The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes

Abstract

We study the structure of mod 2 cohomology rings of oriented Grassmannians $$\widetilde{{\text {Gr}}}_k(n)$$ Gr ~ k ( n ) of oriented k-planes in $${\mathbb {R}}^n$$ R n . Our main focus is on the structure of the cohomology ring $$\textrm{H}^*(\widetilde{{\text {Gr}}}_k(n);{\mathbb {F}}_2)$$ H ∗ ( Gr ~ k ( n ) ; F 2 ) as a module over the characteristic subring C, which is the subring generated by the Stiefel–Whitney classes $$w_2,\ldots ,w_k$$ w 2 , … , 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 $$\widetilde{{\text {Gr}}}_k(2^t)$$ Gr ~ k ( 2 t ) , $$k<2^t$$ k < 2 t , and formulate a conjecture on the exact value of the characteristic rank of $$\widetilde{{\text {Gr}}}_k(n)$$ Gr ~ k ( n ) . For the case $$k=3$$ k = 3 , we use the Koszul complex to compute a presentation of the cohomology ring $$H=\textrm{H}^*(\widetilde{{\text {Gr}}}_3(n);{\mathbb {F}}_2)$$ H = H ∗ ( Gr ~ 3 ( n ) ; F 2 ) for $$2^{t-1}<n\le 2^t-4$$ 2 t - 1 < n ≤ 2 t - 4 for $$t\ge 4$$ t ≥ 4 , complementing existing descriptions in the cases $$n=2^t-i$$ n = 2 t - i , $$i=0,1,2,3$$ i = 0 , 1 , 2 , 3 for $$t\ge 3$$ t ≥ 3 . 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 $$k>3$$ k > 3 , supported by computer calculation.