The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with b₂ > 0 has b₂ curves. By the results of [Ka1]-[Ka3], [Na1]-[Na3], [DOT], [OT] this conjecture (if true) would solve the classification problem completely. We explain a new approach...

For G = SU(n), Sp(n) or Spin(n), let $C_G\left(SU\left(2\right)\right)$ be the centralizer of a certain SU(2) in G. We have a natural map $J:G/C_G\left(SU\left(2\right)\right)\to \Omega ₀³G$. For a generator α of $H⁎(G/C_G\left(SU\left(2\right)\right);\mathbb{Z}/2)$, we describe J⁎(α). In particular, it is proved that $J⁎:H⁎(G/C_G\left(SU\left(2\right)\right);\mathbb{Z}/2)\to H⁎(\Omega ₀³G;\mathbb{Z}/2)$ is injective.