The tensor-algebra injectivity conjecture for the weight-zero cohomology of the moduli of abelian varieties

Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$, let $\mathcal{A}=\{\mathcal{A}_g\}_{g\geq 1}$, and let $W_0H^*_c(\mathcal{A};\mathbb{R})$ be its weight-zero compactly supported cohomology, equipped with its coproduct. Let $\Omega^*_c$ be the space appearing in the canonical map, and let $T(\Omega^*_c[-1])$ denote its tensor algebra. The inclusion of $\Omega^*_c[-1]\otimes\mathbb{R}$ into the primitive elements for this coproduct induces a map

$$T(\Omega^*_c[-1])\otimes\mathbb{R}\longrightarrow W_0H^*_c(\mathcal{A};\mathbb{R}).$$

Tensor-algebra injectivity conjecture. This induced map is injective. The later sections provide various kinds of evidence for this conjecture, while the equivalent formulation in terms of the associated free Lie algebra is stated separately in the paper.