On finitely dominated CW complexes

Abstract:

Let $\mathcal {D}$ be the class of all CW complexes homotopy dominated by finite CW complexes. In this paper we prove the following theorem. Theorem. Suppose a connected CW complex $X \in \mathcal {D}$ is a union of two connected subcomplexes ${X_1}$, ${X_2}$ with ${X_1} \cap {X_2} = {X_0} \in \mathcal {D}$. Then ${X_1}$, ${X_2} \in \mathcal {D}$ if one of the following conditions is satisfied: (i) ${\pi _1}({X_0},x) \to {\pi _1}(X,x)$ is a monomorphism for each $x \in {X_0}$, (ii) ${\pi _1}({X_i}) \to {\pi _1}(X)$ is a monomorphism for $i = 1,2$ and ${\pi _1}({X_1})$, ${\pi _1}({X_2})$ are finitely presented.