Multiple point of self-transverse immesions of certain manifolds

In this paper we will determine the multiple point manifolds of certain self-transverse immersions in Euclidean spaces. Following the triple points, these immersions have a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to Dold manifold $V^5$ or a boundary. We will show there is an immersion of $S^7times P^2$ in $mathbb{R}^{13}$ with double point manifold cobordant to Dold manifold $V^5$, and an immersion of $P^2times P^2times P^2times P^2times P^2$ in $mathbb{R}^{15}$ with double point manifold a boundary and the triple point set is odd number. These will be done by introducing the product technique and reading off the Stiefel-Whitney numbers of the self-intersection manifolds.