Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of Social Choice Theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory preference cycles. Topological methods have since broadened Social Choice Theory and elucidated existing results. However, characterisations of preference cycles in Topological Social Choice Theory are lacking. In this paper, we address this gap by introducing a framework for topologically modelling preference cycles that generalises Baryshnikov's existing topological model of strict, ordinal preferences on 3 alternatives. In our framework, the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein Bottle or Real Projective Plane, depending on how preference cycles are represented. These findings allow us to reduce Arrow's Impossibility Theorem to a statement about the orientability of a surface. Furthermore, these results contribute to existing wide-ranging interest in the relationship between non-orientability, impossibility phenomena in Economics, and logical paradoxes more broadly.

翻译：偏好循环在决策问题中普遍存在，当偏好被假定为可传递时，这些循环是矛盾的。这一矛盾构成了孔多塞悖论的基础，该悖论是社会选择理论的开创性成果，其中对决策过程直观且看似合理的约束必然导致矛盾的偏好循环。拓扑方法此后拓展了社会选择理论并阐明了现有结果。然而，拓扑社会选择理论中缺乏对偏好循环的表征。在本文中，我们通过引入一个拓扑建模偏好循环的框架来填补这一空白，该框架推广了Baryshnikov现有的关于3种备选方案的严格序数偏好拓扑模型。在我们的框架中，孔多塞悖论背后的矛盾在拓扑上对应于一个曲面的不可定向性，该曲面同胚于克莱因瓶或实射影平面，具体取决于偏好循环的表示方式。这些发现使我们能够将阿罗不可能性定理简化为一个关于曲面可定向性的陈述。此外，这些结果有助于深化当前关于不可定向性、经济学中的不可能性现象以及更广泛逻辑悖论之间关系的广泛研究兴趣。