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

Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map $\mathfrak{s} \colon $\mathcal{A}$n^c \to Y_n$\mathcal{I}\mathcal{C}${g,1}/Y_{n+1}$issurjectivefor$n \geq 2$, and its kernel is closely related to the symmetry of Jacobi diagrams. We determine the kernel of $\mathfrak{s}$ 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 $Y_n{\mathcal{I}\mathcal{C}}_{g,1}\mathrm{/}{Y}_{n+2}$ for $n\ge 2$.