Multibranched surfaces in 3-manifolds

Abstract: A multibranched manifold is a second countable Hausdorff
space that is locally homeomorphic to multibranched Euclidean space.
In this talk, we concentrate compact 2-dimensional multibranched
manifolds (multibranched surfaces) embedded in 3-manifolds.
We give a necessary and sufficient condition for a multibranched
surface to be embedded in some closed orientable 3-manifold.
Then we can define the genus of a multibranched surface in virtue of
Heegaard genera of 3-manifolds, and show an inequality between the
genus, the number of branch loci and regions.
We determine whether two multibranched surfaces have same neighborhood
by means of local moves.
Similarly to the graph minor, we also introduce a minor on
multibranched surfaces, and consider the obstruction set for the set
of multibranched surfaces embedded in the 3-sphere.
This talk is a survey including recent joint works with Kazufumi Eto,
Shosaku Matsuzaki, Mario Eudave-Munoz, Kai Ishihara, Yuya Koda, Koya
Shimokawa.