Vanishing $H^1$ for Hurwitz spaces of fully-marked admissible covers of degree 3

We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers $H^1({\bar{\mathcal{H}}}_{\underset{‾}{d},\underset{‾}{g}}(\underset{‾}{\mu }))$ vanishes for covers of degree $d=3$ and deduce the same result for the classical Hurwitz space of simply-branched covers. In degree 4, we compute examples where $H^1({\bar{\mathcal{H}}}_{\underset{‾}{4},\underset{‾}{g}}(\underset{‾}{\mu }))$ is nonzero, which implies that $H^1({\bar{\mathcal{H}}}_{\underset{‾}{d},\underset{‾}{g}}(\underset{‾}{\mu }))$ is nonvanishing for $d\ge 4$. We describe the stratification of the boundary of ${\bar{\mathcal{H}}}_{\underset{‾}{d},\underset{‾}{g}}(\underset{‾}{\mu })$ by lower-dimensional ${\bar{\mathcal{H}}}_{\underset{‾}{d^{\mathrm{\prime }}},\underset{‾}{g^{\mathrm{\prime }}}}(\underset{‾}{{\mu }^{\mathrm{\prime }}})$, and set up an inductive framework which may be used for future arguments involving the odd cohomology of ${\bar{\mathcal{H}}}_{\underset{‾}{d},\underset{‾}{g}}(\underset{‾}{\mu })$.