Let $M$ be a multiple pointed space, i.e. $M$ is a topological space and there is a finite point set $M\supset P=\{p_1,...,p_k\}, k<\infty$. Such a $p_i$ is called a marked point. A map $$\varphi:M,P\to M,P$$ is called a $P$-relative map if $$\varphi|_P=id|_P.$$ Our question is that

**For a disk $D^2$ (moreover we order that all maps are invariant on $S^1=\partial D^2$) or a sphere $S^2$ with a non-empty $P$, if a homeomorphism $h$ relative to $P$ (and $\partial D^2$ for $D^2$ case) is connected to identity via a homotopy $H$ relative to $P$ (and $\partial D^2$ for $D^2$ case), can we improve such homotopy $H$ to an isotopy $I$?**

For example, if there is only one marked point, the answer to such question is **"Yes"** by Alexander, see CURVES ON 2-MANIFOLDS AND ISOTOPIES of Epstein, Section 5.

In generally, we want to study the homotopy classes and the isotopy classes of $P$-relative homeomorphisms for any two dimensional surfaces $S$ (may contain some boundary $\partial S\subset S$ and we order homeomorphisms on boundary are identity, or some puncture points $q\not \in S$). So the question above can be restated as

**When the natural corresponding map $T$ from an isotopy class of homeomorphism of $S$ to a homotopy class of homeomorphism of $S$ $$T:[Homeo(S)]_{isotopy}\to[Homeo(S)]_{homotopy}$$
is an injection or an isomorphism (one-one)?**