TY - JOUR
T1 - Positive flow-spines and contact 3-manifolds
AU - Ishii, Ippei
AU - Ishikawa, Masaharu
AU - Koda, Yuya
AU - Naoe, Hironobu
N1 - Funding Information:
The authors wish to express their gratitude to Riccardo Benedetti for many insightful comments. The second author would like to thank Shin Handa and Atsushi Ichikawa for useful discussions in the early stages of research. The second author is supported by JSPS KAKENHI Grant Numbers JP19K03499, JP17H06128 and Keio University Academic Development Funds for Individual Research. The third author is supported by JSPS KAKENHI Grant Numbers JP15H03620, JP17K05254, JP17H06463, and JST CREST Grant Number JPMJCR17J4. The fourth author is supported by JSPS KAKENHI Grant Number JP19K21019.
Publisher Copyright:
© 2023, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023
Y1 - 2023
N2 - A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form whose Reeb flow is a flow of the flow-spine. It is known by Thurston and Winkelnkemper that any open book decomposition of a closed oriented 3-manifold supports a contact structure. In this paper, we introduce a notion of positivity for flow-spines and prove that any positive flow-spine of a closed, connected, oriented 3-manifold supports a contact structure uniquely up to isotopy. The positivity condition is critical to the existence of the unique, supported contact structure, which is also proved in the paper.
AB - A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form whose Reeb flow is a flow of the flow-spine. It is known by Thurston and Winkelnkemper that any open book decomposition of a closed oriented 3-manifold supports a contact structure. In this paper, we introduce a notion of positivity for flow-spines and prove that any positive flow-spine of a closed, connected, oriented 3-manifold supports a contact structure uniquely up to isotopy. The positivity condition is critical to the existence of the unique, supported contact structure, which is also proved in the paper.
KW - 3-dimensional manifold
KW - Contact structure
KW - Flow
KW - Polyhedron
KW - Spine
UR - http://www.scopus.com/inward/record.url?scp=85151362142&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85151362142&partnerID=8YFLogxK
U2 - 10.1007/s10231-023-01314-1
DO - 10.1007/s10231-023-01314-1
M3 - Article
AN - SCOPUS:85151362142
SN - 0373-3114
VL - 202
SP - 2091
EP - 2126
JO - Annali di Matematica Pura ed Applicata
JF - Annali di Matematica Pura ed Applicata
IS - 5
ER -