Diffeomorphisms isotopic to identity Let $M$ be a compact manifold and $\phi\in\text{Diff}(M)$ a diffeomorphism on $M$. 
Is it true that, if $\phi$ is sufficiently $C^{1}$-small, then is $\phi$ isotopic to $\text{Id}_{M}$? 
Intuitively this seems right to me, but I cannot give a precise proof. Also, it is not clear to me if I have to impose compactness on $M$. Any help or references are much appreciated.
 A: If $M$ is a compact manifold the space $C^1(M, M)$ of $C^1$-self-maps of $M$ with constitutes a Banach manifold with the weak topology; for a proof see, for example, this paper. 
Stability of diffeomorphisms implies that $\text{Diff}(M) \subset C^1(M, M)$ is an open subset. Choose a Banach chart $\mathscr{U}$ around $\text{Id}$, and consider the intersection $\mathscr{U} \cap \text{Diff}(M)$. This is a (not necessarily connected) open subset containing $\text{Id}$, therefore contains a smaller (connected) Banach chart $\mathscr{V}$ around $\text{Id}$. If $\phi_n$ is a sequence of $C^1$-diffeomorphism converging to $\text{Id}$ in the weak topology, it must be eventually contained in $\mathscr{V}$ - which, being homeomorphic to a connected open subset of a Banach space, is path connected. So suppose for all $n > N$, $\phi_n$ is contained in $\mathscr{V}$. Therefore there exists a homotopy $\varphi_t$ serving as a path joining $\varphi_0 = \phi_n$ and $\varphi_1 = \text{Id}$ which stays inside $\text{Diff}(M)$ for all time $t \in [0, 1]$; this implies $\varphi_t$ is an isotopy of $\phi_n$ to the identity for all $n > N$, as required. 
I think compactness of $M$ is necessary for this to hold. As a counterexample consider the two-ended surface of infinite genus $\Sigma$ constructed as a sequence of connected sums $\bigsqcup_{k \in \Bbb Z} T_k^2 \setminus (D^{-}_k \cup D^{+}_k)/\!\!\sim$ where $\sim$ identifies $D^+_k$ with $D^-_{k+1}$ for each $k$. Identify $T^2 \setminus (D^- \cup D^+)$ with $X = S^1 \times S^1 \setminus (D_x \cup D_y)$ where $x$ and $y$ are two points on the diagonal $\Delta \subset S^1 \times S^1$ and $D_x$ and $D_y$ are small open disks around them, and let $f : X \to X$ be the diffeomorphism given by $f(a, b) = (b, a)$. This preserves the boundary of the removed disks. 
Therefore define $\phi_n : \Sigma \to \Sigma$ to be the diffeomorphism which acts on the $n$-th doubly punctured torus by $f$ and is identity on the rest. $(\phi_n)$ converges to $\text{Id}$ in the $C^1$ compact-open topology on $\text{Diff}(\Sigma)$ since by construction for any compact subset $K \subset \Sigma$, $(\phi_n)$ eventually becomes the identity on $K$. None of the $\phi_n$'s are homotopic to identity as it acts nontrivially on $H_1(\Sigma)$ by switching the two generator curves on the $n$-th torus - in fact no two $\phi_n$ belong to the same connected component of $\text{Diff}(\Sigma)$. 
Morally the genus accumulation towards a point in the space of ends of $\Sigma$ forces $\text{Diff}(\Sigma)$ to have infinitely many connected components arbitrarily close to the component of $\text{Id}$.
