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.

PD_4-Complexes: constructions, cobordisms and signatures / Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia. - In: HOMOLOGY, HOMOTOPY AND APPLICATIONS. - ISSN 1532-0073. - STAMPA. - 18:2(2016), pp. 267-281. [10.4310/HHA.2016.v18.n2.a15]

PD_4-Complexes: constructions, cobordisms and signatures

CAVICCHIOLI, Alberto;SPAGGIARI, Fulvia
2016

Abstract

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.
2016
18
2
267
281
PD_4-Complexes: constructions, cobordisms and signatures / Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia. - In: HOMOLOGY, HOMOTOPY AND APPLICATIONS. - ISSN 1532-0073. - STAMPA. - 18:2(2016), pp. 267-281. [10.4310/HHA.2016.v18.n2.a15]
Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11380/1122820
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