Instanton L-spaces and splicing

TL;DR

This paper proves that certain 3-manifolds formed by splicing knot complements in homology spheres cannot be instanton L-spaces, linking to recent results on the existence of nontrivial SU(2) and SL(2,C) representations of their fundamental groups.

Contribution

It establishes a new obstruction to a 3-manifold being an instanton L-space via splicing, connecting to recent theorems on fundamental group representations.

Findings

Spliced manifolds from nontrivial knots are not instanton L-spaces.

Links to the existence of nontrivial SU(2) and SL(2,C) representations.

Supports the conjecture that certain 3-manifolds admit nontrivial representations.

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.