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Is this question acceptable?: TS^2 and S^2 x R^2 not homeomorphic

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    This question:

    http://mathoverflow.net/questions/92337/ts2-and-s2-x-r2-not-homeomorphic-as-topological-spaces-closed

    asks whether the tangent bundle to the two-sphere and the trivial plane-bundle over the two-sphere have homeomorphic total spaces. The question's been closed, but it seems to me to be a perfectly fine MO-level question (though, of course, I might be overlooking something).

    I'd vote to reopen it except for one thing: The poster says he's trying to prove that the two spaces are not homeomorphic, as opposed to asking *whether* they are homeomorphic. This suggests that he already knows the answer, which in turn suggest it' s a homework problem.

    So I guess my twin questions are: 1) Am I wrong in thinking this question is a perfectly good MO-level question? And 2) Am I wrong in thinking that the wording suggests homework and that this is sufficient not to reopen it?
    • CommentAuthorRyan Budney
    • CommentTimeMar 26th 2012 edited
     

    I replied in the question. The question is more subtle than I first thought, but it is at the level of a 1st year grad algebraic topology course out of a book like Hatcher's.

    I still don't think it's appropriate for MO, though.

    • CommentAuthordeane.yang
    • CommentTimeMar 27th 2012
     
    I must say that I agree with Steve, as well as the comment by George to the question. This is a nice question, even if it can be solved using 1st year graduate algebraic topology.
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    I agree with George and Deane that the sophistication of Ryan's answer is a good argument for reopening. The counterargument is that I still suspect the question is homework (not because of its level but because the wording suggests that the questioner knows what answer to expect). On balance, I voted for reopening after seeing Ryan's argument.
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    I can't imagine offering this as a homework problem in first year algebraic topology without a hint or two. This is in no way obvious unless you know the right sort of tricks, which might or might not get dealt with in a first year course, depending on the level. I think it is a perfectly fine question.

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