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maybe this is a stupid question. I know how to prove that the connected sum $M\#N$ of two topological manifolds is a topological manifold, however I don´t know how to prove that the connected sum of two differentiable manifolds is , in fact, differentiable.

Thanks in advance.

user40276
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1 Answers1

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This is covered for instance in Section 3 of J.Milnor "Lectures on the h-cobordism theorem" (combine corollary 3.6 and theorem 1.4).

Moishe Kohan
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  • There isn´t any direct proof? This reference looks like kinda advanced. – user40276 Aug 23 '13 at 14:08
  • @user40276: Read the proof of theorem 1.4 and see if you understand it. If not, then chances are that any proof will be too advanced for you. A side remark: The differentiable structure will depend on the choice of diffeomorphism between the codimension 1 spheres. For instance, Milnor's exotic 7-dimensional spheres are constructed as connected sums of ordinary spheres. – Moishe Kohan Aug 23 '13 at 14:26