We prove that the mapping class group of a closed surface $S$ does not
lift to the diffeomorphism group $\Diff(S)$ of $S$ if the genus of $S$ is greater
than or equal to $3$. The proof makes uses of the Thurston stability theorem and
a fixed point theorem for certain subgroups of $\Homeo(D^2)$. If time permits, I
will also discuss a semi-conjugacy result for homeomorphisms in a pseudo-Anosov
mapping class.

Lecture #12582

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