Question

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy

1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.

2) $ a\bar{b}+c\bar{d}=0 $

There is a (component-wise) $S^1$ action on $C$ and let $S$ be the quotient ($S$ is a 3-manifold).

Is $S$ orientable or not ?

Thanks.

Answer 1

Identifying your points with matrices $M$ with column vectors $(a,b)^T$ and $(c,d)^T$, your equations come from the components of $M M^\dagger=I$ where $M^\dagger$ denotes the conjugate transpose. So $C$ is $U(2)$ and $S$ is $SU(2)$. Since $SU(2)$ is diffeomorphic to $S^3$, it is orientable.