On the topology of $B\Gamma_n^\mathbb{C}$ and its application to complex structures on open manifolds
Abstract
Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8 and 10. The result is proved by applying Gromov's h-principle in order to adapt a method of Haefliger, originally used to study foliations, to the holomorphic setting. For dimension 12 and greater, the conjecture remains open.
Keywords
Cite
@article{arxiv.2504.10610,
title = {On the topology of $B\Gamma_n^\mathbb{C}$ and its application to complex structures on open manifolds},
author = {Filip Samuelsen},
journal= {arXiv preprint arXiv:2504.10610},
year = {2025}
}
Comments
This paper has been withdrawn, due to a mistake in the proof of theorem 2.1 which then invalidates the proof of theorem 1.4. Theorem 1.5 is independent from theorem 1.4 and deemed to be correct. A new preprint containing only Theorem 1.5 (and its consequences) will appear at a later time