Nonabelian Torus Exteriors in π4 and Torus Replacement in Four-Manifolds
Abstract
We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kanenobu-Kazama, with explicitly controlled fundamental groups and peripheral systems, we construct smooth homotopy 4-spheres and manifolds homeomorphic to S^2 x S^2 and \#_3(S^2 x S^2). We also study direct gluings of two torus exteriors. A swap gluing of two one-null exteriors kills both meridians and gives a simply connected manifold with Euler characteristic 4 and signature 0, while a meridian-preserving gluing of doubly-null exteriors retains the complement groups as an amalgam over the meridian subgroup. Twisted peripheral gluings give cyclic and explicit finite nonabelian fundamental groups. We also consider the full Kanenobu-Kazama 1-handle family and Litherland tori with full-rank peripheral subgroup; the latter have no null primitive boundary slope and lead naturally to amalgamated-product fundamental groups.
Related
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BibTeX
@misc{akhmedov2026nonabelian,
title = {{Nonabelian Torus Exteriors in $\mathbb S^4$ and Torus Replacement in Four-Manifolds}},
author = {Anar Akhmedov and Azer Akhmedov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25080},
eprint = {2609.25080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25080},
note = {CC0},
}
RIS
TY - UNPB
TI - Nonabelian Torus Exteriors in π^4 and Torus Replacement in Four-Manifolds
AU - Anar Akhmedov
AU - Azer Akhmedov
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.25080
UR - https://arxiv.org/abs/2609.25080
AB - We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kanenobu-Kazama, with explicitly controlled fundamental groups and peripheral systems, we construct smooth homotopy 4-spheres and manifolds homeomorphic to S^2 x S^2 and \#_3(S^2 x S^2). We also study direct gluings of two torus exteriors. A swap gluing of two one-null exteriors kills both meridians and gives a simply connected manifold with Euler characteristic 4 and signature 0, while a meridian-preserving gluing of doubly-null exteriors retains the complement groups as an amalgam over the meridian subgroup. Twisted peripheral gluings give cyclic and explicit finite nonabelian fundamental groups. We also consider the full Kanenobu-Kazama 1-handle family and Litherland tori with full-rank peripheral subgroup; the latter have no null primitive boundary slope and lead naturally to amalgamated-product fundamental groups.
ER -