Filtration of cohomology via symmetric semisimplicial spaces

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 $m$.