Let $M$ be a topological manifold and $N$ is a subset of $M$. Let $f_t$ be an isotopy from $id $ to $f$ which is a homeomorphism on $M$. Suppose $f(N)=N$ and there is an isotopy $g_t$ on $N$ such that $g_0=id|_N$ and $g_1=f|_N$. Assuming $f_t$ and $g_t$ satisfy the following properties.
- For any $x\in N$, $\gamma_x: [0,1] \to M$ defined by $\gamma_x(t):=\begin{cases} f_{2t}(x) & \text{if } t\in [0,1/2]\\ g_{2-2t}( x)&\text{if } t\in[1/2,1] \end{cases}$
is a trivial loop in $\pi_1(M,x)$.
- For any $x,y\in N,\;x\neq y,$ $\beta_x:=\{(\gamma_x(t),t)\in M\times S^1\,|\,t\in [0,1]\}$ is a trivial loop in $\pi_1(M\times S^1-\beta_y,(x,0))$. ($\beta_y$ is defined similarly.)
Then I want to prove:
There is an extension of $g_t$ on $M$. Precisely, there is an isotopy $ \hat g_t $ from $id$ to $f$ such that when restricted in $N$, $\hat g_t|_N=g_t$.
I am not sure the conditions are sufficient to prove the conclusion. So proof or counter example are both welcome.
Thanks a lot.