Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 7 3 2026 07 01 Homology groups and decomposition of the game complex 283 291 5506 10.22060/ajmc.2024.20822.1216 EN Neda Shojaee Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran Aidin Fallah Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran Journal Article 2024 02 17 In this paper, we introduce a novel simplicial complex named Game Complex for finite non-cooperative games in the strategic form. We prove that the number of Nash equilibrium in non-cooperative games with more than two players is the rank of the first homology group of the game complex. Furthermore, we give a decomposition of the game complex. https://ajmc.aut.ac.ir/article_5506_99b410aa504a6f67da128d333896ecd4.pdf