2-torsion in instanton Floer homology

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⁽Sₙ³(K);Z) of any nonzero integral n-surgery along a knot K in S³ would imply that K is fibered. Also, we show that I⁽Sᵣ³(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⁽S³,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 S³, if there is a nonzero rational number r such that the dual knot Kᵣ inside S³ᵣ(K) is Floer simple, then S³ᵣ(K) must be an L-space and K must be an L-space knot.