I drew a deformation retract of the twice punctured genus 2-torus. It would seem that this should deformation retract to a bouquet of 3 circles.
One the other hand, computing the Euler-characteristics shows that it should deformation retract to a bouquet of 5 circles.
If anyone could explain what I'm doing wrong, I would appreciate it!
The shaded areas seem to ignoring the center of each torus as you move the boundary of the puncture points past them and so the third and fourth step are erroneous.