Characterize the reduced weight-two bridgeless graph complex

Determine whether the cohomology of the reduced weight-two bridgeless Getzler–Kapranov graph complex for the moduli space of stable n-pointed curves of compact type admits a description in terms of weight-zero compactly supported cohomology, and construct analogous injections from weight-zero-derived terms if they exist.

Background

The paper decomposes the weight-two compactly supported cohomology of the compact-type moduli space into two contributions: a genus-one contribution identified with shifted weight-zero cohomology in lower genus, and the reduced weight-two bridgeless Getzler–Kapranov graph complex. The genus-one contribution is therefore understood in terms of weight zero, while the reduced summand remains unexplained by the methods developed in the paper.

For ordinary, non-bridgeless graph complexes, prior work gives descriptions of the weight-two cohomology in terms of several weight-zero contributions and constructs injections from weight-zero-derived terms in the marked case. The unresolved problem is to obtain comparable descriptions or injections for the reduced bridgeless complex, which would clarify the remaining contribution to weight-two compactly supported cohomology of the compact-type locus.