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 the 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 the 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.
