Hidden singularities and the Vasil'ev homology complex of singularity classes

In order to determine the cohomology classes dual to the degeneracy cycles of various structures on manifolds, V. Vasil'ev introduced the so-called universal cohomology complex of singularity classes. The formal reversal of arrows, that is, the passage from the cohomology complex to the homology complex, leads to new characteristic classes, called the 'classes of hidden singularities'. We describe here this Vasil'ev homology complex and compute the characteristic classes of the Lagrange and Legendre immersions determined by it.