The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

Abstract

We study the structure of mod 2 cohomology rings of oriented Grassmannians $\tilde{\operatorname{Gr}}_k(n)$ of oriented $k$-planes in $\mathbb{R}^n$. Our main focus is on the structure of the cohomology ring ${\rm H}^*(\tilde{\operatorname{Gr}}_k(n);\mathbb{F}_2)$ as a module over the characteristic subring $C$, which is the subring generated by the Stiefel-Whitney classes $w_2,\ldots, w_k$. We identify this module structure using Koszul complexes, which involves the syzygies between the relations defining $C$. We give an infinite family of such syzygies, which results in a new upper bound on the characteristic rank of $\tilde{\operatorname{Gr}}_k(2^t)$, and formulate a conjecture on the exact value of the characteristic rank of $\tilde{\operatorname{Gr}}_k(n)$. For the case $k=3$, we use the Koszul complex to compute a presentation of the cohomology ring $H={\rm H}^*(\tilde{\operatorname{Gr}}_3(n);\mathbb{F}_2)$ for $2^{t-1}<n\leq 2^t-4$, complementing existing descriptions in the $n=2^t-3,...,2^t$ cases. More precisely, as a $C$-module, $H$ splits as a direct sum of the characteristic subring $C$ and the anomalous module $H/C$, and we compute a complete presentation of $H/C$ as a $C$-module from the Koszul complex. We also discuss various issues that arise for the cases $k>3$, supported by computer calculation.