PD_4-Complexes: constructions, cobordisms and signatures

The oriented topological cobordism group $\Omega_4 (P)$ of an oriented $\operatorname{PD}_4$--complex $P$ is isomorphic to $\Bbb Z \oplus \Bbb Z$. The invariants of an element $\{ f : X \to P \} \in \Omega_4 (P)$ are the signature of $X$ and the degree of $f$. We prove an analogous result for the Poincar\' e duality cobordism group $\Omega_{4}^{\operatorname{PD}} (P)$: If $\pi_1 (P)$ does not contain nontrivial elements of order $2$, then $\Omega_{4}^{\operatorname{PD}} (P)$ is isomorphic to $L^{0} (\Lambda) \oplus \Bbb Z$, where $L^{0} (\Lambda)$ is the Witt group of non-degenerated hermitian forms on finitely generated stably free $\Lambda$--modules. The component of an element $\{ f : X \to P \} \in \Omega_{4}^{\operatorname{PD}} (P)$ in $L^{0} (\Lambda)$ is related to the symmetric signature of $X$. Then we construct explicitly $\operatorname{PD}_4$--complexes, define the well--known map $L_4 (\pi_1 (P)) \to \Omega_{4}^{\operatorname{PD}} (P)$, and characterize the image of the map $\Omega_{4}^{\operatorname{PD}} (P) \to \Omega_{4}^{N} (P)$. The results are summarized in Theorems 1.1 and 1.2 stated in the introduction.