$PD_4$-complexes: constructions, cobordisms and signatures

The oriented topological cobordism group $\Omega_4 (P)$ of an oriented $\operatorname{PD}_4$-complex $P$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$. The invariants of an element {$\{ f \colon X \to P \} \in \Omega_4 (P)$} are the signature of $X$ and the degree of $f$. We prove an analogous result for the Poincaré 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 $L^{0} (\Lambda) \oplus \mathbb{Z}$, 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 $\{ f \colon X \to P \} \in \Omega_{4}^{\operatorname{PD}} (P)$ 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.

Keywords

Poincaré duality complex, signature, cobordism group, surgery sequence, Witt group, homotopy type, Whitehead quadratic group, spectral sequence, obstruction theory, homology with local coefficients, total surgery obstruction

2010 Mathematics Subject Classification

57N65, 57Q10, 57R67

Full Text (PDF format)

Published 29 November 2016