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.