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

• Published in: Journal of topology Vol. 15; no. 2; pp. 587 - 619
• Main Authors: Nozaki, Yuta, Sato, Masatoshi, Suzuki, Masaaki
• Format: Journal Article
• Language: English
• Published: 01.06.2022
• ISSN: 1753-8416; 1753-8424
• DOI: 10.1112/topo.12233

Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map s:Anc→YnICg,1/Yn+1$\mathfrak {s}\colon \mathcal {A}_n^c \rightarrow Y_n\mathcal {I}\mathcal {C}_{g,1}/Y_{n+1}$ is surjective for n⩾2$n\geqslant 2$, and its kernel is closely related to the symmetry of Jacobi diagrams. We determine the kernel of s$\mathfrak {s}$ 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 YnICg,1/Yn+2$Y_n\mathcal {I}\mathcal {C}_{g,1}/Y_{n+2}$ for n⩾2$n\geqslant 2$.