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...

We first construct closed spherical CR manifolds of dimension at least five having non-trivial Chern class with real coefficients. next prove a constraint on classes coefficients (not necessarily closed) manifolds. Finally, we obtain topological obstruction to the existence structures co-oriented contact

This note is about spherical classes in H∗Q0S 0. A conjecture, due to Ed. Curtis, predicts that only Hopf invariant one and Kervaire invariant one elements will give rise to spherical classes in H∗Q0S 0. Yet, there has been no proof of this conjecture around. Assuming that this conjecture fails, there must exist some other spherical classes in H∗Q0S 0. This note determines the form of these pot...