A 2-Person Game Decomposing 2-Manifolds

Two players play a game by alternately splitting a surface of a compact 2-manifold along a simple closed curve that is not null-homotopic and attaching disks to the resulting boundary; the last player who can move wins. Starting from an orientable surface, the G-series is 012̇0̇ according to increasing genus. Starting from a nonorientable surface, the G-series is 0124̇603̇ according to increasing genus. Nim addition determines the G-values of the remaining compact 2-manifolds.