Recall that the topological rigidity of the closed orientable surface of genus $g(\geq 2)$ requires the topological rigidity of pair of pants. We will give the sketch of proof of the same thing in a more general setting, namely:

If a homotopy equivalence between two compact bordered orientable surfaces induces homeomorphism between their boundaries, then it is homotopic to a homeomorphism relative to the boundary.

One application of the topological rigidity of closed orientable surfaces is the following:

Any degree one self-map of a closed orientable surface is homotopic to a homeomorphism. To prove this, we need the Hopficity of the fundamental group of a closed orientable surface.