Double point self-transverse immersions of

A self-transverse immersion of a smooth manifold M 8 k in R 16 k − 4 has a double point self-intersection set which is the image of an immersion of a smooth four-dimensional manifold, cobordent to P 4 , P 2 × P 2 , P 4 + P 2 × P 2 or a boundary. We will prove that for any value of k > 1 the double point self-intersection set is a boundary. If k = 1 , then there exists an immersion of P 2 × P 2 × P 2 × P 2 in R 12 with double point manifold boundary and odd number of triple points. In particular any immersion of oriented manifold in this dimension has double point manifold cobordant to a boundary.