Filtration of cohomology via symmetric semisimplicial spaces

TL;DR

This paper introduces a new approach using symmetric semisimplicial spaces to simplify the computation of cohomology for complex moduli spaces and configuration spaces, providing both known and new results.

Contribution

It develops a spectral sequence framework based on symmetric semisimplicial filtrations, simplifying cohomology calculations and extending results to new moduli spaces.

Findings

Simplified computation of singular and étale cohomology of moduli spaces

Unified proof techniques bypassing combinatorial complexities

New results on the cohomology of moduli spaces of smooth sections

Abstract

In the simplicial theory of hypercoverings, we replace the indexing category $\Delta$ by the \emph{symmetric simplicial category} $\Delta S$ and study (a class of) $\Delta S$-hypercoverings, which we call \emph{spaces admitting symmetric (semi)simplicial filtration}. For $\Delta S$-hypercoverings we construct a spectral sequence, somewhat like the \v{C}ech-to-derived category spectral sequence. The advantage of working on $\Delta S$ is that all of the combinatorial complexities that come with working on $\Delta$ are bypassed, giving simpler, unified proof of known results like the computation of (in some cases, stable) singular cohomology (with $\mathbb{Q}$ coefficients) and et al e cohomology (with $\mathbb{Q}_{\ell}$ coefficients) of the moduli space of degree $n$ maps $C\to \mathbb{P}^r$, $C$ a smooth projective curve of genus $g$, of unordered configuration spaces etc. as well as new: that of the moduli space of smooth sections of a fixed $\mathfrak{g}^r_d$ that is $m$-very ample for some $m$.