Attaching cells in the Morse Theory Every critical point of a Morse function tells us to glue a cell of a dimension equal to the index of the critical point. Is there any information about how to glue this cell? For instance, does the attaching map have to be an embedding? To be more precise, can we change  $M^{c-\epsilon}$ up to homotopy type so that the attaching map of the cell is an embedding?
Here $M^a=f^{-1}(-\infty,a]$ and $c$ is a critical value. $\epsilon$ is small enough not to provide any new critical points.

If $p \in M$ is the critical point then the standard Morse theory argument shows that if $a=c-\epsilon$ with $\epsilon>0$ sufficiently close to $0$, then there exists a neighborhood $U$ of $p$, and a diffeomorphism $f : U \to B(0,1) \subset \mathbb{R}^m$ (where $M$ has dimension $m$), such that $f(U \cap M^a)$ is the intersection with $B(0,1)$ of the set 
$$\{x=(x_1,...,x_m) \,\bigm|\, -x_1^2-...-x_k^2+x_{k+1}^2+...+x_{m}^2 \le -(.5)^2\}
$$
(where $k$ is the index of the singularity). In this situation, the cell you want to attach is the $k$-dimensional ball of radius $.5$ in the $k$-dimensional coordinate hyperplane $x_{k+1}=...=x_{m}=0$. The boundary of that ball is embedded in $U$, and is contained in the set
$$\{x=(x_1,...,x_m) \,\bigm|\, -x_1^2-...-x_k^2+x_{k+1}^2+...+x_{m}^2 = -(.5)^2\}
$$ 
which is equal to $f(U \cap \partial M^a)$. The restriction of $f^{-1}$ to the boundary of that ball is the attaching map of the cell, so that attaching map is indeed an embedding.
