Endomorphisms of real Grassmannians that commute with Steenrod squares

Abstract

For $k\in\{2,3\}$ we classify endomorphisms of $H^*(G_{k,n};\mathbb Z_2)$ that commute with Steenrod squares (here, $G_{k,n}$ denotes the Grassmann manifold of $k$-dimensional subspaces in $\mathbb R^{n+k}$). Additionally, for all positive integers $k$ and $n$ we classify endomorphisms of $H^*(G_{k,n};\mathbb Z_2)$ that commute with Steenrod squares and such that the image of each class is a polynomial in the (nonzero) class of $H^1(G_{k,n};\mathbb Z_2)$.