Double point self-transverse immersions of M8k↬R16k−4

A self-transverse immersion of a smooth manifold M8k in R16k−4 has a double point self-intersection set which is the image of an immersion of a smooth four-dimensional manifold, cobordent to P4, P2×P2, P4+P2×P2 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 P2×P2×P2×P2 in R12 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.