Rational cohomology algebra of mapping spaces between complex Grassmannians
Summary: We consider the complex Grassmannian \(\mathrm{Gr} (k, n)\) of \(k\)-dimensional subspaces of \(\mathbb{C}^n\). There is a natural inclusion \(i_{n, r} : \mathrm{Gr} (k, n) \hookrightarrow \mathrm{Gr} (k, n + r)\). Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion \(i_{n, r}\) in the space of mappings from \(\mathrm{Gr} (k, n)\) to \(\mathrm{Gr} (k, n + r)\) for \(r \geq 1\) and in particular to show that the cohomology of \(\text{map} (\mathrm{Gr} (n, k), \mathrm{Gr} (n, k + r); i_{n, r})\) contains a truncated algebra \(\mathbb{Q} [x] / x^{r + n + k^2 - n k} \), where \(|x| = 2\), for \(k \geq 2\) and \(n \geq 4\).
scientific article; zbMATH DE number 7193178
scientific article