  Every homology cylinder is obtained from Jacobi diagrams by clasper surgery.
The surgery map $\mathfrak{s} \colon \mathcal{A}_n^c \to
Y_n\mathcal{IC}_{g,1}/Y_{n+1}$ is surjective for $n \geq 2$, 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 $Y_n\mathcal{IC}_{g,1}/Y_{n+2}$ for
$n \geq 2$.
