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A hypergraph model for the cyclic BV operad and its applications

Sergei Merkulov

Abstract

A dg cyclic operad $BVHgraphs$ of hypergraphs is introduced which comes equipped with an explicit quasi-isomorphism $BV\rightarrow BVHgraphs$ from the { cyclic} operad $BV$ of Batalin-Vilkovisky algebras. A proof that the cohomology of $BVHgraphs$ equals $BV$ occupies most of this paper. We use this model to construct an explicit quasi-isomorphism $Chains(FFM_2) \rightarrow BVHgraphs$ from the chain operad of the cyclic operad $FFM_2$ of the compactified moduli spaces of genus zero curves with marked framed points to the dg cyclic operad $BVHgraphs$ which, combined with the main result mentioned above, gives a new proof of the cyclic formality of $FFM_2$.

A hypergraph model for the cyclic BV operad and its applications

Abstract

A dg cyclic operad of hypergraphs is introduced which comes equipped with an explicit quasi-isomorphism from the { cyclic} operad of Batalin-Vilkovisky algebras. A proof that the cohomology of equals occupies most of this paper. We use this model to construct an explicit quasi-isomorphism from the chain operad of the cyclic operad of the compactified moduli spaces of genus zero curves with marked framed points to the dg cyclic operad which, combined with the main result mentioned above, gives a new proof of the cyclic formality of .
Paper Structure (82 sections, 300 equations)

This paper contains 82 sections, 300 equations.

Table of Contents

  1. Introduction
  2. Operads of (framed) little disks
  3. An outline of main results
  4. Why hypergraphs?
  5. Some notation
  6. Cyclic operads: definitions and examples
  7. Cyclic (co)operads
  8. Derivations of cyclic operads
  9. Example: a cyclic operad of commutative algebras
  10. Example: cyclic homology operad $H_\bullet(S^1)$
  11. Example: a cyclic operad of $BV$-algebras
  12. Example: a topological cyclic operad $\mathsf{FFM}_2$
  13. A cyclic operad of homotopy Lie algebras $\mathcal{H} \mathit{olie}^{\mathfrak c}_{d}$ for $d$ odd
  14. Example: a cyclic operad of graphs $\mathcal{G} ra^{\mathfrak{c}}_d$
  15. Example: a cyclic operad of infinitesimal ribbon braids
  16. ...and 67 more sections

Theorems & Definitions (13)

  • proof
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  • ...and 3 more