Getzler's relation by localization (Q935594)

\textit{E. Getzler} [Integrable systems and algebraic geometry. Proc. 41st Taniguchi symp. 73--106 (1998; Zbl 1021.81056)] has found a genus 2 topological recursive relation for the cohomology ring of the compactified moduli space of pointed Riemann surfaces of genus two. He also conjectured that this relation is actually a rational equivalence. In this short article, the author uses the localization method to reproduce Getzler's relation. This approach also shows that this is a rational equivalence relation. A theoretical proof was also given by \textit{T. Graber} and \textit{R. Vakil}, Duke Math. J. 130, No. 1, 1--37 (2005; Zbl 1088.14007)].