Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on the boundary and the ones that are the identity on some (non-fixed) neighborhood of the boundary. Equip them with the usual smooth Whitney topology. We have inclusions $$Diff_{U\partial}(M)\subseteq Diff_\partial(M) \subseteq Diff(M)$$ and I am interested in the effect of those in homotopy groups.

The second inclusion is usually far away from being a (weak) homotopy equivalence. In the case $M=D^2$, $Diff_\partial(M)$ is contractible and $Diff(M)$ has the homotopy type of $O(2)$.

The situation of the first inclusion seems to be more subtle. In the case of low dimensions, I believe that it should be a weak homotopy equivalence in general and I expect that this is no longer true in higher dimensions. This fits to the literature in the sense that people are usually very vague between the difference of $Diff_{U\partial}(M)$ and $Diff_\partial(M)$ in low dimensions and mostly emphasize the use of $Diff_{U\partial}(M)$ opposite of $Diff_\partial(M)$ when working in higher dimensions.

- Is my guess correct?
- Is there anything else useful to say about the relation of the homotopy type between those three groups?