Boundaries of π1-injective surfaces

Ulrich Oertel

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 languageEnglish (US)
Pages (from-to)215-234
Number of pages20
JournalTopology and its Applications
Issue number3
StatePublished - 1997

ASJC Scopus subject areas

  • Geometry and Topology


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


Dive into the research topics of 'Boundaries of π1-injective surfaces'. Together they form a unique fingerprint.

Cite this