Classifying spaces for cobordisms of projective immersions with restricted singularities

We give an explicit construction for classifying spaces of mildly singular maps
obtained as hyperplane projections of immersions. We prove strucutre theorems for these classifying spaces.
As a byproduct we get a certain relatively explicit description of the generator of the third stable homotopy group of spheres
(which is known to be Z_{24}).