### Abstract

It is our goal to determine whether a given multibranched surface can be embedded in the 3-sphere or not.

In a similar way to the forbidden graph characterization in Graph Theory, we define a minor for multibranched surfaces and consider the forbidden minor.

On the other hand, we define a critical 2-complex.

It turns out that any forbidden multibranched surface contains a critical 2-complex.

In this talk, we exhibit forbidden minor multibranched surfaces which are $K_5 \times S^1$ and $K_{3,3} \times S^1$ families and critical 2-complexes which are contained in that.

This is a joint work with Mario Eudave-MuĂ±oz.