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?
  • $\begingroup$ The first inclusion is always a weak equivalence. $\endgroup$ Mar 14 '16 at 11:48
  • $\begingroup$ Thanks for your comment. Could you provide me an idea of proof or a reference? $\endgroup$
    – Andrzej
    Mar 14 '16 at 11:50
  • 1
    $\begingroup$ 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). $\endgroup$ 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.