A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres

We determine the $RO(C_2)$-graded Hurewicz images of the $C_2$-equivariant Eilenberg–MacLane spectra $H\underset{‾}{{\mathbb{F}}_2}$, $H\underset{‾}{\mathbb{Z}}$ and $H\underset{‾}{A}$, where $\underset{‾}{{\mathbb{F}}_2}$ and $\underset{‾}{\mathbb{Z}}$ denote the constant Mackey functors with values in ${\mathbb{F}}_2$ and $\mathbb{Z}$, respectively, and $\underset{‾}{A}$ denotes the Burnside Mackey functor. Surprisingly, the answer is closely tied to the problem of vector fields on spheres: the element $\frac{\theta }{{\rho }^k{\tau }^n}$ in the negative cone of the homotopy groups of $H\underset{‾}{{\mathbb{F}}_2}$ lies in the Hurewicz image if and only if $S^n$ admits $k$ linearly independent vector fields. Moreover, using the Generalized Leibniz Rule and the Generalized Mahowald Trick introduced by arXiv:2412.10879, we show that there are nonzero Adams differentials of arbitrary length supported by filtration-$0$ elements in the genuine $C_2$-equivariant Adams spectral sequence.