A Sign-Reversing Involution on Genus-One Boundary Strata

We give an elementary calculation of the Euler characteristic of the boundary complex of the moduli space of genus-one stable curves. Our result agrees with the work of Chan-Galatius-Payne, who proved that this boundary complex is homotopy equivalent to a particular wedge of spheres, but the proof technique in our case is purely combinatorial: we construct a sign-reversing involution on genus-one boundary strata modeled on a similar calculation in genus zero due to Blankers-Gillespie-Levinson.