zero geometric intersection number between two simple closed curves on an oriented connected closed surface of genus $g \ge 1$

Question

Let $l_1$ and $l_2$ be two simple closed curves in $S$, where $S$ is a closed connected oriented surface of genus $g \geq 1$. $l_1$ and $l_2$ intersect transversally at exactly two crossing points such that the geometric intersection number is zero. My question is, what is the geometric interpretation of this zero intersection number. I know that if $g=1$ (Torus), then zero intersection number between two simple closed curves means that the two curves belong to the same homology class if they are both not homologous to zero or means one of the two curves must be homologous to zero. I can see that this is not accurate for $g>1$. Is it true that if intersection number is zero between two simple closed curves on any surface of any genus, then one of the curves is homologous to zero ?

Answer 1

(I'm assuming that by "simple" you mean "non-self-intersecting" rather than "boundary of an embedded disk"...)

The curves certainly need not belong to the same homology class, even if they are homologically nontrivial, at least for $g > 3$. Consider a ring of donuts. Let me be more explicit:

Place 12 donuts on a table with the hole in each donut sitting above one of the numbers in a picture of a clock, so that I can call them donuts 1 through 12. Now merge each donut with its two neighbors (perhaps by filling in the gap with a little frosting); that's now a 13-holed torus. Draw a curve on the top of this thing, starting at 12 and running along the inner edge (but near the top, i.e., away from the table) past 1, 2, ..., all the way to 6 and a little beyond, heading out to the outer edge, and then back up to a little before 12, and across to the inner edge, and close up the curve. (Basically...the obvious curve that "encloses" all the holes on the right-hand side of the clock-face). Call this curve $B$.

Now draw an analogous curve on the left-hand side: Start on the inner edge, near 12 o-clock, so that you're on curve $B$, and continue around, on the inner edge, towards 11, 10, all the way to 5:30, where you again cross curve $B$. But now continue the curve down through the 5-o'clock hole, and then, near the tabletop, on the outer (but low down) portion, head back towards 6, 7, 8, ..., until you reach 12:30, and then come UP through the 1-o'clock hole and connect up. Call this curve $Q$.

$Q$ and $B$ intersect at exactly two points, but are clearly not homologous. (Consider the intersection curve with a "meridian" that simply goes "down through the hole" at 9-o'clock. Curve $Q$ intersects it once, curve $B$ intersects it $0$ times.)

(Yeah, I could have drawn a picture. But I figured it would be of some value for you to draw one, so I described it instead.)

This leaves open your question for $g = 2, 3$, but perhaps suggests how you might look for answers in those cases.

Aw hell...let's go ahead and finish it off. The curves I described above are interesting because their oriented intersection numbers is $+2$. But let's put a 2-holed torus on a table, and draw a circle $K$ on the top enclosing the left hole, and a circle $L$ enclosing the right hole. It's easy to make these disjoint, or to make them tangent at the "midpoint" of the top surface, or, by enlarging them just a little more, to make them intersect at two points. Once again, neither is null-homologous, nor are the curves homologous to each other (as intersections with meridians shows again).