Parametric decompositions into products of involutions

Abstract

It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-homotopy characterizes exactly those matrices in the special linear group over rings of continuous or holomorphic functions that are finite products of involutions. We give an upper bound $I(n, d)$, depending only on the size $n$ of the matrices and the dimension $d$ of the parameter space, for the number of involutions needed in the factorization. Our upper bound is not optimal, however we show that for fixed $n$ the optimal upper bound tends to infinity as $d\to \infty$. In some cases we are able to determine the optimal bound.