Framed instanton homology and Fr{\o}yshov's invariant

Abstract

We determine the framed instanton homology with coefficients in $\mathbb F = \mathbb Z/2$ for Dehn surgeries on a knot in the $3$-sphere. The dimension of these groups is seen to have a close relationship with a homology cobordism invariant due to Froyshov. As an application, we show that $r$-surgery on a non-trivial knot cannot be nondegenerate $SU(2)$-abelian for any $|r| \le 4\lceil g(K)/2\rceil$, which is $2g(K)$ for $g$ even and $2g(K) + 2$ for $g$ odd.