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 restate Arrow's Impossibility Theorem in terms of the orientability of a surface as well.

翻译：偏好循环在决策问题中普遍存在，并且在偏好被假定为可传递的情况下具有矛盾性。这一矛盾构成了孔多塞悖论的基础——社会选择理论的一项开创性成果，其中直观且看似合理的决策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论。然而，拓扑社会选择理论中缺乏对偏好循环的特征刻画。在本文中，我们通过引入一个拓扑建模偏好循环的框架来弥补这一空白，该框架推广了Baryshnikov关于3个选项上严格序数偏好的现有拓扑模型。在我们的框架中，孔多塞悖论所蕴含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根据曲面的可定向性重新表述阿罗不可能定理。