Finiteness Conditions in Covers of Poincaré Duality Spaces
نویسنده
چکیده
A closed 4-manifold (or, more generally, a finite PD4space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a PD3-complex. A Poincaré duality space is a space X of the homotopy type of a cell complex which satisfies Poincaré duality with local coefficients. It is finite if the singular chain complex of the universal cover X̃ is chain homotopy equivalent to a finite free Z[π1(X)]-complex. (The PD-space X is homotopy equivalent to a Poincaré duality complex ⇔ it is finitely dominated ⇔ π1(X) is finitely presentable. See [2].) In this note we show that finiteness hypotheses in two theorems about covering spaces of PD-complexes may be relaxed. Theorem 4 extends a criterion of Stark to all Poincaré duality groups. The main result is Theorem 5, which characterizes finite PD4-spaces with finitely dominated infinite regular covering spaces. We shall often write “vPD-group” instead of “virtual Poincaré duality group”, and similarly vPDr, vFP , etc. We say also that a group G is a weak PDr-group if H (G;Z[G]) ∼= Z and H(G;Z[G]) = 0 for q 6= r.
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