Moduli space actions on the Hochschild co-chains of a Frobenius algebra. I: Cell operads.

*(English)*Zbl 1145.55008This is the first of two papers to prove that the cell model of the moduli space of curves with marked points and tangent vectors acts on the Hochschild cochains of a Frobenius algebra. There are several combinatorially defined objects that underlie operads and PROPs which act on chain and cochain modules. These include Cacti, Arcs, chord diagrams, little discs, etc. Interest is strong for these structures because they induce operations on homology under the right circumstances. The second part of this series promises the definitions of the actions.

In this paper, the constructions and relations between constructions are given of the various operads and PROPs, sometimes defined weakly to gain flexibility. The starring role is played by versions of the Arc operad, studied by R. M. Kaufmann, M. Livernet and R. C. Penner [Geom. Topol. 7, 511–568 (2003; Zbl 1034.18009)] which is then related to moduli spaces via a dual graph construction taking us through marked metric ribbon graphs.

In this paper, the constructions and relations between constructions are given of the various operads and PROPs, sometimes defined weakly to gain flexibility. The starring role is played by versions of the Arc operad, studied by R. M. Kaufmann, M. Livernet and R. C. Penner [Geom. Topol. 7, 511–568 (2003; Zbl 1034.18009)] which is then related to moduli spaces via a dual graph construction taking us through marked metric ribbon graphs.

Reviewer: John McCleary (Poughkeepsie)

##### MSC:

55P48 | Loop space machines and operads in algebraic topology |

18D50 | Operads (MSC2010) |

16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

17A99 | General nonassociative rings |

32G81 | Applications of deformations of analytic structures to the sciences |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

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