AI 中文总结
研究强自吸收C* - 代数上群作用等的提升障碍，证明相关交叉模分类空间有张量积诱导的无限循环空间结构，证实猜想并扩展与稳定同伦理论联系，通过构造\(\mathbb{I}\) - FCPs等方法证明，确定自然变换取值于上同调群。
AI 中文摘要
Izumi、Giron - Pacheco和第一作者最近通过拓扑交叉模对群作用、上循环作用和Γ - 核的提升障碍给出了上同调描述。对于强自吸收C* - 代数A，我们证明了分别控制上循环作用和Γ - 核的交叉模的分类空间\(\mathcal{B}^D\mathcal{G}_A\)和\(\mathcal{B}^DP\mathcal{G}_A\)具有由张量积诱导的无限循环空间结构；当\(U(A)\)连通时，交叉模\(B^D\tilde{\mathcal{G}}_A\)也如此。这证实了上述工作中的一个猜想，并将与稳定同伦理论的联系扩展到Γ - 核和上循环作用。我们通过从相关交叉模构造\(\mathbb{I}\) - FCPs，过渡到Γ - 空间，并应用May - Thomason无限循环空间机器来证明。因此，自然变换\(H^1(\Gamma,\mathcal{G}) \to [\mathcal{B}\Gamma,\mathcal{B}^D\mathcal{G}]\)取值于上同调群。
英文摘要
Lifting obstructions for group actions, cocycle actions, and $Γ$-kernels admit a cohomological description via topological crossed modules, as recently developed by Izumi, Giron-Pacheco, and the first named author. For a strongly self-absorbing $C^*$-algebra $A$, we show that the classifying spaces $\mathcal{B}^D\mathcal{G}_A$ and $\mathcal{B}^DP\mathcal{G}_A$ of the respective crossed modules governing cocycle actions and $Γ$-kernels, respectively, carry infinite loop space structures induced by the tensor product; the same holds for the crossed module $B^D\tilde{\mathcal{G}}_A$ whenever $U(A)$ is connected. This confirms a conjecture from the aforementioned work and extends to $Γ$-kernels and cocycle actions the connection with stable homotopy theory. For the proof we construct $\mathbb{I}$-FCPs from the relevant crossed modules, pass to $Γ$-spaces, and apply the May-Thomason infinite loop space machine. Consequently, the natural transformation $H^1(Γ,\mathcal{G}) \to [\mathcal{B}Γ,\mathcal{B}^D\mathcal{G}]$ takes values in cohomology groups.
Comments29 pages
