Hodge diamond: Hopf surfaces are quotients of C2−(0,0) by a discrete group G acting freely, and have vanishing second Betti numbers.
The simplest example is to take G to be the integers, acting as multiplication by powers of 2; the corresponding Hopf surface is diffeomorphic to S1×S3.
Those with b2=1 were classified by Nakamura (1984b) under an additional assumption that the surface has a curve, that was later proved by Teleman (2005).
The global spherical shell conjecture claims that all class VII0 surfaces with positive second Betti number have a global spherical shell.
Conversely Georges Dloussky, Karl Oeljeklaus, and Matei Toma (2003) showed that if a minimal class VII surface with positive second Betti number b2 has exactly b2 rational curves then it has a global spherical shell.