[Paper Review] Holomorphic factorization of mappings into SL_n(C)

Björn Ivarsson, Frank Kutzschebauch|ArXiv.org|Dec 1, 2008

Homotopy and Cohomology in Algebraic Topology6 references7 citations

TL;DR

This paper solves Gromov's Vaserstein problem by proving that any null-homotopic holomorphic map from a finite-dimensional reduced Stein space into SLₙ(ℂ) factors into a finite product of unipotent matrices with holomorphic entries. The proof uses the Oka-Grauert-Gromov h-principle applied to stratified fibrations, establishing a uniform bound on the number of factors depending only on the dimension of the domain and size of the matrix.

ABSTRACT

We solve Gromov's Vaserstein problem. Namely, we show that a null-homotopic holomorphic mapping from a finite dimensional reduced Stein space into SL_n(C) can be factored into a finite product of unipotent matrices with holomorphic entries.

Motivation & Objective

• To resolve Gromov's Vaserstein problem concerning holomorphic factorization of maps into SLₙ(ℂ).
• To establish that null-homotopic holomorphic mappings from finite-dimensional reduced Stein spaces into SLₙ(ℂ) admit factorization into holomorphic unipotent matrices.
• To prove the existence of a uniform upper bound on the number of unipotent factors required, depending only on the dimension of the domain and the matrix size.

Proposed method

• Application of the Oka-Grauert-Gromov h-principle to stratified fibrations associated with matrix factorization problems.
• Construction of a fibration over a Stein space with fibers isomorphic to level sets of certain polynomial maps, enabling the use of holomorphic sprays.
• Use of globally integrable vector fields and sprays on smooth strata to construct holomorphic sections from topological ones.
• Leveraging Vaserstein’s result on continuous factorization to deduce the existence of a topological section, which lifts to a holomorphic section via the h-principle.
• Inductive argument over matrix size and stratification of the base space to handle singular fibers and ensure global holomorphic factorization.
• Use of contradiction to establish a uniform bound on the number of unipotent factors across all such maps.

Experimental results

Research questions

• RQ1 Can every null-homotopic holomorphic map from a finite-dimensional reduced Stein space into SLₙ(ℂ) be factored into a finite product of unipotent matrices with holomorphic entries?
• RQ2 Is there a uniform upper bound on the number of unipotent factors required for such factorizations, depending only on the dimension of the domain and the matrix size?
• RQ3 How do the numbers of factors required in the continuous case (K_C(m,n)) and holomorphic case (K_O(m,n)) relate, particularly for n=2?

Key findings

• Every null-homotopic holomorphic map f: X → SLₙ(ℂ), where X is a finite-dimensional reduced Stein space, factors into a finite product of unipotent matrices with holomorphic entries.
• A uniform upper bound K exists such that any such map factors into at most K unipotent matrices, with K depending only on the dimension m of X and the size n of the matrix.
• For n=2, the holomorphic factorization number K_O(m,2) satisfies K_O(m,2) ≤ K_C(m,2) + 4, where K_C(m,2) is the continuous case bound.
• The proof establishes that the fibration used in the h-principle argument admits a holomorphic section, lifting a topological section guaranteed by Vaserstein’s result.
• The number of factors is bounded uniformly across all Stein spaces of dimension m and all null-homotopic maps into SLₙ(ℂ), even in the holomorphic category.
• The Cohn counterexample for polynomial maps in SL₂(ℂ[z₁,z₂]) requires 5 holomorphic unipotent factors, though it factors with 4 continuous ones, showing a strict gap between continuous and holomorphic settings.