The nilCoxeter algebra $\mathcal{N}S_n$ of the symmetric group $S_n$ is the algebra over $\mathbb{Z}$ with generators $Y_i$ ($1\leqslant i\leqslant n-1$), satisfying the braid relations $Y_iY_{i+1}Y_i=Y_{i+1}Y_iY_{i+1}$, $Y_iY_j=Y_jY_i$ ($|j-i|\geqslant 2$), together with the relations $Y_i^2=0$. We describe an explicit presentation for the cohomology ring $Z\cong\mathsf{Ext}^*_{\mathcal{N}S_n}(\mathbb{Z},\mathbb{Z})$, with $n-i$ new generators in degree $i$ for $0< i

In their homonymous article, Sam Payne and Thomas Willwacher construct a combinatorial graph complex to compute the weight 11 part of the compactly supported cohomology of the moduli space of curves $\cM_{g,n}$ and compute explicitly the cohomology of the introduced graph complexes in cases of complexes of excess zero, one, two, and three. In this paper, we extend the computation the cohomology to excess four graph complexes. Along the way, we give more details on generators, the computation process, and the definition of the graph complex.

The Cartan formula relates the cup product and the action of the Steenrod algebra on mod p cohomology. For any pair of mod p cocycles in a simplicial set, where p is an odd prime, we explicitly construct a natural coboundary descending to this relation in cohomology