AI 中文总结
本文研究三角剖分闭紧曲面的2-回路分解问题，定义可定向曲面的一阶上同调障碍，证明其为零等价于存在该分解，且非可定向曲面存在该分解当且仅当其可定向2-重覆盖存在该分解。
AI 中文摘要
图的欧拉回路是恰好经过图中每条边一次的闭合路径，欧拉回路与回路分解问题也可推广到高维单纯复形中。复形K中的欧拉k-回路是顶点序列v₁…vₙ，其中每k+1个相邻项{v_i,v_{i+1},…,v_{i+k}}（下标模n）构成一个k-单形，且K的每个k-单形恰好出现在序列v₁v₂…vₙ(v₁v₂…v_k)中一次。本文研究三角剖分闭紧曲面的2-回路分解问题：对可定向三角剖分曲面，利用路径的内角定义一个障碍，该障碍属于曲面的一阶上同调，其为零当且仅当该曲面存在2-回路分解；还证明非可定向三角剖分曲面存在2-回路分解当且仅当它的可定向2-重覆盖存在2-回路分解。
英文摘要
An Euler circuit of a graph is a closed path that visits every edge of the graph exactly once. Euler circuit and circuit decomposition problems can also be formulated for higher dimensional simplicial complexes. An Euler k-circuit in K is a cyclic sequence of vertices v_1...v_n such that every k+1 adjacent terms { v_i,v_{i+1},...,v_{i+k} } (indexed modulo n) form a k-simplex, and every k-simplex of K appears exactly once in the sequence v_1v_2...v_n(v_1v_2...v_k). We investigate the 2-circuit decomposition problem for triangulated closed compact surfaces. For an orientable triangulated surface we use interior angles of paths to define an obstruction that lives in the first cohomology of the surface. It vanishes if and only if the surface has a 2-circuit decomposition. We also show that a non-orientable triangulated surface has a 2-circuit decomposition if and only if its orientable 2-fold cover does.
