This one is going to be more tricky.

Imagine a sphere with two linked handles. Is it homeomorphic to a sphere with two unlinked handles? Well, you know the classification theorem, so you know that the answer is “yes”. But can you actually imagine a gradual, continuous deformation that unlinks two handles? (If you do, you must have an extraordinarily good spatial imagination.)

Solution (it involves creepy, googly-eyed men with no legs):

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Very very nice. I kind of spoiled it for myself — first I though you meant a sphere with two handles that joined somewhere, which clearly has a different fundamental group than a sphere with two handles — and once I’d looked at the first one, I’d already seen enough of the others for it to be easy to work out. Still fun though. I need to read more algebraic topology…

Much more challenging than your previous problem. 🙂

I knew I should have put a picture in the description of the problem, to make it clear, but I was too lazy.