Boundaries of π1-injective surfaces

Ulrich Oertel

doi:https://doi.org/10.1016/s0166-8641(96)00124-1

Boundaries of π₁-injective surfaces

Ulrich Oertel

Rutgers Newark, Math & Computer Science

Rutgers, The State University

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

According to a result of A. Hatcher, just finitely many boundary slopes (isotopy classes of simple closed curves) can be realized as boundaries of incompressible, ∂-incompressible surfaces in a closed, compact, orientable, irreducible 3-manifold with boundary a single torus. We consider, in this paper, proper maps of surfaces (S, ∂S) into a 3-manifold (M, ∂M) which are injective on π₁ and on relative π₁, and which are embeddings on ∂S. We show that there exists a 3-manifold M, with boundary a single torus, in which every boundary slope is realized by the boundary of such a map. We prove a result interpreting the significance of boundary slopes of such surfaces for Dehn filling. More generally, we consider maps of surfaces S which are injective on π₁ and on relative π₁ as before, and which embed each component of ∂S, but do not necessarily embed all of ∂S. We show that there exists a 3-manifold with boundary a single torus admitting such a map of a connected surface simultaneously realizing an arbitrary finite set of boundary slopes. We also give examples generalizing the preceding constructions to the case where ∂M is a surface of higher genus.

Original language	English (US)
Pages (from-to)	215-234
Number of pages	20
Journal	Topology and its Applications
Volume	78
Issue number	3
DOIs	https://doi.org/10.1016/s0166-8641(96)00124-1
State	Published - 1997

ASJC Scopus subject areas

Geometry and Topology

Keywords

3-manifolds
Boundary slopes
Branched surfaces
Dehn filling
Incompressible surfaces
Injective surfaces

Access to Document

https://doi.org/10.1016/s0166-8641(96)00124-1

Cite this

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title = "Boundaries of π1-injective surfaces",

abstract = "According to a result of A. Hatcher, just finitely many boundary slopes (isotopy classes of simple closed curves) can be realized as boundaries of incompressible, ∂-incompressible surfaces in a closed, compact, orientable, irreducible 3-manifold with boundary a single torus. We consider, in this paper, proper maps of surfaces (S, ∂S) into a 3-manifold (M, ∂M) which are injective on π1 and on relative π1, and which are embeddings on ∂S. We show that there exists a 3-manifold M, with boundary a single torus, in which every boundary slope is realized by the boundary of such a map. We prove a result interpreting the significance of boundary slopes of such surfaces for Dehn filling. More generally, we consider maps of surfaces S which are injective on π1 and on relative π1 as before, and which embed each component of ∂S, but do not necessarily embed all of ∂S. We show that there exists a 3-manifold with boundary a single torus admitting such a map of a connected surface simultaneously realizing an arbitrary finite set of boundary slopes. We also give examples generalizing the preceding constructions to the case where ∂M is a surface of higher genus.",

keywords = "3-manifolds, Boundary slopes, Branched surfaces, Dehn filling, Incompressible surfaces, Injective surfaces",

author = "Ulrich Oertel",

note = "Funding Information: {\textquoteright} E-mail: oertel@andromeda.rutgers.edu. {\textquoteright} Supported by the NSF; UniversitC Louis Pasteur, Strasbourg;",

year = "1997",

doi = "https://doi.org/10.1016/s0166-8641(96)00124-1",

language = "English (US)",

volume = "78",

pages = "215--234",

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TY - JOUR

T1 - Boundaries of π1-injective surfaces

AU - Oertel, Ulrich

N1 - Funding Information: ’ E-mail: oertel@andromeda.rutgers.edu. ’ Supported by the NSF; UniversitC Louis Pasteur, Strasbourg;

PY - 1997

Y1 - 1997

N2 - According to a result of A. Hatcher, just finitely many boundary slopes (isotopy classes of simple closed curves) can be realized as boundaries of incompressible, ∂-incompressible surfaces in a closed, compact, orientable, irreducible 3-manifold with boundary a single torus. We consider, in this paper, proper maps of surfaces (S, ∂S) into a 3-manifold (M, ∂M) which are injective on π1 and on relative π1, and which are embeddings on ∂S. We show that there exists a 3-manifold M, with boundary a single torus, in which every boundary slope is realized by the boundary of such a map. We prove a result interpreting the significance of boundary slopes of such surfaces for Dehn filling. More generally, we consider maps of surfaces S which are injective on π1 and on relative π1 as before, and which embed each component of ∂S, but do not necessarily embed all of ∂S. We show that there exists a 3-manifold with boundary a single torus admitting such a map of a connected surface simultaneously realizing an arbitrary finite set of boundary slopes. We also give examples generalizing the preceding constructions to the case where ∂M is a surface of higher genus.

AB - According to a result of A. Hatcher, just finitely many boundary slopes (isotopy classes of simple closed curves) can be realized as boundaries of incompressible, ∂-incompressible surfaces in a closed, compact, orientable, irreducible 3-manifold with boundary a single torus. We consider, in this paper, proper maps of surfaces (S, ∂S) into a 3-manifold (M, ∂M) which are injective on π1 and on relative π1, and which are embeddings on ∂S. We show that there exists a 3-manifold M, with boundary a single torus, in which every boundary slope is realized by the boundary of such a map. We prove a result interpreting the significance of boundary slopes of such surfaces for Dehn filling. More generally, we consider maps of surfaces S which are injective on π1 and on relative π1 as before, and which embed each component of ∂S, but do not necessarily embed all of ∂S. We show that there exists a 3-manifold with boundary a single torus admitting such a map of a connected surface simultaneously realizing an arbitrary finite set of boundary slopes. We also give examples generalizing the preceding constructions to the case where ∂M is a surface of higher genus.

KW - 3-manifolds

KW - Boundary slopes

KW - Branched surfaces

KW - Dehn filling

KW - Incompressible surfaces

KW - Injective surfaces

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