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 $$\textsf{BV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BV</mml:mi> </mml:math> and $$\textsf{HyCom}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>HyCom</mml:mi> </mml:math> , and the top $$-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> weight cohomology of the Feynman transforms of $$D\textsf{BV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mi>BV</mml:mi> </mml:mrow> </mml:math> and $$\textsf{Grav}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Grav</mml:mi> </mml:math> . Using a result of Giansiracusa, we compute, in particular, the top $$-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> weight cohomology of the handlebody group. We compare the result to the top $$-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> weight cohomology of the moduli space of curves $$\mathcal {M}_{g,n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> , 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.