Applications of the algebraic geometry of the Putman–Wieland conjecture

Abstract

We give two applications of our prior work toward the Putman-Wieland conjecture. First, we deduce a strengthening of a result of Marković-Tošić on virtual mapping class group actions on the homology of covers. Second, let g ⩾ 2 and let Σ g ′ ,𝑛 ′ → Σ g,𝑛 be a finite 𝐻cover of topological surfaces. We show the virtual action of the mapping class group of Σ g,𝑛+1 on an 𝐻-isotypic component of 𝐻 1 (Σ g ′ ) has nonunitary image.

Generating the homology of covers of surfaces

“…Let $$\pi \colon \widetilde{\Sigma }\rightarrow \Sigma$$ be a finite branched cover between closed oriented surfaces. The homology of $$\widetilde{\Sigma }$$ encodes subtle information about the mapping class group of $$\Sigma$$, and over the last decade has been intensely studied [5, 9–14, 16–18, 20]. Much of this is motivated by a conjecture of Putman–Wieland [20] we discuss below.…”

“…Conjecture 1.1 has been proved in a variety of cases; see, for example, [5, 11–14]. However, we know very little about when $$\operatorname{H}^{\operatorname{scc}}_1(\widetilde{\Sigma };\mathbb {Q}) = \operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$$.…”

“…It turns out that there are also nonzero $$\vec{v} \in \operatorname{H}^{\operatorname{scc}}_1(\widetilde{\Sigma };\mathbb {Q})^{\perp }$$ whose $$\operatorname{Mod}(\Sigma,\widetilde{\Sigma })$$‐orbits are infinite. Indeed, letting $$D$$ be the deck group of $$\pi \colon \widetilde{\Sigma }\rightarrow \Sigma$$, it follows from the constructions in [16] and [9] that some $$D$$‐isotypic subspace $$V$$ of $$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$$ lies in $$\operatorname{H}^{\operatorname{scc}}_1(\widetilde{\Sigma };\mathbb {Q})^{\perp }$$, and Landesman–Litt [12] proved that some nonzero $$\vec{v} \in V$$ has an infinite $$\operatorname{Mod}(\Sigma,\widetilde{\Sigma })$$‐orbit.…”