I am trying to visualize a seven sphere here: http://www.youtube.com/watch?v=II-maE5HEj0. What is $(u,v)\to(u, u^hvu^j)$ in the attached picture?
Are quaternion maps commonly used to glue boundaries of topological spaces? How do we know that this quaternion is gluing the boundaries? Is $M$ a 7-sphere?
Any answers/insights would be helpful.
In the picture, $M$ is being described as the total space of an $S^3$-bundle over $S^4$ with structure group $\rm{SO}(4)$, which is denoted by $\xi_{h,j}$. The bundle $\xi_{h,j}$ is trivialized over two hemispheres of $S^4$, each of which are diffeomorphic to the disk $D^4$. Hence to completely determine $\xi_{h,j}$, we just need to say what the transition function $\theta: \partial D^4 \longrightarrow \rm{SO}(4)$ on the equator is. Here the equator is $S^3$, which is identified as the group of unit quaternions. The transition function is then given by $$\theta(u)v = u^h v u^j,$$ where the multiplication is quaternion multiplication. Since unit quaternions form a group under quaternion multiplication, $\theta(u)$ maps $S^3$ to $S^3$.
The fact that the total space $M$ is a $7$-sphere (when $h + j = 1$) follows after constructing a Morse function on $M$ that has only two critical points (any closed, smooth manifold admitting a Morse function with only two critical points is homeomorphic to a sphere).
The construction in your picture is featured heavily in the classic paper On manifolds homeomorphic to the $7$-sphere by Milnor.