$PD_4$-complexes: constructions, cobordisms and signatures

Alberto Cavicchioli ; Friedrich Hegenbarth ; Fulvia Spaggiari

Homology, homotopy, and applications, Tome 18 (2016) no. 2, pp. 267-281.

Résumé

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.

Détail

DOI : 10.4310/HHA.2016.v18.n2.a15

Classification : 57N65, 57Q10, 57R67
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