English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The real locus of the moduli space of stable genus-zero curves with markedpoints, $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$, is known to be asmooth manifold and is the Eilenberg-MacLane spaces for the so-called pureCactus groups. We describe the operad formed by these spaces in terms of ahomotopy quotient of an operad of associative algebras. Using this model, weidentify various Hopf models for the algebraic operad of chains and homologiesof $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$. In particular, we showthat the operad $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal.As an application of these operadic constructions, we prove that for each $n$,the cohomology ring $H^{\bullet}(\overline{{\mathcal M}_{0,{n+1}}}({\mathbbR}), {\mathbb{Q}})$ is a Koszul algebra, and that the manifold$\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal for $n\geq 6$but is a rational $K(\pi,1)$-space. Additionally, we describe the Lie algebrasassociated with the lower central series filtration of the pure Cactus groups.

Real moduli space of stable rational curves revised