Homotopy type of diffeomorphism which are the identity on and near the boundary 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?

 A: The first inclusion $Diff_{U\partial} M \to Diff_{\partial} M$ is a homotopy-equivalence provided you do not let the neighbourhood get "too big".  If you fix the neighbourhood there is a fibre sequence
$$Diff_{U\partial} M \to Diff_{\partial} M \to Emb(U, M)$$
where the embeddings of the collar neighbourhood $U$ of $\partial M$ in $M$ are required to be the identity on the boundary.  The fact that this embedding space is contractible boils down to the uniqueness of tubular neighbourhoods theorem + the convexity of small linear collars. 
I almost never see people use the space where you allow the neighbourhood $U$ to vary.  What literature are you reading?   But you can adapt this argument to describe the homotopy-type of that space, as well.  As yours is basically the union of these spaces.  As these spaces all intersect over a homotopy-equivalent subspace, you get the result at the weak homotopy-type level. 
Regarding the last inclusion $Diff_{\partial} M \to Diff M$ this also sits in a fibre sequence
$$Diff_\partial M \to Diff M \to Diff(\partial M)$$
this space is more closely related to the idea of pseudoisotopy.  In particular, this fibre-sequence is not always onto as not every diffeomorphism of the boundary extends to the interior of the manifold.  
