Spanning bipartite quadrangulations in even triangulations of real projective spaces

Abstract

We study spanning quadrangulations of even triangulations of real projective spaces, using the higher-dimensional notion of quadrangulation introduced by Kaiser and Stehl\'ık. We prove that every even triangulation of $\RP^d$, where $d\ge 2$, admits a spanning bipartite quadrangulation. Moreover, such a triangulation admits a spanning non-bipartite quadrangulation if and only if it is not balanced; if it is balanced, every spanning quadrangulation is bipartite. Our proofs use coloring monodromy: its orbits yield bipartite quadrangulations, while a covering-space construction gives non-bipartite ones. %In dimension three, a bipartite quadrangulation can always be chosen to induce $K_{2,2}$ on each tetrahedron. More generally, we establish a sufficient condition for the existence of spanning bipartite quadrangulations on closed manifolds in terms of the monodromy action, and derive a consequence involving the fundamental group.