Three-dimensional Poincaré complexes: Homotopy classification and splitting. (Russian) Zbl 0693.57013

For a 3-dimensional Poincaré complex (3-PC for short), X, with fundamental group \(\pi_ 1(X)=\pi\), orientation class \(w(X)\in H^ 1(X;{\mathbb{Z}}_ 2)=H^ 1(\pi;{\mathbb{Z}}_ 2)\), and fundamental class [X]\(\in H_ 3(X;{\mathbb{Z}}^ w)\) (where \({\mathbb{Z}}^ w\) are integer coefficients twisted by w(X)), denote by \(\mu\) (X)\(\in H_ 3(\pi;{\mathbb{Z}}^ w)\) the image of [X] under the homomorphism induced by the canonical map \(X\to K(\pi,1)\). The triple \((\pi_ 1(X),w(X),\mu (X))\) determines X up to homotopy: H. Hendriks [J. Lond. Math. Soc., II. Ser. 16, 160-164 (1977; Zbl 0366.57002)] proved that two 3-PC’s X and Y are homotopy equivalent if and only if there exists an isomorphism of \(\pi_ 1(X)\) onto \(\pi_ 1(Y)\) which respects w and \(\mu\). In the present paper the author gives an algebraic (exact) criterion for a triple (\(\pi\),w,\(\mu)\) consisting of a finitely presented group \(\pi\) and elements \(w\in H^ 1(\pi;{\mathbb{Z}}_ 2)\) and \(\mu \in H_ 3(\pi;{\mathbb{Z}}^ w)\) to be realizable by a 3-PC. The criterion is too complicated to be explained here and does not seem easy to apply in concrete cases. However, it enabled the author to extend to 3-PC’s some well known theorems on connected sum decompositions of closed 3- manifolds. In particular, every 3-PC is homotopy equivalent to a connected sum of prime (i.e. connected-sum-indecomposable) 3-PC’s, and this decomposition is unique up to order of the summands, homotopy equivalence of the summands, and the relation X#(S\({}^ 1\times S^ 2)\simeq X\#(twisted\) \(S^ 2\)-bundle over \(S^ 1)\), which holds for every 3-PC X with w(X)\(\neq 0\).
Reviewer: J.Vrabec


57P10 Poincaré duality spaces
57M99 General low-dimensional topology
55P15 Classification of homotopy type


Zbl 0366.57002
Full Text: EuDML