My question focuses on the relationship of the Chern-Simons theories about a Seifert fibration and the trivial product space. $ Sigma_g times S ^ 1 $, and its implication for the instantaneous homology of Floer. Here, $ Sigma_g $ It is a Riemann gender surface. $ g $.

Leave $ M_ {g, p} $ be the Seifert variety that presents itself as a degree $ -P $, $ U (1) $

package on a Riemann surface of the genus $ g $.

In this Thompson article, it is explained that the Chern-Simons theory for $ M_ {g, p} $ Y $ Sigma_g times S ^ 1 $ They are closely related. This is essentially due to the fact that each 3-manifold has a contact structure, and for $ M_ {g, p} $ the natural contact structure, $ kappa $, is the $ U (1) $ connection in the $ U (1) $ Package used in its definition. As explained earlier in equation 2.2 of Thompson's paper, this corresponds to the obvious structure in $ 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 fundamental beam power in the space of flat connection modules in $ Sigma_g times S ^ 1 $, While for $ M_ {g, p} $ There is a simple generalization.

This brings me to my question. Since the Instanton Floer homology is defined using the Chern-Simons function, **Is there a simple relationship between the Floer instant homology of $ M_ {g, p} $ Y $ Sigma_g times S ^ 1 $?**