Low-weight compactly supported cohomology of ℳg,nct and the extended Torelli morphism
Abstract
We study the weight zero and weight two graded pieces of the compactly supported cohomology of the moduli space ℳg,nct of stable n-pointed curves of genus g of compact type and show that they are computed by bridgeless graph complexes. In weight two, we construct a natural embedding of shifted weight zero cohomology from lower genus. For unmarked curves, we construct canonical classes represented by pinched odd wheel graphs and prove that distinct wheel monomials give linearly independent classes in the dual of W0Hc2g(ℳgct;ℚ). By relating the bridgeless graph complex to the K-theory of graphs and the Quillen spectral sequence, we show that these classes remain linearly independent after pushforward along the extended Torelli morphism ℳgct→𝒜g. Finally, we prove that the realizations of the motivic Lie algebra arising from ℳg and 𝒜g have the same image in the weight zero compactly supported cohomology of ℳgct.
Related
- Sharp Quantitative Convergence of Compactly Supported Aggregation-Diffusion
- Compactly Supported Solutions near a Nehari Critical Value
- Compactly supported instability of Riemannian Schwarzschild–Tangherlini metrics
- A new approach to compactly supported stationary Euler flows
- A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in ℝ2
BibTeX
@misc{bradford2026lowweight,
title = {{Low-weight compactly supported cohomology of $\mathcal M_g,n^ct$ and the extended Torelli morphism}},
author = {Amy Bradford},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22809},
eprint = {2610.22809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22809},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Low-weight compactly supported cohomology of ℳ_g,n^ct and the extended Torelli morphism
AU - Amy Bradford
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22809
UR - https://arxiv.org/abs/2610.22809
AB - We study the weight zero and weight two graded pieces of the compactly supported cohomology of the moduli space ℳ_g,n^ct of stable n-pointed curves of genus g of compact type and show that they are computed by bridgeless graph complexes. In weight two, we construct a natural embedding of shifted weight zero cohomology from lower genus. For unmarked curves, we construct canonical classes represented by pinched odd wheel graphs and prove that distinct wheel monomials give linearly independent classes in the dual of W_0H_c^2g(ℳ_g^ct;ℚ). By relating the bridgeless graph complex to the K-theory of graphs and the Quillen spectral sequence, we show that these classes remain linearly independent after pushforward along the extended Torelli morphism ℳ_g^ct→𝒜_g. Finally, we prove that the realizations of the motivic Lie algebra arising from ℳ_g and 𝒜_g have the same image in the weight zero compactly supported cohomology of ℳ_g^ct.
ER -