Welcome to MathOverflow

Question

0

Hi!

As the title says i'm wondering how can i realize a real vector bundle rank 2 on $S^2$ of euler class $k\in\mathbb{Z}$ as a submersion of $S^3\times\mathbb{R}^2$. My initial idea was to realize it as a quotient of an action of $S^1$, more precisely i consider the following action of $S^1$ on $S^3\times\mathbb{R}^2$: i consider $S^3$ as a subset of $\mathbb{C}^2$ and i identify $\mathbb{R}$ with $\mathbb{C}$ $$\rho:S^1\times S^3\times\mathbb{R}^2\rightarrow S^3\times\mathbb{R}^2 $$ $$\rho(\theta,z_1,z_2,re^{i\alpha})=(z_1e^{i\theta},z_2e^{i\theta},re^{i(\alpha-k\theta)})$$ I can't see if $S^3\times\mathbb{R}^2/S^1$ is what i'm looking for because i can't se if this object has a vector bundle structure on $S^2$. Please can anyone help?

Thank you in advance and sorry if it is a silly question.

flag	asked Nov 17 2010 at 20:09 Italo 721●4●7

1

This is a homework level question in an introductory course on vector bundles, and not really appropriate for MO. Have you considered reading something like Steenrod's book on fibre bundles, or Hatcher's vector bundle notes (available on-line)? – Ryan Budney Nov 17 2010 at 20:16

I will do it, if necessary i'll delete the question. – Italo Nov 17 2010 at 20:26

For future reference, for a question like this the math.stackexchange site (listed in the FAQ) is a more appropriate venue. – Ryan Budney Nov 17 2010 at 20:36

1

I am not going to explain why $(S^3\times\mathbb R^2)/S^1$ is a vector bundle (this is indeed a a homework problem) but computing the Euler class is less trivial. I suggest you look at the associated circle bundles. Namely, $S^2$ can be described $(S^3\times S^1)/T^2$, and replacing $T^2$ with the circle $S^1_{1,k}\subset T^2$ gives the circle bundle $(S^3\times S^1)/S^1_{1,k}\to (S^3\times S^1)/T^2$ whose "$k$th root" is the Hopf bundle with Euler number $1$, or perhaps $-1$. Now you need tos how that taking "$k$th power" is multiplicative for the Euler number. – Igor Belegradek Nov 17 2010 at 20:55

How can i realize a real rank 2 vector bundle on $S^2$ of euler class $k\in\mathbb{Z}$ as a submersion of $S^3\times\mathbb{R}^2$?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.