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^ (Sₑ³ (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.