# computing $\pi_1 S^1$ from a spectral sequence

In most of my calculations that I have done based off of mosher and tangora, calculations have proceeded by knowing for example that the fiber of some fibration is say a $1-sphere$. From this we are able to deduce more complicated information.

My topology professor once joked that kids in his class would need a spectral sequence to compute $\pi_1 S^1$. How could this be done?

• Why do I have 2 votes on this question that did not require any research effort, but I have 0 votes on all my other questions that I teared my hair out over? – Hari Rau-Murthy Apr 16 '16 at 2:40
• Using the fact that $S^1$ is a topological group (hence its $\pi_1$ is automatically abelian) plus Hurewicz and universal coefficients, you can reduce it to computing $H^1(S^1; \mathbb{Z})$. There's an example in Bott & Tu's section about the Leray construction (page 180 in my edition) where this calculation is done with a spectral sequence for $\mathbb{R}$ coefficients. I think the same argument works, though now you have to worry about extension problems; it seems easy though because in the diagonal of $E_2$ corresponding to $H^1$ there's only one non-zero term ($\mathbb{Z}$). – Pedro Apr 16 '16 at 16:58