We show that computation of the SPERNER problem is PPA-complete on the Möbius band under proper boundary conditions, settling a long term open problem. Further, the same computational complexity results extend to other discrete fixed points on the Möbius band, such as the Brouwer fixed point problem, the DPZP fixed point problem and a simple version of the Tucker problem, as well as the projective plane and the Klein bottle. We expect it opens up a new route for further studies on the related combinatorial structures.

中文翻译：

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了解PPA的完整性

我们证明，在适当的边界条件下，SPERNER问题的计算在莫比乌斯带上是PPA完全的，从而解决了一个长期的开放问题。此外，相同的计算复杂度结果还扩展到了Möbius波段上的其他离散不动点，例如Br​​ouwer不动点问题，DPZP不动点问题和Tucker问题的简单版本，以及射影平面和Klein瓶。我们希望它为有关组合结构的进一步研究开辟一条新途径。