Elementary calculation of the cohomology rings of real Grassmann manifolds (Q2363197)

In [Ann. Math. (2) 57, 115--207 (1953; Zbl 0052.40001)] \textit{A. Borel} developed a general technique of computing cohomology rings of compact symmetric spaces using invariant theory. However, there are some exceptional cases including those of real Grassmann manifolds of odd dimension that do not immediately fit the Borel theory. In these cases the cohomology rings with real coefficients were determined by \textit{M. Takeuchi} [J. Fac. Sci., Univ. Tokyo, Sect. I 9, 313--328 (1962; Zbl 0108.35802)]. Let \(G(m,n)\) be the Grassmann manifold of oriented planes of dimension \(m\) in \(\mathbb R^{m+n}\). Its tautological \(m\)- and \(n\)-vector bundles support the total Pontrjagin classes \(p\) and Pontryagin classes \(p_i\), \(\bar{p_j}\) and Euler classes \(e_m\), \(\bar{e_n}\). If \(mn\) is odd, there is also a cohomology class \(r\) in \(G(m,n)\) of degree \(m + n - 1\). Let \(\mathbb P = \mathbb P (m,n)\) denote the symmetric algebra over \(\mathbb R\) on the Pontrjagin classes \(p_i\), \(\bar{p_j}\) with the relation \(p . \bar{p} = 1\). Takeuchi [loc. cit.] proved that for \(m,n > 1\) the cohomology algebra \(H^* (G(m,n),\mathbb R)\) over \(\mathbb R\) is isomorphic to one of the symmetric algebra generated by Euler cdlasses. The algebra \(H^* (G(m,n),\mathbb R)\) as well as its equivariant version were also recently computed by means of the GKM theory by \textit{C. He} [``GKM theory, characteristic classes and the equivariant cohomology ring of real Grassmannian'', Preprint, \url{arXiv:1609.06243}]. Furthermore, there is an elegant computation of these algebras by means of pure Sullivan models by \textit{J. D. Carlson} [``The Borel equivariant cohomology of real Grassmannians'', Preprint, \url{arXiv:1611.01175}] whose work relies on a model constructed by \textit{V. Kapovitch} [``A note on rational homotopy of biquotients'', Preprint, \url{http://www.math.toronto.edu/vtk/biquotient.pdf}]. In this paper the authors give a short elementary proof of theorem based on an observation that the total spaces of the tautological \((m- 1)\)-sphere bundle over \(G(m,n)\) and \(n\)-sphere bundle over \(G(n + 1, m - 1)\) are isomorphic. They also deduce from the theorem its equivariant version.