Oka realizations, fundamental groups, and holomorphic cofibrations
Abstract
The homotopy types of Oka manifolds are exactly those of countable finite-dimensional CW complexes. We construct realizations as open subsets of complex Euclidean spaces with explicit dimension bounds. Other realizations have good open covers by sets biholomorphic to CN, with prescribed finite nerves, or are unions of Fatou--Bieberbach domains whose nonempty finite intersections are again Fatou--Bieberbach domains. Every nonempty Stein manifold admits a proper holomorphic embedding into an Oka open subset of some CN, and this embedding is a homotopy equivalence. We obtain holomorphic representatives of ordinary and generalized cohomology classes, and realize every countable group as the fundamental group of an Oka domain in C5. The fundamental groups of algebraically elliptic manifolds are precisely the finite groups, and every virtually abelian Kähler group is the fundamental group of a projective Oka manifold. Finally, we classify holomorphic cofibrations in Lárusson's intermediate model structures and establish factorizations over aspherical targets into proper holomorphic embeddings and holomorphic fibre bundles with Oka fibre.
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BibTeX
@misc{guo2026realizations,
title = {{Oka realizations, fundamental groups, and holomorphic cofibrations}},
author = {Bin Guo and Yuta Kusakabe and Peng-Chao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17044},
eprint = {2610.17044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17044},
}
RIS
TY - UNPB
TI - Oka realizations, fundamental groups, and holomorphic cofibrations
AU - Bin Guo
AU - Yuta Kusakabe
AU - Peng-Chao Wang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17044
UR - https://arxiv.org/abs/2610.17044
AB - The homotopy types of Oka manifolds are exactly those of countable finite-dimensional CW complexes. We construct realizations as open subsets of complex Euclidean spaces with explicit dimension bounds. Other realizations have good open covers by sets biholomorphic to C^N, with prescribed finite nerves, or are unions of Fatou--Bieberbach domains whose nonempty finite intersections are again Fatou--Bieberbach domains. Every nonempty Stein manifold admits a proper holomorphic embedding into an Oka open subset of some C^N, and this embedding is a homotopy equivalence. We obtain holomorphic representatives of ordinary and generalized cohomology classes, and realize every countable group as the fundamental group of an Oka domain in C^5. The fundamental groups of algebraically elliptic manifolds are precisely the finite groups, and every virtually abelian Kähler group is the fundamental group of a projective Oka manifold. Finally, we classify holomorphic cofibrations in Lárusson's intermediate model structures and establish factorizations over aspherical targets into proper holomorphic embeddings and holomorphic fibre bundles with Oka fibre.
ER -