Multiple points of immersion

Given smooth manifolds Vn and Mm, an integer k > 1, and an immersion f : V ↬ M, we have constructed an obstruction for existence of a regular homotopy of f to an immersion f' : V ↬ M without k-fold self-intersection points. It takes values in certain twisted bordism group, and for ( k + 1)n + 2 < km turns out to be complete. As a byproduct, under certain dimensional restrictions, we also constructed a complete obstruction for eliminating by regular homotopy the points of common intersection of several immersions f 1 : V1 ↬ M,...,fk : Vk ↬ M.