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arXiv · 2610.04516

Realization of permutation modules as homology of CW-complexes

Yang Bai·Chaoshun Yang·Xiugui Liu
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Abstract

In this paper, we investigate the realizability problem, which asks to realize groups acting on graded modules as the groups of self-homotopy equivalences of CW-complexes acting on their homology groups. We provide a partial answer in the case of arbitrary groups and permutation modules concentrated in certain degrees. As a consequence, we also provide a complete answer to Kahn's realizability problem for arbitrary groups in the categories of CW-complexes with arbitrary connectivity.

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Yang Bai, Chaoshun Yang, Xiugui Liu. 2026-10-03. Realization of permutation modules as homology of CW-complexes. https://arxiv.org/abs/2610.04516

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