2024/12/04 by Tommaso Rossi, Paolo Salvatore, Rossi, Tommaso +1 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2412.03474
Let M0,n+1 be the moduli space of genus zero stable curves with (n+1)-marked points. The collection M=\M0,n+1\n≥ 2 forms an operad in the category of complex projective varieties; its homology Hycom= H_*(M) is called the Hypercommutative operad. In this paper we construct a chain model for the hypercommutative operad, i.e. an operad of chain complexes C_*dual(M) which is weakly equivalent to the operad of singular chains C_*(M). We prove that C_*dual(M) is the linear dual of the bar construction B(grav), where grav is a chain model of the gravity operad based on cacti without basepoint. This shows that the Gravity and Hypercommutative operad are Koszul dual also at the chain level, refining a previous result of Getzler. The construction is topological, since C_*dual(M)(n) is the cellular complex associated to a regular CW-decomposition of M0,n+1.