In this talk, we will show how the Goldman bracket (see
previous talk for definition, etc.) helps to solve the rigidity problem of compact orientable connected bordered surfaces.
As discussed earlier, every closed, orientable, connected surface is topologically rigid.
But in the case of compact, orientable, bordered surfaces, this is not true;
for example, the one-holed torus and three-holed sphere are both homotopy equivalent to figure eight but are not homeomorphic (as the number of boundary components of these two surfaces is different).
That means we need to impose some extra conditions on the homotopy equivalence so that it can preserve the number of boundary components,
and one way to achieve this is to assume that the homotopy equivalence preserves the Goldman bracket.
More precisely, we prove the following:
A homotopy equivalence $f \colon \Sigma'\to \Sigma$ between compact, connected,
oriented bordered surfaces is homotopic to a homeomorphism if and only if it commutes with the Goldman bracket,
i.e., for all $x,y\in \Bbb Z\big[\widehat\pi(\Sigma')\big]$, we have,
$$\big[f_*(x), f_*(y)\big] = f_*\big([x, y]\big),$$
where $f_*\colon \Bbb Z\big[\widehat\pi(\Sigma')\big]\to \Bbb Z\big[\widehat\pi(\Sigma)\big]$ is the induced function given by
$$f_*\left(\sum_\textup{finite} n_i\cdot \textup{cls}(\alpha_i)\right)=\sum_\textup{finite} n_i\cdot \textup{cls}(f\alpha_i).$$
Reference