# the local structure of an immersed incompressible surface

Assume that $M$ is a closed, irreducible, orientable 3-manifold. Suppose that we have a closed, immersed, incompressible surface $F$ of genus at least 1. Since we only required $F$ to be immersed in $M$, there may be some self-intersections along arcs or curves. Of course, we can assume that all these intersections are efficient, i.e. transverse. So my first question is:

Are these self-intersecting components some graphs without valence-1 vertices?

My second question is:

If we remove all these self-intersection components from $F$, what does the complement look like?

As for the second question, the complement of the self-intersection locus will just be subsurfaces, and can change quite a bit under homotopy. If the surface is made minimal area (in its homotopy class), then some information can be gleaned from a paper of Freedman-Hass-Scott. Minimal area maps of $\pi_1-$injective surfaces are immersions, and may factor through a finite covering. They show that the minimal immersions intersect minimally in a certain sense. From their results, one can show that the complementary regions of the singular locus will be $\pi_1$-injective (of course, this is meaningless if all of the complementary regions are disks). I think one could perturb this immersion to be generic while retaining this property. Not sure what exactly you're fishing for though.
• thank you, this is a great answer, specially for the figure, May I ask that if we require that $F$ is least area, is there any disk in the complement of the self intersection locus? – yanqing Nov 16 '16 at 6:46