We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers $H^1(\overline{ \mathcal{ H}}_{ \underline{ d},\underline{ g}}(\underline{ \mu}))$ 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 $H^1(\overline{ \mathcal{ H}}_{ \underline{ 4},\underline{ g}}(\underline{ \mu}))$ is nonzero, which implies that $H^1(\overline{ \mathcal{ H}}_{ \underline{ d},\underline{ g}}(\underline{ \mu}))$ is nonvanishing for $d \geq 4$. We describe the stratification of the boundary of $\overline{ \mathcal{ H}}_{ \underline{ d},\underline{ g}}(\underline{ \mu})$ by lower-dimensional $\overline{ \mathcal{ H}}_{ \underline{ d'},\underline{ g'}}(\underline{ \mu'})$, and set up an inductive framework which may be used for future arguments involving the odd cohomology of $\overline{ \mathcal{ H}}_{ \underline{ d},\underline{ g}}(\underline{ \mu})$.
Categories: math.AG