On the number of factors in the unipotent factorization of holomorphic mappings into \(\mathrm{SL}_{2}(\mathbb {C})\) (Q2880642)

The starting point is Vaserstein's theorem: For any natural number \(n\) and integer \(d \geq 0\) there is a natural number \(K\) such that for any finite dimensional normal topological space \(X\) of dimension \(d\) and a null-homotopic, that is homotopic to the identity, continuous mapping \(f : X \rightarrow \mathrm{SL}_{n}(\mathbb {C})\), \(f\) can be written as a finite product of no more than \(K\) unipotent matrices. That is, one can find continuous mappings \(F_l : X \rightarrow \mathbb {C}^{n(n-1)/2}\), \(1 \geq l \geq K\), such that \(f(x) = M_1\big(F_1(x)\big)\dots M_K\big(F_K(x)\big)\), where \(M_j(F_j)\) are upper or lower triangular idempotent matrices.NEWLINENEWLINEIn their previous paper [the authors, Ann. Math. (2) 175, No. 1, 45--69 (2012; Zbl 1243.32007)] the authors have proved that if \(X\) is a \(d\)-dimensional Stein space then there exists a number \(K'>K\) depending on \(d\) and \(n\) such that there exists such a factorization by holomorphic upper or lower triangular idempotent matrices. The main tool in the proof is the homotopy principle and the construction of a suitable submersion with a spray. However, due to the choice of the fibration, the number of factors needed in the holomorphic case is much larger than in the continuous one. Therefore the authors reprove their factorization theorem in the case of \(SL_2(\mathbb{C})\) by using a different submersion. As a result they get \(K' = K + 2\) and prove that \(K' = 4\) if \(n = 1\) and \(K' = 5\) if \(n = 2\).