  We prove the existence of homeomorphisms of a closed, orientable surface of
genus 3 or greater that do not extend to any handlebody bounded by the surface.
We show that such homeomorphisms exist arbitrarily deep in the Johnson
filtration of the mapping class group. The second and third terms of the
Johnson filtration are the well-known Torelli group and Johnson subgroup,
respectively. Richard Hain has obtained very similar results by different
methods.
