3

I want to understand and try to give a proof of the following claim:

Let $B$ be a compact, connected topological $2$-manifold (surface) with nonempty boundary, then the $S^1$-bundles over $B$ with structure group $O(2)$ are in 1-1 correspondence with $\mathrm{Hom}(\pi_1(B), \mathbb{Z}_2)$.

I am familiar with the basic properties of bundles but not very familiar with how to classify them. It would be extremely helpful if you could help me prove this step by step so I can understand what is the idea here, so I am not completely lost when another similar thing comes up.

Sak
  • 3,866
  • 2
    Step 1: prove that any such bundle has a section. Step 2: argue if the bundle is orientable it is trivial. Step 3: The homomorphism to $\mathbb Z_2$ is the orientability obstruction for the bundle. Perhaps as a baby case, use this kind of argument to classify circle bundles over the circle. – Ryan Budney Jan 08 '14 at 20:25
  • Thank you for your response @Ryan, I'm trying to follow the steps and I thought of something. I remember something like this: Are all bundles $P\rightarrow B$ with fibre $F$ and structure group $G$ twisted products $P\times_G F$? – Sak Jan 09 '14 at 01:18

0 Answers0