# Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration).

I know that $S^3$ is simply connected. Can I use this?

Edit: Maybe I was a bit confused. The base space of the Hopf fibration is S^3. Which is simply connected. So I am done?

• glimpse at en.wikipedia.org/wiki/… – janmarqz Jun 15 '16 at 10:07
• The base space of the Hopf fibration is $S^2$. The total space is $S^3$. You can only use that $S^3$ is simply connected if you actually know that the total space is $S^3$. – user98602 Jun 15 '16 at 13:36
• @MikeMiller I'm sorry, I ment to say that S^3 is the total space. Why would it be a mistery that the total space is S^3? I can just construct the hopf fibration that way, as is done for example on wikipedia.. (?) – Timon van der Berg Jun 17 '16 at 13:12
• @TimonvanderBerg I certainly don't know the context in which your question arises. I could easily believe it possible that your course says "The Hopf bundle is the circle bundle over $S^2$ with Euler class 1. Prove that the total space is simply connected." (Of course, it seems this is not the case.) – user98602 Jun 18 '16 at 18:29
• So, you're trying to calculate $\pi_1(S^3)$? What's the difference between "simply connected" and "has trivial fundamental group"; aren't those the same thing? – arctic tern Feb 2 at 19:54