Instanton L-spaces and splicing

Abstract

We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton L-spaces, by a map which identifies meridians with Seifert longitudes, cannot be an instanton L-space. This recovers the recent theorem of Lidman, Pinzon-Caicedo, and Zentner that the fundamental group of every closed, oriented, toroidal 3-manifold admits a nontrivial SU(2)-representation, and consequently Zentner's earlier result that the fundamental group of every closed, oriented 3-manifold besides the 3-sphere admits a nontrivial SL(2,C)-representation.

18 pages, 8 figures; v2: accepted version

References in corpus (2)

• Instanton Floer homology of almost-rational plumbings
• Toroidal homology spheres and SU(2)-representations

Cited by in corpus (6)

• Small Dehn surgery and SU(2)
• SU(2) representations and a large surgery formula
• Taut foliations, braid positivity, and unknot detection
• Rational homology 3-spheres and SL(2,$\mathbb{C}$) representations
• Small Heegaard genus and SU(2)
• Framed instanton homology and concordance, II