# Prof. dr. Frank Kutzschebauch: Holomorphic factorization of maps into the special linear group

Source: Mathematics colloquium

**Frank Kutzschebauch**

Univerza v Bernu, Švica

It is standard material in a Linear Algebra course that the group SL_{m}(**C**) is generated by elementary matrices *E* + *α* *e*_{ij} *i* ≠ *j*, i.e., matrices with 1's on the diagonal and all entries outside the diagonal are zero, except one entry. The same question for matrices in SL_{m}(*R*) where *R* is a commutative ring instead of the field **C** is much more delicate, interesting is the case that *R* is the ring of complex valued functions (continuous, smooth, algebraic or holomorphic) from a space *X*.

For *m*≥ 3 (and any *n*) it is a deep result of Suslin that any matrix in SL_{m}(**C**[**C**^{n}]) decomposes as a finite product of unipotent (and equivalently elementary) matrices. In the case of continuous complex valued functions on a topological space *X* the problem was studied and solved by Thurston and Vaserstein. For rings of holomorphic functions on Stein spaces, in particular on **C**^{n}, this problem was explicitly posed as the **Vaserstein problem** by Gromov in the 1980's. In this talk we explain a complete solution to Gromov's *Vaserstein Problem*; from a joint work with B. Ivarsson. The proof uses a very advanced version of the Oka-principle proposed by Gromov and proved in recent years by Forstnerič: An elliptic stratified submersion over a Stein space admits a holomorphic section iff it admits a continuous section.