Euler's theorem for regular CW-complexes (Q6548024)

As it is well-known, the celebrated Euler's theorem for connected multigraphs states the equivalence of even-degree vertices, decomposition into edge-disjoint cycles, and existence of a closed trail using all edges, see page 64 of \textit{F. Harary} [Graph Theory. Addison-Wesley Publishing Company (1969; Zbl 0182.57702)].\par The present paper follows the work og \textit{B. Grünbaum} [Graphs, complexes, and polytopes. Recent Prog. Comb., Proc. 3rd Waterloo Conf. 1968, 85-90 (1969; Zbl 0197.49903)], in order to give a higher-dimensional topological analogue of cycle decomposition and closed Euler trail. The same task has already been realized in dimension two by the same authors in [Math. Mag. 97, No. 1, 23-35 (2024; Zbl 1547.57051)], via the notions of \textit{circlet} and \textit{Euler cover}. In general dimension, they work in the setting of pseudomanifolds, making use in particular of the property of ``sphericity of codimension-2 intervals in the face poset of regular CW-complexes''.\par The main theorem of the paper states that, for each pure \(s\)-connected \(n\)-complex, the following are equivalent:\par \(K\) is even;\par \(K\) is a faced-disjoint union of circlets;\par there exists an \(n\)-pseudomanifold \(M\) and an Euler cover \(\phi: M \to K\).