A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference that essentially states this as a result? It has been difficult to locate precise definitions. In addition, a sketch of how this works would be very helpful.
$\begingroup$
$\endgroup$
3
-
7$\begingroup$ The canonical reference is Taubes' "Casson's invariant and gauge theory". $\endgroup$– Chris GerigCommented May 24, 2019 at 3:44
-
2$\begingroup$ Hmmm, @ChrisGerig, I would have thought that Floer's paper the the canonical reference ;) projecteuclid.org/euclid.cmp/1104161987 $\endgroup$– Ian AgolCommented May 24, 2019 at 4:21
-
3$\begingroup$ @IanAgol Maybe canonical "up to a sign". Their papers refer to each other, but Floer only defines his homology, of which Taubes gives the relation to Casson. $\endgroup$– Chris GerigCommented May 24, 2019 at 4:35
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Just to finalize comments since people are upvoting the question: The canonical reference is Taubes' "Casson's invariant and gauge theory" which makes the statement rigorous and has all the definitions (based on the relevant Chern-Simons functional). Floer's paper "An instanton-invariant for 3-manifolds" does build the Floer homology using this functional, and asserts that its Euler characteristic is (twice) the Casson invariant, but he only quotes Taubes' paper (that's what the reference [3] in Floer's paper really is).
Although I think Taubes' paper is beautiful enough that a sketch isn't needed, you can find a background sketch/summary in Saveliev's book ("Invariants of homology 3-spheres"), Chapter 5: "Casson Invariant and Gauge Theory".
answered May 24, 2019 at 14:45