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How to prove connect sum of two manifolds doesn't depend on the choices of balls(which would be cutted) and different gluing of boundary spheres? Is it still true in differential category? Thanks!

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    The connected sum of what are you talking about?2010-10-01
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    It sounds like he's talking about surfaces. In that case the theorem is called the "Tubular Neighbourhood Theorem".2010-10-01
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    The only proof I know can be found in Kosinski's book "Differential Manifolds". It's now published by dover, so quite cheap. He argues that the answer to your second question is "yes".2010-10-01
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    And of course, the answer does depend on the "choice of balls" provided your manifolds are orientable and do not admit an orientation-reversing diffeomorphism. If you restrict your balls to be orientation-preserving embeddings, this is not an issue.2010-10-01
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    For connected compact surfaces this is a consequence of the classification theorem for those surfaces.2010-10-01

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