Systole and $\lambda_{2g-2}$ of a hyperbolic surface

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${\lambda_{2g-2}}(S)$: ${\lambda_{2g-2}}(S) > 1/4 + {\epsilon_0}(S)$, where ${\epsilon_0}(S) > 0$ is an explicit constant which depends only on the systole of $S$