## 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

### MSC:

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

Zbl 0366.57002
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