PreprintCC BY 4.0arXiv
Realization of permutation modules as homology of CW-complexes
Abstract
In this paper, we investigate the realizability problem, which asks whether prescribed group actions on graded modules can be realized by the groups of self-homotopy equivalences of CW-complexes acting on their homology. We provide a partial answer in the case of arbitrary groups and permutation modules concentrated in certain degrees. As a consequence, we realize every group as the group of self-homotopy equivalences of an R-local CW-complex with arbitrary prescribed connectivity, where R may be chosen with ρ(R) sufficiently large. In particular, this provides a complete answer to Kahn's realizability problem.
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BibTeX
@misc{bai2026realization,
title = {{Realization of permutation modules as homology of CW-complexes}},
author = {Yang Bai and Chaoshun Yang and Xiugui Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04516},
eprint = {2610.04516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04516},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Realization of permutation modules as homology of CW-complexes
AU - Yang Bai
AU - Chaoshun Yang
AU - Xiugui Liu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.04516
UR - https://arxiv.org/abs/2610.04516
AB - In this paper, we investigate the realizability problem, which asks whether prescribed group actions on graded modules can be realized by the groups of self-homotopy equivalences of CW-complexes acting on their homology. We provide a partial answer in the case of arbitrary groups and permutation modules concentrated in certain degrees. As a consequence, we realize every group as the group of self-homotopy equivalences of an R-local CW-complex with arbitrary prescribed connectivity, where R may be chosen with ρ(R) sufficiently large. In particular, this provides a complete answer to Kahn's realizability problem.
ER -