# 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.

1. Is my guess correct?
2. Is there anything else useful to say about the relation of the homotopy type between those three groups?
• The first inclusion is always a weak equivalence. – Tom Goodwillie Mar 14 '16 at 11:48
• Thanks for your comment. Could you provide me an idea of proof or a reference? – Andrzej Mar 14 '16 at 11:50
• Proposition 1.3 of Igusa's "Stability theorem for smooth pseudoisotopies" deals with closely related matters, and probably there is no better reference. The paper can be found at people.brandeis.edu/~igusa/Papers/Selected.htm as 12Mb file (which is why I did not link it). – Igor Belegradek Mar 15 '16 at 11:57

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.