TITLE: Verifying 'standup maths' homeomorphism claim
QUESTION [33 upvotes]: In his newest video, Matt Parker claims that a sphere with three holes (a pair of trousers) and a torus with one hole (a pair of trousers with the legs sewn) are homeomorphic. I assume he meant removing closed discs, since he wanted them to be manifolds.
As justification, he shows that they're homotopy equivalent (to $S^1\vee S^1$) and then claims 'because they have thickness' they are homeomorphic.
I'm not fully convinced, but cannot show either way. My thoughts are that they are homeomorphic, but that it's not as simple as he suggested (he's essentially suggesting 'rotating' the perpendicularly glued annuli so that they are glued parallel to each other, in some sort of projection, I'm not convinced this is well-defined).
Thoughts?

REPLY [24 votes]: Watching the video, he seems to be aware that they are not homeomorphic as 2-manifolds.
He says that to justify the crucial step of going from two perpendicularly glued cylinders to two annuli joined on the side within a plane, you have to consider the thickness of the fabric, so that they are 3D objects, rather than the thickness in going from 1D (2 joined circles) to 2D.
See this point in the video.