A Sperner lemma complete for PPA

Abstract The class PPA characterizes search problems whose solution is guaranteed by the lemma that “every finite graph with an odd degree vertex has another”. The smaller class PPAD is defined similarly for directed graphs. While PPAD has several natural complete problems corresponding to classical existence theorems in topology, no such complete problems were known for PPA. Here we overcome the difficulty by considering non-orientable spaces: Sperner's lemma for non-orientable 3-manifolds is complete for PPA.