A 2-Person Game Decomposing 2-Manifolds

TL;DR

This paper introduces a combinatorial game played on 2-manifolds, analyzes its structure through a series of invariants called the G-series, and determines the game values for various surfaces.

Contribution

It defines a new two-player game on surfaces and computes the G-series for orientable and nonorientable cases, linking game values to topological properties.

Findings

G-series for orientable surfaces is 0120

G-series for nonorientable surfaces is 0124603

G-values are determined by Nim addition

Abstract

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 $01\dot{2}\dot{0}$ according to increasing genus. Starting from a nonorientable surface, the $G$-series is $012\dot{4}60\dot{3}$ according to increasing genus. Nim addition determines the $G$-values of the remaining compact $2$-manifolds.