Let's say I have a closed Riemannian manifold $m$ that admits $p$ spin structures each with a section of the associated spinor bundle $\psi_{p}$.
Let us suppose our manifold of interest $M$ is the connected sum of $k$ manifolds described above $M=\#_{k} m_{k}$
For example, this might be a prime decomposition of a closed three-manifold.
Question:
How can we decompose the spin structure (or, more to the point, section of the associated spinor bundle) on $M$ to reflect the components of it's prime decomposition?
Say for example we have $m=S^{1}\times S^{2}$ which admits two spin structures $P_{1}$ and $P_{2}$. For a connected sum $M=\#_{k}\left(S^{1}\times S^{2}\right)$ The set of spin structures on $M$ will be in bijection with the free product of the set of spin structures on each of the $m$. (see for example here )
I would expect (intuitively) then that the section $\Psi(M)$ of a particular spin structure on M to be of the form
$\Psi(M)=\psi(m_{1})\psi(m_{2})....\psi(m_{k-1})\psi(m_{k})$
Where each $\psi\left(m_{k}\right)$ is a section of one of the two spin structures on the kth $S^{1}\times S^{2}$. How would I go about showing such a result is true (if it is) in general?