János Kollár : Cremona transformations and homeomorphisms of topological surfaces.

Posted by Roman Shvydkoy , part of the Departmental Colloquium.

At

Oct. 3, 2008, 3 p.m.

In

SEO 636

Abstract

The simplest Cremona transformation of projective 3-space is the involution $\sigma:(x_0:x_1:x_2:x_3)\mapsto \left(\frac1{x_0}:\frac1{x_1}:\frac1{x_2}:\frac1{x_3}\right),$ which is a homeomorphism outside the "coordinate tetrahedron" $(x_0x_1x_2x_3=0)$. By studying the action of $\sigma$ on real quadric surfaces, we show that $\sigma$ and its conjugates generate a dense subgroup of $Homeo(S^2)$, the group of homeomorphisms of the 2-sphere. Then we show that the same holds if the 2-sphere is replaced by the torus or by any non-orientable surface and explain why there can not be similar results for orientable surfaces of genus $\geq 2$. (Joint work with Frédéric Mangolte.)