In "Local Invariants of Mappings of Oriented Surfaces Into 3-Space", V.Goryunov classified singular maps of surface into 3-space and considered their resolution and local invariants.
It is natural for thinking about cohomology of the generic maps, using filtration of compactification singular sets, $ \Sigma_1^{*} \subset \Sigma_2^{*} \cdots$ by calculating spectral sequence and using Alexander duality. Are there any results for this problem?