5
$\begingroup$

When Floer defined instanton Floer homology for an integer homology sphere. He chose a finite set of loops in the manifold and consider the holonomy along these loops as a perturbation on Chern-Simons functional. He proved we can choose the perturbation such that all irreducibles critical points of the perturbed CS functional are "non-degenerate" .

But his argument used the fact that the reducible flat connection over an integer homology sphere is isolated with the irreducible flat connections. So all irreducible flat connections modulo gauge transformation form a compact set.

For rational homology sphere, this may not be true. Can we still do holonomy perturbation to get this transverslity for irreducible critical points? How to control things near the reducible connections? (Of course, we can't define an Floer homology in this way.)

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ 1. For some Seifert-fibered rational homology spheres, irreducible flat connections do not form a closed set. 2. You should take a look at Walker's and Lescop's books on Casson invariant to see how they handled this issue. 3. There is a trick due to Kronheimer and Mrowka (I think) which allows one to work with the space $R'$ of $SU(2)$-representations of a central extension of the surface group instead of representations of the surface group. This trick eliminates the reducible connections issue since $R'$ is smooth, so you can do intersection theory there. $\endgroup$
    – Misha
    Commented Mar 19, 2013 at 19:09

0

Reset to default

You must log in to answer this question.