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Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group

Michael Borinsky, Benjamin Brück, Thomas Willwacher

Abstract

We compute the weight 2 cohomology of the Feynman transforms of the cyclic (co)operads $\mathsf{BV}$ and $\mathsf{HyCom}$, and the top$-2$ weight cohomology of the Feynman transforms of $D\mathsf{BV}$ and $\mathsf{Grav}$. Using a result of Giansiracusa, we compute, in particular, the top$-2$ weight cohomology of the handlebody group. We compare the result to the top$-2$ weight cohomology of the moduli space of curves $\mathcal{M}_{g,n}$, recently computed by Payne and the last-named author. We also provide another proof of a recent result of Hainaut-Petersen identifying the top weight cohomology of the handlebody group with the Kontsevich graph cohomology.

Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group

Abstract

We compute the weight 2 cohomology of the Feynman transforms of the cyclic (co)operads and , and the top weight cohomology of the Feynman transforms of and . Using a result of Giansiracusa, we compute, in particular, the top weight cohomology of the handlebody group. We compare the result to the top weight cohomology of the moduli space of curves , recently computed by Payne and the last-named author. We also provide another proof of a recent result of Hainaut-Petersen identifying the top weight cohomology of the handlebody group with the Kontsevich graph cohomology.
Paper Structure (49 sections, 44 theorems, 313 equations, 6 tables)

This paper contains 49 sections, 44 theorems, 313 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Results about the Feynman transform of DBV*
  3. Results on Feyn(HyCom) and Feyn(BV)
  4. Corollaries for the handlebody groups
  5. Euler characteristics
  6. Acknowledgements
  7. Recollections and setup
  8. Basic notation
  9. A lemma for dg vector spaces
  10. Operads and cyclic operads
  11. Modular operads
  12. Vector spaces of decorated graphs
  13. Feynman transform
  14. Feynman transform for weight-graded cooperads
  15. Cyclic (co)bar construction and dg dual (co)operads
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $V_{g,n}^m$ be a genus $g$ handlebody with $m$ distinct marked disks and $n$ marked points on the boundary. Let $\mathrm{HMod}_{g,n}^m:=\pi_0 \mathrm{Diff}(V_{g,n}^m)$ be the handlebody group. Then as long as $(g,m)\neq (1,0)$, where $\mathrm{Feyn}_\mathfrak{k}(-)$ denotes the Feynman transform of 1-shifted cyclic operads. Furthermore, with $\mathrm{AFeyn}_\mathfrak{k}$ the amputated Feynman

Theorems & Definitions (80)

  • Theorem 1.1: Giansiracusa Giansiracusa
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.7: cf. also PayneWillwacher
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 70 more