Consider the following question:
Let $K\subset S^{3}$ be a nontrivial knot, and let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ vanishes?


The answer seems to be no. Indeed by Gordon-Luecke's theorem, $Y$ is not $S^{3}$, while as far as I know, people don't have any example of an integer homology 3-sphere other than $S^{3}$ having vanishing instanton Floer homology.


My question is: is there a rigorous proof of this statement? (for example, using instanton suture Floer homology?)



Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?
Kronheimer and Mrowka first proved that 1-surgery on a knot has non-trivial instanton Floer homology using Witten's conjecture (see their paper <http://arxiv.org/abs/math/0311489>). They later proved this using sutured instanton Floer homology as you suggested (see Section 7 of <http://arxiv.org/abs/0807.4891>).


As far as I know, for $n$ at least 2, this is unknown in general. K-M's first proof of Property P uses a long exact sequence in instanton Floer homology relating the instanton Floer homologies of $S^3$, $S^3\_1(K)$, and $S^3\_0(K)$. Some total speculation would be to try to construct a long exact sequence relating the instanton Floer homologies of $S^3$, $S^3\_{1/n}(K)$, and $S^3\_0(K)$ with some sort of twisted coefficients (analogous to Theorem 9.14 in <http://arxiv.org/abs/math/0105202>) and try to repeat their argument.


