Homotopically trivial vs isotopically trivial diffeomorphisms

Question

Let $M$ be a manifold. Let's say $M$ is smooth, connected, oriented. We can also assume that $M$ is closed if that makes things easier.

Let $\mathit{Diff}(M)$ denote the group of diffeomorphisms of $M$ and $\mathit{Diff}_0(M)$ denote its identity component, consisting of the isotopically trivial diffeomorphisms of $M$. Let us also denote by $\mathit{Diff}_1(M)$ the subgroup of homotopically trivial diffeomorphisms of $M$.

I know that $\mathit{Diff}_0(M) \subsetneq \mathit{Diff}_1(M)$ in general and that they are the same for surfaces and some hyperbolic $3$-manifolds, but that's about all I know.

Can we say more? In particular, I would like to know if $\mathit{Diff}_0(M)$ is always a subgroup of $\mathit{Diff}_1(M)$ such that the quotient is discrete? (maybe with assuming some extra conditions on $M$?)

Thank you for your insights.

Answer 1

The quotient group $Diff_1(M)/Diff_0(M)$ is a discrete group since $ Diff_0(M)$ is a path component of $Diff(M)$, hence also a connected component since $Diff(M)$ is locally path-cconnected, and $Diff_1(M)$ is a union of components of $Diff(M)$, making the quotient discrete. [It is easy to see that $Diff_0(M)$ is a normal subgroup of $Diff(M)$, hence also of $Diff_1(M)$, so the quotient $Diff_1(M)/Diff_0(M)$ is indeed a group.]

For closed 3-manifolds it is usually true that $Diff_0(M)=Diff_1(M)$, but there are some exceptions such as the connected sum of two suitably chosen lens spaces, if I'm remembering correctly. Dimension 4 is far more subtle and much less is known. I have a dim recollection that examples are known where $Diff_0(M)\neq Diff_1(M)$, but someone like Danny Ruberman would know for sure. In dimension 5 there are many simply-connected closed manifolds for which $Diff_0(M)= Diff_1(M)$ such as the sphere $S^5$, but this often fails in the nonsimply-connected case. For example when $M$ is the 5-dimensional torus the quotient group $Diff_1(M)/Diff_0(M)$ is the direct product of an infinite (but countable) number of cyclic groups of order 2. However for the connected sum of the 5-torus and $S^2\times S^3$ the quotient $Diff_1(M)/Diff_0(M)$ is trivial.

These 5-dimensional phenomena persist in higher dimensions as well, and there are additional reasons why $Diff_0(M)$ can differ from $Diff_1(M)$ due to the existence of exotic spheres. For example $Diff_1(S^6)/Diff_0(S^6)$ is isomorphic to the group of exotic 7-spheres, a cyclic group of order 28. More generally $Diff_1(S^n)/Diff_0(S^n)$ is isomorphic to the group of exotic $(n+1)$-spheres whenever $n\geq 5$, and it seems that this group is almost always nontrivial. The known exceptions where no exotic $n$-spheres exist are $n\leq 3$ and $n=5$, $6$, $12$, $56$, and $61$. This list is complete up to $n=61$ except that the case $n=4$ is still unknown. The case $n=56$ was only added to the list in the past year due to recent calculations of Dan Isaksen and Zhouli Xu. There is a nice discussion of this question in an article by Milnor in the 2011 Notices of the A.M.S. called "Differential topology 46 years later", though this predates the discovery that 56 is also an exceptional dimension.