Joshua Lehman

Joshua Lehman

Research

• Flexible surfaces in $\mathbb{C}P^2$ and $S^2\times S^2$. [pdf, arXiv, abstract]
  A surface $\Sigma$ in a 4-manifold $M$ is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism $(M,\Sigma) \to (M,\Sigma)$. We construct flexible surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$ within any prescribed non-characteristic homology class. Within characteristic homology classes there is a spin structure obstructing flexibility and we construct so-called spin-flexible representatives.
• Flexible algebraic curves and fake vanishing cycles. [pdf, arXiv, abstract]
  We prove that complex curves in sufficiently ample linear systems on simply connected projective surfaces are topologically (spin) flexible: any (spin) mapping class of the underlying real surface extends to a topological mapping class of the ambient 4-manifold. As a corollary, we construct fake vanishing cycles, that is, simple closed curves on the underlying real surface that are not vanishing cycles, yet still bound a topologically embedded $(\pm1)$-framed disk in the surface exterior. We also give examples of pushed in Seifert surfaces in $D^4$ that are not topologically spin-flexible, but become spin-flexible after a single internal stabilization. Our methods utilize the equivariant intersection form on the surface exterior.

Links

• Notre Dame Topology Seminar
• What's that in TeX?