Instantons and L-space surgeries

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the Spinc decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for r a positive rational number and K a nontrivial knot in the 3-sphere, there exists an irreducible homomorphism π1(S3r(K))→SU(2) unless r≥2g(K)−1 and K is both fibered and strongly quasipositive, broadly generalizing results of Kronheimer and Mrowka. We also answer a question of theirs from 2004, proving that there is always an irreducible homomorphism from the fundamental group of 4-surgery on a nontrivial knot to SU(2). In another application, we show that a slight enhancement of the A-polynomial detects infinitely many torus knots, including the trefoil