what is the fundamental group of a torus with $k$ points removed

Question

I was about to use the Seifert-van Kampen to compute the fundamental group of a torus of genus 2. In the process, I need to know the fundamental group of a torus (of genus one) with a hole removed. is there an obvious deformation retraction ? (I don't see it ... ) In general : How can I find the fundamental group of a torus with $k$ holes removed ?

Thank you in advance,

Answer 1

Hint: Draw the torus as a square with opposite sides identified. If I remove one point in the interior of a convex polygon, what is the obvious deformation retraction to the boundary?

Answer 2

To figure out the Torus with a point removed, you might try drawing the torus as a rectangle with opposite edges "glued" in appropriate ways: