# Making diffeomorphism of submanifolds boring

This is probably very well known in surgery theory... I'm looking for a modern reference on the following questions (the only one I know is Browder's "Diffeomorphism of 1-connected manifolds" from 1976 which is nice but probably a little outdated):

Let $M$ be a smooth manifold and let $N \hookrightarrow M$ be a smooth submanifold. We assume that $\dim(N) \geq 2$, $\dim(M) \geq \dim(N)+3$ and that $M$ is 1-connected (or has a finite fundamental group).

Let $\phi : N \to N$ be an orientation preserving diffeomorphism which represents a non-trivial element in the smooth mapping class group $\pi_0 \mathrm{Diff}^+(N)$.

Question. What are the conditions/obstructions to lifting $\phi$ to:

1. A homotopy equivalence $\Phi : M \to M$?

(Perhaps more generally - how does one relate the groups of self-homotopy equivalences of $M$ and $N$?)

2. An orientation preserving diffeomorphism $\Phi : M \to M$?

3. A diffeomorphism $\Phi : M \to M$ which is trivial in $\pi_0 \mathrm{Diff}^+(M)$?

(More generally - what can be said about the topology of the space of pairs which consist of: orientation preserving diffeomorphism $\Phi : M \to M$ which lift $\phi$ and a choice of trivializing isotopy?)

• What do you mean by compatibly oriented here? – Greg Friedman Jun 15 '17 at 8:13
• Hmm ... I guess it was just a usual caveat for the case of codimension $=0$, but I've already ruled it out by the dimension condition above ... Thx, removed. – Nati Jun 15 '17 at 12:10