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Homotopy G-algebras and moduli space operad

Murray Gerstenhaber, Alexander A. Voronov

Abstract

This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy Gerstenhaber (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex $Ω^\bullet X$ of a Kähler manifold $X$, thereby yielding the most general type of homotopy G-algebra structure on $Ω^\bullet X$.

Homotopy G-algebras and moduli space operad

Abstract

This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy Gerstenhaber (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex of a Kähler manifold , thereby yielding the most general type of homotopy G-algebra structure on .

Paper Structure

This paper contains 13 sections, 9 theorems, 36 equations.

Table of Contents

  1. Homotopy G-algebra structure on an operad
  2. The brace structure on an operad
  3. Homotopy G-algebras
  4. Applications: Hochschild cochains and singular cochains
  5. Hochschild complex
  6. G-algebras
  7. Singular cochain complex
  8. Generalities
  9. $A_\infty$ version
  10. Bar interpretation
  11. Topological and mirror reflections
  12. Moduli spaces and little squares
  13. The construction

Key Result

Proposition 1

For every operad $\mathcal{O}$ of vector spaces, the braces brace define the natural structure of a brace algebra on the underlying graded vector space $\mathcal{O}$.

Theorems & Definitions (20)

  • Conjecture $\! \! \!$: Deligne
  • Remark $\! \!$
  • Remark $\! \!$
  • Definition 1
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Remark $\! \!$
  • Theorem 3
  • proof
  • ...and 10 more