"Thin position of a pair (3-manifold, 1-submanifold)"
- abstract
-
For 1-submanifolds T in compact orientable 3-manifolds M
possibly with non-empty boundaries,
we introduce the notion of Heegaard splittings of the pair (M,T)
which is an amalgamation of
the notions of bridge decompositions defined by H. Schubert
and Heegaard splittings,
following the study of H. Doll.
We also introduce the notion of thin multiple Heegaard splittings of (M,T)
which is an amalgamation of the notions of
thin positions for links in the 3-sphere S^3 defined by D. Gabai
and thin positions for 3-manifolds defined by M. Scharlemann and A. Thompson
.
Thus (M,T) is regarded
as a union of pairs of (compression bodies, trivial arcs)
obtained by gluing positive boundaries to positive boundaries
and negative boundaries to negative boundaries.
We show
that the negative surfaces of a thin multiple Heegaard splitting of (M,T)
are "T-essential" in (M,T),
following the study of M.Scharlemann and A. Thompson.
We also show
that a T-irreducible and "weakly T-reducible" Heegaard splitting
of (M,T) can be "untelescoped"
into a multiple Heegaard splitting
containing a T-incompressible negative surface
which is not a sphere disjoint from T.
This is a generalization of a result of A.J. Casson and C.McA. Gordon.
you can download the paper here
