AI 中文总结
本文证明了tom Dieck同态及其因子符号同态在限制到p-群的融合稳定子群时仍满射，进而解决了Mazza-Miller的问题，确定了Lefschetz同态满射当且仅当域为F₂。
AI 中文摘要
Tornehave与Yalçin证明了将实虚表示映射到Burnside环单位元的tom Dieck同态，对任意p-群S都是满射的。我们证明，该同态及其分解出的符号同态，在限制到与S上饱和融合系统相关的融合稳定子群时，仍保持满射性。作为推论，我们解决了Mazza-Miller在arXiv:2508.07404中提出的主要问题：给定特征为正的域k，从p-置换模有界同伦范畴的皮卡群到其Grothendieck环单位群的Lefschetz同态，对所有有限群都是满射的当且仅当k = F₂。
英文摘要
Tornehave and Yalçin proved that the tom Dieck homomorphism, which sends a virtual real representation to a unit of the Burnside ring, is surjective for any $p$-group $S$. We prove that this homomorphism, and the sign homomorphism it factors through, remain surjective when restricted to fusion-stable subgroups associated to a saturated fusion system on $S$. As a corollary, we close the main question posed by Mazza--Miller in arXiv:2508.07404 by showing that given a field $k$ of positive characteristic, the Lefschetz homomorphism from the Picard group of the bounded homotopy category of $p$-permutation modules to the unit group of its Grothendieck ring is surjective for all finite groups if and only if $k = \mathbb{F}_2$.
Comments12 pages, comments welcome
