Toroidal integer homology three-spheres have irreducible \(SU(2)\)-representations. (English) Zbl 1530.57029
The article under discussion is motivated by Problem 3.105(A) of Kirby’s problem list [R. Kirby (ed.), AMS/IP Stud. Adv. Math. 2, 35–473 (1997; Zbl 0888.57014)]. This problem is equivalent to the statement that the fundamental groups of all integer homology three-spheres other than \(S^3\) admit irreducible \(\mathrm{SU}(2)\)-representations.
The authors give an affirmative answer to this problem in the case where the integer homology three-sphere contains an embedded incompressible torus. As an corollary they obtain that the fundamental group of every integer homology three-sphere other than \(S^3\) has an irreducible representation into \(\mathrm{SL}(2,\mathbb{C})\). This simplifies the original proof of R. Zentner [Duke Math. J. 167, No. 9, 1643–1712 (2018; Zbl 1407.57008)].
The authors give an affirmative answer to this problem in the case where the integer homology three-sphere contains an embedded incompressible torus. As an corollary they obtain that the fundamental group of every integer homology three-sphere other than \(S^3\) has an irreducible representation into \(\mathrm{SL}(2,\mathbb{C})\). This simplifies the original proof of R. Zentner [Duke Math. J. 167, No. 9, 1643–1712 (2018; Zbl 1407.57008)].
Reviewer: Michael Heusener (Clermont-Ferrand)
Keywords:
three-dimensional homology sphere; instanton Floer homology; representations into \(\mathrm{SU}(2)\); holonomy perturbationsReferences:
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