Here is a question that has been haunting me for a while: Let $\mathbb{R}^{n-1} \times [0, \infty)$ be the upper half space of $\mathbb{R}^n$ and suppose we have a smooth homeomorphism (not a diffeo)
$f : \mathbb{R}^{n-1} \times [0, \infty) \rightarrow \mathbb{R}^{n-1} \times [0, \infty)$
such that $f$ is the identity on $\mathbb{R}^{n-1} \times \{0\}$ and $f$ restricts to a diffeomorphism from the punctured (at the origin) half-space to itself. In other words, $f$ is the identity on the boundary and it would be a diffeomorphism from the half-space to itself except that the derivative is singular at the origin.
Question : Is the restriction of $f$ to the punctured half space smoothly isotopic to the identity?
A little background: If $Df$ were nonsingular at 0 this would just be an application of the uniqueness of collars, which is a special case of the uniqueness of tubular neighborhoods. That proof uses the morse lemma to write the coordinates of the imbedding as a linear combination of coordinate functions. The coefficients are given by smooth functions whose values at zero are the entries of the Jacobian of the imbedding at the origin. The isotopy one constructs takes the imbedding to the linear map $Df_0$. The issue here is of course that this linear map is degenerate. But perhaps we can construct a different isotopy if we remove the bad point?
This is mainly motivated by my (borderline obsessive and unhealthy) thinking about this question: Different definitions of handle attachment
Thanks for reading my question and I hope It's not too cranky.
I don't really see the connection to your handle attachment question -- it looks to me like you're approaching it as a more general problem than it actually is.
But this question has a reasonable answer. Think of your half-space as sitting inside the Euclidean space $\mathbb R^n$. Then you can intersect the half-space with spheres of radius $r$ centred around the origin. This converts your punctured half-space (via a diffeomorphism) to:
$$ D^{n-1} \times \mathbb R $$
So you are studying the group of diffeomorphisms of this manifold where the diffeomorphism restricts to the identity on the boundary. I think in full generality this space is not connected. Aside from dimension $4$ I think this space is known to have the homotopy-type of $\Omega Diff(D^n)$, where $Diff(D^n)$ is the group of diffeomorphisms of the $n$-disc $D^n$ which restrict to the identity on the boundary.
The basic idea for why you'd expect something like this is that a diffeomorphism of $D^{n-1} \times \mathbb R$ restricts to an embedding of $D^{n-1} \times \{0\}$ into $D^{n-1} \times \mathbb R$. That map is a fibre bundle. There's a standard sequence of relations between this embedding space and $Diff(D^n)$, it's outlined at the end of Hatcher's paper on the Smale conjecture. Needless to say, the spaces $Diff(D^n)$ for $n$ large tend not to be contractible, and the fundamental groups are also known to be non-trivial in a large number of cases. Partial computations of the fundamental groups were done by Milnor and Kervaire.