6 and let f: M → S 1 be a smooth map which induces an isomorphism of fundamental groups. In this situation, a classical result of surgery theory [W. Browder and J. Levine, Comment. Math. Helv. 40 (1966), 153–160; MR0195104 (33 #3309)] implies that f is homotopic to a submersion if and only if the homotopy groups of M are finitely generated in every degree. The paper under review isolates the homotopy theoretic essence of this statement by proving fibering results in the Poincaré duality category. Recall that a space X is called a Poincaré duality space if it has a homological fundamental class (possibly with twisted coefficients) such that the cap product with this class induces an isomorphism between cohomology and homology groups of X for all local coefficient systems. Furthermore, a space is called finitely dominated if it is a retract of a space which has the homotopy type of a finite CW complex. Let f: X → P be a map of connected, finitely dominated Poincaré duality spaces of (formal) dimensions d and p respectively. Let F be the homotopy fibre of f. The goal is to find conditions under which f is a Poincaré submersion, i.e. when F is a Poincaré duality space. Theorem A: Let P = Bπ1(X) and f classify the universal cover of X. Then F is a Poincar

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Year: 2013

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