3
votes
1answer
368 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
2
votes
1answer
280 views

Cohomology rings of (some) sphere bundles over spheres

Recall that 2-dimensional complex vector bundles over $S^4$ are classified by $\pi_4(BU(2))=\pi_4(BU)=\mathbb Z$. For any integer $\lambda$ one can consider projectivisation of the corresponding ...
2
votes
1answer
246 views

Octonionic Hopf fibration and $\mathbb HP^3$

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations ...