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 there 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.

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.

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 there 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 discrete subgroup of $\mathit{Diff}_1(M)$ (maybe with assuming some extra conditions on M). Thank you for your insights.

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 there 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.