Triviality of the 𝐽₄-equivalence among homology 3-spheres

Abstract

We prove that all homology 3-spheres are J 4 J_4 -equivalent, i.e. that any homology 3-sphere can be obtained from one another by twisting one of its Heegaard splittings by an element of the mapping class group acting trivially on the fourth nilpotent quotient of the fundamental group of the gluing surface. We do so by exhibiting an element of J 4 J_4 , the fourth term of the Johnson filtration of the mapping class group, on which (the core of) the Casson invariant takes the value 1 1 . In particular, this provides an explicit example of an element of J 4 J_4 that is not a commutator of length 2 2 in the Torelli group.

Surgery equivalence relations for 3-manifolds

“…In a very recent paper [18], Faes proved the next step for S. But, in contrast with Pitsch's proof of Theorem 3.12, his arguments require the classification of the Y k -equivalence on S for k ∈ {2, 3, 4}, which was obtained by Habiro [30].…”

“…Course n o I-Surgery equivalence relations for 3-manifolds Theorem 3.13 (Faes 2022). Any homology 3-sphere is J 4 -equivalent to S 3 .…”

“…In genus g = 0, Theorem 3.12 is thus recovered with a completely different proof than [87]. Besides, the same strategy of proof (i.e., use Y k to understand J k ) is used in [18] for proving Theorem 3.13. Remark 3.22.…”

“…This seems to be currently out of reach, as revealed already by the case of homology 3-spheres. Indeed, the methods for proving the triviality of the J 3 -equivalence (resp., J 4 -equivalence) in [87] (resp., in [18]) seem to be hard to adapt to arbitrary high degrees.…”