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## Main Question or Discussion Point

If you have a strip and you bring it around so that the ends join, that is a manifold, call it X for convenience. If instead, you put a single twist in it before joining the ends, that is a Mobius strip, which is not homeomorphic to X. If you instead put

My intuition says that it probably is. Y can be obtained from X by cutting X, adding the two twists and then sticking the ends back together. It can be done so that in the neighbourhood of the cut, the transformation is the identity, so it doesn't alter the topology. And around the rest of the manifold, all you've done is warp it a bit which also doesn't alter the topology - it would appear that this transformation should indeed be a homeomorphism. The two manifolds both have two edges, they come around and join themselves and they don't have any holes or singularities or anything like that in them.

On the other hand, Y can't be obtained from X just by stretching and bending, which usually means that the two should not be topologically equivalent - is this a counterexample that highlights a limitation of that understanding of topological equivalence? Or is the reasoning from the former paragraph just wrong? There is a clear qualitative difference between the two manifolds when they are viewed embedded in 3d space.

*two*twists in it before you join the ends, that is another manifold, call it Y for convenience. Is Y homeomorphic to X?My intuition says that it probably is. Y can be obtained from X by cutting X, adding the two twists and then sticking the ends back together. It can be done so that in the neighbourhood of the cut, the transformation is the identity, so it doesn't alter the topology. And around the rest of the manifold, all you've done is warp it a bit which also doesn't alter the topology - it would appear that this transformation should indeed be a homeomorphism. The two manifolds both have two edges, they come around and join themselves and they don't have any holes or singularities or anything like that in them.

On the other hand, Y can't be obtained from X just by stretching and bending, which usually means that the two should not be topologically equivalent - is this a counterexample that highlights a limitation of that understanding of topological equivalence? Or is the reasoning from the former paragraph just wrong? There is a clear qualitative difference between the two manifolds when they are viewed embedded in 3d space.