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 .