# How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration

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 $$\Sigma_g\times S^1$$?

• 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 $SU(2)$ Apr 21 '19 at 12:35