2-torsion in instanton Floer homology

Abstract

This paper studies the existence of $2$-torsion in instanton Floer homology with $\mathbb{Z}$ coefficients for closed $3$-manifolds and singular knots. First, we show that the non-existence of $2$-torsion in the framed instanton Floer homology $I^{♯}(S_n^3(K);\mathbb{Z})$ of any nonzero integral $n$-surgery along a knot $K$ in $S^3$ would imply that $K$ is fibered. Also, we show that $I^{♯}(S_r^3(K);\mathbb{Z})$ 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 $I^{♯}(S^3,K;\mathbb{Z})$ 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 $S_r^3(K)$ is Floer simple, then $S_r^3(K)$ must be an L-space and $K$ must be an L-space knot.

arXiv commentv2, 41 pages, 17 figures: published version in Adv. Math. with some further fypos fixed; comments are welcome

References in corpus (20)

• Knot Floer homology and the four-ball genus
• Knot Floer homology detects fibred knots
• Bordered Heegaard Floer homology: Invariance and pairing
• On the Khovanov and knot Floer homologies of quasi-alternating links
• Knot homology groups from instantons
• Floer Simple Manifolds and L-Space Intervals
• Link cobordisms and functoriality in link Floer homology
• Moving basepoints and the induced automorphisms of link Floer homology
• Instantons and odd Khovanov homology
• Instanton Floer homology and the Alexander polynomial
• Instantons and L-space surgeries
• Quasi-stabilization and basepoint moving maps in link Floer homology
• Sutured Floer homology and invariants of Legendrian and transverse knots
• Naturality in sutured monopole and instanton homology
• Instanton Floer homology, sutures, and Heegaard diagrams
• Cabling in terms of immersed curves
• Stein fillings and SU(2) representations
• Contact structures, excisions, and sutured monopole Floer homology
• Gluing maps and cobordism maps for sutured monopole Floer homology
• Instanton Floer homology, sutures, and Euler characteristics