Assuming $M$ to be a connected oriented PL 4-manifold, our main results are the following: (1) if $M$ is compact with (possibly empty) boundary, there exists a simple branched covering $p : M o S^4 - Int(B^4_1 cup dots cup B^4_n)$, where the $B^4_i$'s are disjoint PL 4-balls, $n geq 0$ is the number of boundary components of $M$; (2) if $M$ is open, there exists a simple branched covering $p : M o S^4 - End (M)$, where $End (M)$ is the end space of $M$ tamely embedded in $S^4$. In both cases, the degree $d(p)$ and the branching set $B_p$ of $p$ can be assumed to satisfy one of these conditions: (1) $d(p) = 4$ and $B_p$ is a properly self-transversally immersed locally flat PL surface; (2) $d(p) = 5$ and $B_p$ is a properly embedded locally flat PL surface. In the compact (resp. open) case, by relaxing the assumption on the degree we can have $B^4$ (resp. $R^4$) as the base of the covering. A crucial technical tool used in all the proofs is a quite delicate cobordism lemma for coverings of $S^3$, which also allows us to obtain a relative version of the branched covering representation of bounded 4-manifolds, where the restriction to the boundary is a given branched covering. We also define the notion of branched covering between topological manifolds, which extends the usual one in the PL category. In this setting, as an interesting consequence of the above results, we prove that any closed oriented emph{topological} 4-manifold is a 4-fold branched covering of $S^4$. According to almost-smoothability of 4-manifolds, this branched covering could be wild at a single point.

On branched covering representation of 4-manifolds

Zuddas D
2019-01-01

Abstract

Assuming $M$ to be a connected oriented PL 4-manifold, our main results are the following: (1) if $M$ is compact with (possibly empty) boundary, there exists a simple branched covering $p : M o S^4 - Int(B^4_1 cup dots cup B^4_n)$, where the $B^4_i$'s are disjoint PL 4-balls, $n geq 0$ is the number of boundary components of $M$; (2) if $M$ is open, there exists a simple branched covering $p : M o S^4 - End (M)$, where $End (M)$ is the end space of $M$ tamely embedded in $S^4$. In both cases, the degree $d(p)$ and the branching set $B_p$ of $p$ can be assumed to satisfy one of these conditions: (1) $d(p) = 4$ and $B_p$ is a properly self-transversally immersed locally flat PL surface; (2) $d(p) = 5$ and $B_p$ is a properly embedded locally flat PL surface. In the compact (resp. open) case, by relaxing the assumption on the degree we can have $B^4$ (resp. $R^4$) as the base of the covering. A crucial technical tool used in all the proofs is a quite delicate cobordism lemma for coverings of $S^3$, which also allows us to obtain a relative version of the branched covering representation of bounded 4-manifolds, where the restriction to the boundary is a given branched covering. We also define the notion of branched covering between topological manifolds, which extends the usual one in the PL category. In this setting, as an interesting consequence of the above results, we prove that any closed oriented emph{topological} 4-manifold is a 4-fold branched covering of $S^4$. According to almost-smoothability of 4-manifolds, this branched covering could be wild at a single point.
2019
Pubblicato
https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.12187
File in questo prodotto:
File Dimensione Formato  
Branched cover represent.pdf

Accesso chiuso

Descrizione: Articolo
Tipologia: Documento in Versione Editoriale
Licenza: Copyright Editore
Dimensione 474.45 kB
Formato Adobe PDF
474.45 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
wild-covers-v2.pdf

accesso aperto

Descrizione: Articolo (versione libera)This is the accepted version of the following article: FULL CITE, which has been published in final form at https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.12187
Tipologia: Bozza finale post-referaggio (post-print)
Licenza: Creative commons
Dimensione 612.37 kB
Formato Adobe PDF
612.37 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/2957300
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 2
social impact