$4$ planes in ${\mathbb R}^4$

Abstract:

We establish a homeomorphism between the moduli space $A_{4,k}^\textrm {ord}(\mathbb {R})$ of ordered $k$-tuples $(H_1,\ldots ,H_k)$ of 2-dimensional linear subspaces $H_i \subset \mathbb {R}^4$ (mod $\textrm {GL}_4(\mathbb {R})$) and the quotient by simultaneous conjugation of a certain open subset $(\textrm {GL}_2^{k-3})^* \subset (\textrm {GL}_2(\mathbb {R}))^{k-3}$. For $k=4$, this leads to an explicit computation of the moduli space $A_{4,4}(\mathbb {R})$ of central 2-arrangements in $\mathbb {R}^4$ mod $\textrm {GL}_4(\mathbb {R})$ and its subspace $A_{2,4}({\mathbb C})$ of those classes that contain a complex hyperplane arrangement.