My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. Here, $\Sigma_g$ is a Riemann surface of genus $g$.
Let $M_{g,p}$ be the Seifert manifold that is presented as a degree $−p$, $U(1)$ bundle over a Riemann surface of genus $g$. In this paper by Thompson, it is explained that Chern-Simons theory for $M_{g,p}$ and $\Sigma_g \times S^1$ are closely related. This is essentially due to the fact that every 3-manifold has a contact structure, and for $M_{g,p}$ the natural contact structure, $\kappa$, is the $U(1)$ connection on the $U(1)$ bundle used in its definition. As explained above equation 2.2 of the paper by Thompson, this corresponds to the obvious structure on $\Sigma_g\times S^1$.
This leads to similar forms for the partition functions of the two theories, as shown on page 2 of Thompson. That is, for $\Sigma_g\times S^1$, one finds the Hirzebruch-Riemann-Roch theorem for a power of the fundamental line bundle on the moduli space of flat connections on $\Sigma_g\times S^1$, while for $M_{g,p}$ one finds a simple generalization.
This brings me to my question. Since instanton Floer homology is defined using the Chern-Simons functional, is there a simple relationship between the instanton Floer homology of $M_{g,p}$ and $\Sigma_g\times S^1$?
