Instanton $L$-spaces and splicing
Annales Henri Lebesgue, Volume 5 (2022), pp. 1213-1233.

### Metadata

KeywordsInstanton Floer homology, L-spaces, incompressible tori

### 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–Pinzón-Caicedo–Zentner that the fundamental group of every closed, oriented, toroidal 3-manifold admits a nontrivial $\mathit{SU}\left(2\right)$-representation, and consequently Zentner’s earlier result that the fundamental group of every closed, oriented $3$-manifold besides the 3-sphere admits a nontrivial $\mathit{SL}\left(2,ℂ\right)$-representation.

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