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M' is a manifold with boundary. One can attaching a handle $h:=D^k\times D^{n-k}$ along $f:S^{k-1}\times D^{n-k}\rightarrow M'$ forming $M=M'\cup_f h$. Suppose f is homotopic with f', and that f,f' are smooth embeddings. Is there a theorem concluding $M=M'\cup_f h$ is diffeomorphic with $M=M'\cup_f' h$ by imposing some sort of restriction on the homotopy between f and f'(Or ideally, no restriction)?

Actually I'm just asking for theorems that make attaching handles more flexible...Any theorem helpful is quite welcomed.


By the way, I wonder whether there is such kind of more general theorem: $f_0,f_1:N\rightarrow M$ are embeddings(topologically or smoothly), $F:N\times I\rightarrow M$ is a homotopy connecting $f_0, f_1$ satisfying some condition(for example $F_t$ is an embedding for any t). Suppose $N\subset N'$ and there is an embedding $g_0:N'\rightarrow M$ extending $f_0$. Then F extends to a $G:N'\times I\rightarrow M$ with $G_0=g_0$, $G_1$ still an embedding .

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    It is rarely a good idea to use abbreviations in titles.2011-08-03
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    I was being lazy... Fixed. Thanks.2011-08-03
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    I am pretty sure the answer is, yes, homotopic attaching maps give you diffeomorphic manifolds, but I'm at home and can't look it up. Have you checked any standard references on handle theory?2011-08-03
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    Actually no. What are the standard references on handle theory? I didn't learn this kind of theory systematically.2011-08-03
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    lethe: I agree; I too have been groping my way thru the dark trying to learn these topics.2011-08-03
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    As PseudoNeo points out, this question should be about smooth homotopy.2011-08-03

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