Suppose we had a 2D surface with g holes in it, and suppose a child drew closed loops on that surface. How many topologically distinct loops can be drawn on the surface? Two loops are equivalent if they can be continuously deformed into each other.
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2Why do you phrase the question that way...? There's an infinite number of them, look up "fundamental group of genus g surface". – Najib Idrissi Sep 05 '15 at 09:04
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1Got it. You can wind around a hole any integer number of times. Except a sphere. – ChickenGod Sep 05 '15 at 09:28