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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 $\Sigma_g\times S^1$?

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    $\begingroup$ As far as I know the instanton Floer homology is well defined only for $3$-manifolds $M$ with $b_1(M)\leq 1$ and one has to be careful about the symmetry groups $SO(3)$ vs $SU(2)$ $\endgroup$ – Liviu Nicolaescu Apr 21 '19 at 12:35

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