I have a presumably basic question concerning spin structures that has me a bit confused.
Let $C$ be a circle embedded in a spun manifold $X^n$. Given a choice of trivialisation of the normal bundle of $C$ in $X$ together with an orientation of $C$, the spin structure on $X$ induces a spin structure on $C$ which is then either the trivial element or the nontrivial in $\Omega_1^{Spin}$.
Question. Does the resulting element of $\Omega_1^{Spin}$ depend on the trivialisation of the normal bundle or the orientation of $C$?
(I am mainly interested in the case where $n \geq 4$ in case there is any funny stuff going on for n=2,3.)
