2-torsion in instanton Floer homology

Abstract

This paper studies the existence of 2-torsion in instanton Floer homology with $Z$ coefficients for closed 3-manifolds and singular knots. First, we show that the non-existence of 2-torsion in the framed instanton Floer homology $I^{♯}\left(S_n^3\right(K);Z)$ of any nonzero integral n-surgery along a knot K in $S^3$ would imply that K is fibered. Also, we show that $I^{♯}\left(S_r^3\right(K);Z)$ for any nontrivial K with $r=1,1/2,1/4$ always has 2-torsion. These two results indicate that the existence of 2-torsion is expected to be a generic phenomenon for Dehn surgeries along knots. Second, we show that for genus-one knots with nontrivial Alexander polynomials and for unknotting-number-one knots, the unreduced singular instanton knot homology $I^{♯}(S^3,K;Z)$ always has 2-torsion. Finally, some crucial lemmas that help us demonstrate the existence of 2-torsion are motivated by analogous results in Heegaard Floer theory, which may be of independent interest. In particular, we show that, for a knot K in $S^3$, if there is a nonzero rational number r such that the dual knot ${\tilde{K}}_r$ inside $S_r^3\left(K\right)$ is Floer simple, then $S_r^3\left(K\right)$ must be an L-space and K must be an L-space knot.