Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces

Abstract

A seamless parametrization of a closed oriented surface carries a discrete invariant, its holonomy signature: the cone angles, all multiples of $\pi/2$, together with the rotational holonomy $\rho\colon H_1(M\setminus C)\to\mathbb{Z}_4$ of the induced cross field. This is the datum a quadrangulation prescribes, and it decides whether any parametrization exists at all. Shen, Zhu, Capouellez, Panozzo, Campen and Zorin asked which signatures occur and gave a sufficient condition of gcd type; which signatures are realizable has remained open. We answer the question. A Reduction Lemma shows that the mapping class group acts on signatures with fixed cone angles with orbits classified by the subgroup $\mathrm{im}\,\rho\le\mathbb{Z}_4$ alone, so at most three cases survive per angle multiset instead of $4^{2g}$. A dictionary then identifies seamless parametrizations with meromorphic 4-differentials, under which $\mathrm{im}\,\rho$ measures primitivity, and realizability becomes non-emptiness of a stratum of primitive $k$-differentials with $k=4/d$ and $\mathrm{im}\,\rho=\langle d\rangle$. Unwinding this against the known classification of such strata leaves exactly five exceptional families; every other admissible signature is realizable, in every genus. Two of the five appear to be new, and both live in genus two. Four of the five lie outside the gcd condition, and the whole region it leaves open is settled here. The non-emptiness half is made constructive by an explicit one-vertex square-tiled surface in every genus together with a local surgery that splits one cone into two of prescribed angles, leaving the genus, the other cones and $\mathrm{im}\,\rho$ untouched. Two extensions follow: surfaces with boundary, the feature-aligned setting, and the relation to the Abel-Jacobi criterion at a fixed conformal structure.