The mod 2 cohomology rings of oriented Grassmannians via Koszul complexes (Q6585672)

Let \(\mathrm{Gr}_{k} (n)\) denote the real Grassmannian of \(k\)-planes in \(\mathbb{R}^{n}\). It is classical and well-known that the mod \(2\) cohomology ring \(H^{*}(\mathrm{Gr}_{k}(n); \mathbb{F}_{2})\) has a presentation in terms of the Stiefel--Whitney classes of the tautological \(k\)-plane bundle over \(\mathrm{Gr}_{k}(n)\) (see e.g., [\textit{A. Borel}, Comment. Math. Helv. 27, 165--197 (1953; Zbl 0052.40301)]).\NLet \(\widetilde{\mathrm{Gr}}_{k} (n)\) be the \textit{oriented} real Grassmannian of \textit{oriented} \(k\)-planes in \(\mathbb{R}^{n}\). Clearly we have a double covering \(\pi: \widetilde{\mathrm{Gr}}_{k}(n) \longrightarrow \mathrm{Gr}_{k} (n)\).\NIt might be surprising that the mod \(2\) cohomology structure of \(\widetilde{\mathrm{Gr}}_{k}(n)\) is still unclear except for \(k = 1, 2\). Even the mod \(2\) Betti numbers for \(\widetilde{\mathrm{Gr}}_{k}(n)\) have been unknown until recently (see [\textit{T. Ozawa}, Osaka J. Math. 59, No. 4, 843--851 (2022; Zbl 1540.57051)]).\NThe ``standard'' approach to investigate the mod \(2\) cohomology ring \(H^{*}(\widetilde{\mathrm{Gr}}_{k}(n); \mathbb{F}_{2})\) is perhaps to making use of the Gysin exact sequence associated to the above double covering.\NThen we know that \(H^{*}(\widetilde{\mathrm{Gr}}_{k}(n); \mathbb{F}_{2})\) sits in the folloiwng short exact sequence\N\[\N0 \longrightarrow C\N\overset{\pi^{*}}{\longrightarrow}\NH^{*}(\widetilde{\mathrm{Gr}}_{k}(n); \mathbb{F}_{2})\N\overset{\delta}{\longrightarrow}\NK \N\longrightarrow 0,\N\]\Nwhere \(C\) and \(K\) are respectively the cokernel and the kernel of the map \(w_{1}: H^{*}(\mathrm{Gr}_{k}(n); \mathbb{F}_{2}) \longrightarrow H^{* + 1}(\mathrm{Gr}_{k}(n); \mathbb{F}_{2})\) given by the multiplication of the first Stiefel--Whitney class \(w_{1}\) of the tautological \(k\)-plane bundle over \(\mathrm{Gr}_{k}(n)\).\NThe ring \(C\) is fairy easy to describe; It is just the quotient ring of \(H^{*}(\mathrm{Gr}_{k}(n); \mathbb{F}_{2})\) by the ideal generated by \(w_{1}\).\NThe real difficulty is that one does not know how to find cohomology classes that generate the \(C\)-module \(K\).\N\NIn the paper under review, the authors identify this module using certain Koszul complexes, which involves the syzygies between the relations defining \(C\). Using this new technique, the authors obtain the presentation of \(H^{*}(\widetilde{\mathrm{Gr}}_{3}(n); \mathbb{F}_{2})\) for \(2^{t-1} < n \leq 2^{t} - 4\) with \(t \geq 4\), which complements known descriptions in the cases \(n = 2^{t} - i\), \(i = 0, 1, 2, 3\) with \(t \geq 3\).\NThe authors also discuss various issues that arise for the cases \(k > 3\), supported by computer calculation.