Hi all, as the title suggests I'd like to identify the fundamental group of the sphere bundle of the line bundle $$\eta_x \colon K(D_{2n},1)\to BO(1)$$ which is the non-trivial line bundle s.t. $w_1(\eta_x)=x$, where $x$ is a generator in $$H^*(D_{2n};\Bbb Z_2)\cong \Bbb Z_2[ x,y,w]/\langle xy=x^2\rangle, \ \ \ \ \ |x|=|y|=1, |w|=2$$ for $D_{2n}$ the dihedral group with $2n$ elements, with $n$ a multiple of $4$.
By the l.e.s. of the fibration $S^0\to S(\eta_x) \to K(D_{2n},1)$ one immediately has the s.e.s. $$ 0 \to \pi_1(S(\eta_x)) \to D_{2n}\to \Bbb Z_2 \to 0$$ so $\pi_1(S(\eta_x))$ is an index two subgroup of the dihedral group $D_{2n}$ so is either cyclic or the dihedral group (of order $n$). I tried using the fact that the sphere bundle is a normal covering of $K(D_{2n},1)$ but I don't see how this might help
I'd like to be a little more precise, because I'm dealing with a specific case, i.e. the space is the sphere bundle of a precise bundle, but I don't know how to make use of these informations.