  The fundamental, or first, band gap is of unmatched importance in the study
of photonic crystals. Here, we address precisely where this gap can be opened
in the band structure of three-dimensional photonic crystals. Although strongly
constrained by symmetry, this problem cannot be addressed directly with
conventional band-symmetry analysis due to the existence of a photonic
polarization vortex at zero frequency. We develop an approach for overcoming
the associated symmetry singularity by incorporating fictitious, auxiliary
longitudinal modes. Our strategy also enables us to extend recent developments
in symmetry-based topological analysis to the fundamental gap of
three-dimensional photonic crystals. Exploiting this, we systematically study
the topology of the minimal fundamental gaps. This reveals the existence of
topological gap-obstructions that push the fundamental gap higher than what a
conventional analysis would suggest. Our work demonstrates that topology can
play a crucial role in the opening of the fundamental photonic gap and informs
future theoretical and experimental searches for conventional and topological
band gaps in three-dimensional photonic crystals.
