On the kernel of the surgery map restricted to the 1‐loop part

Abstract

Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map 𝔰 ∶  𝑐 𝑛 → 𝑌 𝑛  g,1 ∕𝑌 𝑛+1 is surjective for 𝑛 ⩾ 2, and its kernel is closely related to the symmetry of Jacobi diagrams. We determine the kernel of 𝔰 restricted to the 1-loop part after taking a certain quotient of the target. Also, we introduce refined versions of the AS and STU relations among claspers and study the abelian group 𝑌 𝑛  g,1 ∕𝑌 𝑛+2 for 𝑛 ⩾ 2. M S C ( 2 0 2 0 ) 57K16, 57K31 (primary), 57K20 (secondary) Contents

Surgery equivalence relations for 3-manifolds

“…Course n o I-Surgery equivalence relations for 3-manifolds Currently, the GHC is only known to be true up to degree d = 4, the most recent result in this direction being obtained in [82]. By comparing Lemma 3.31 to Proposition 3.32, we see that the GHC is an analogue of the following problem in group theory, which can be stated for any group G.…”

“…Thus the "Lie algebra of homology cylinders" Gr Y IC() is highly related to the "Torelli Lie algebra" Gr  I(), which has been reviewed at (2.8). We refer to the works [30,26,29,9,31,67,81,82]; see also the end of §3.5 in this connection.…”

“…But computing those universal invariants is a challenge in high degrees (despite their combinatorial construction) and, moreover, it is not known whether they dominate all finite-type invariants (including those with values in torsion abelian groups). Nevertheless, recent works of Nozaki, Sato & Suzuki provide encouraging perspectives [81,82].…”