Multiple point‎ ‎self-transverse immersions 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 show that there is an immersion of $S^7\times P^2$ in‎ ‎$\mathbb{R}^{13}$ with double point manifold cobordant to Dold‎ ‎manifold $V^5$‎, ‎and an immersion of $P^2\times P^2\times P^2\times‎ ‎P^2\times P^2$ in $\mathbb{R}^{15}$ with double point manifold a‎ ‎boundary and the triple point set with odd number‎. ‎These are obtained‎ ‎by introducing the product technique and reading off the‎ ‎Stiefel-Whitney numbers of the self-intersection manifolds‎.