In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a $p$‐local finite group $(S,\mathcal{F},\mathcal{L})$ in terms of representation rings of subgroups of $S$. We also prove a stable elements formula for generalized cohomological invariants of $p$‐local finite groups, which is used to show the existence of unitary embeddings of $p$‐local finite groups. Finally, we show that the augmentation $C^{\ast }{(|\mathcal{L}|}_p^{\wedge };{\mathbb{F}}_p)\to {\mathbb{F}}_p$ is Gorenstein in the sense of Dwyer–Greenlees–Iyengar and obtain some consequences about the cohomology ring of ${|\mathcal{L}|}_p^{\wedge }$.

中文翻译：

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p局部有限群和Benson-Carlson对偶性的分类空间上的向量束

在本文中，我们获得了在向量的分类空间上的复矢量束的Grothendieck群的描述。 $p$-局部有限群 $（\mathrm{小号}，\mathcal{F}，\mathcal{大号}）$ 就子群的表示环而言 $\mathrm{小号}$。我们还证明了广义同调不变量的稳定元素公式$p$-局部有限群，用于显示存在一元嵌入 $p$-局部有限群。最后，我们证明了增强$C^{\ast }{（|\mathcal{大号}|}_p^{\wedge };{\mathbb{F}}_p）\to {\mathbb{F}}_p$ 是Dwyer–Greenlees–Iyengar的Gorenstein，并且对 ${|\mathcal{大号}|}_p^{\wedge }$。