Perron's conjecture on the mod-p extension of the Casson invariant

Let g3g\geq 3 and let p5p\geq 5 be prime. Let Mg,1[p]\mathcal{M}_{g,1}[p] be the mod-pp Torelli group, Tg,1\mathcal{T}_{g,1} the Torelli group, and Dg,1[p]D_{g,1}[p] the subgroup generated by pp-powers of Dehn twists. For Sφ3S3[p]S^3_\varphi\in\mathcal{S}^3[p] with φ=fm\varphi=f\cdot m, where fTg,1f\in\mathcal{T}_{g,1} and mDg,1[p]m\in D_{g,1}[p], define γp:Mg,1[p]Z/p\gamma_p:\mathcal{M}_{g,1}[p]\to\mathbb{Z}/p by

γp(φ)=λ(Sf3)(modp).\gamma_p(\varphi)=\lambda(S^3_f)\pmod p.

Perron's conjecture. The map γp\gamma_p is a well-defined invariant on S3[p]\mathcal{S}^3[p]. Perron proposed this as an extension of the Casson invariant to the mod-pp Torelli group. The supplied text does not state whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Ricard Riba and Wolfgang Pitsch, “Invariants of Z/p-Homology 3-Spheres from the Abelianization of the Level-p Mapping Class Group”, arXiv:2103.15519 (2023).

Additional references

2 papers in this index state this conjecture (2020–2021). The statement above is taken from the most recent of them; the others are arXiv:2002.10589.

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