Hi All, This is a follow-up to another post. Question is: Is the restriction of an isotopy that is the identity on the boundary (working with MCG) an isotopy? First, let me try to answer the case I am most interested in: Isotopies of the (closed) n-disk D^n, and the restriction to its interior, the open disk, say D_^n: By Alexander's trick http://en.wikipedia.org/wiki/Alexander's_trick, there is only one isotopy class for D^n . I think this restricts to D_^n: Now, working in this class , specially since , in an isotopy, the boundary is sent onto the boundary in each embedding in the path, (so that the interior is sent to the interior )I think this map restricts to an isotopy in the interior . Is this right ? Related question :any two contractible subspaces of the same space are homotopic (homotopy is an equiv. rel. ). Are they also isotopic?