AI 中文总结
研究环面\(\Sigma_1\)上无序点配置空间\(B_k(\Sigma_1)\)同调中的奇素数挠率。通过证明不同\(k\)值和奇素数\(p\)下\(H_*(B_k(\Sigma_1);\mathbb{Z})\)的\(p\) - 挠率情况，得出特定条件下的相关结论。
AI 中文摘要
设\(B_k(\Sigma_1)\)表示环面\(\Sigma_1\)上\(k\)个点的无序配置空间。对于每个奇素数\(p\)，我们证明当\(k\leq2p - 1\)时，\(H_*(B_k(\Sigma_1);\mathbb{Z})\)没有\(p\) - 挠率。在\(k = 2p\)时，证明了\(H_{2p - 2}(B_{2p}(\Sigma_1);\mathbb{Z})\)当且仅当\(p\geq5\)时有\(p\) - 挠率等情况。还证明了对于每个奇\(p\)，\(H_{2p}(B_{2p}(\Sigma_1);\mathbb{Z})\)和\(H_{2p + 1}(B_{2p}(\Sigma_1);\mathbb{Z})\)没有\(p\) - 挠率。
英文摘要
The main object of study of this paper is $B_k(Σ_1)$, the unordered configuration space of $k$ points in the torus $Σ_1$. First, we produce a marked-point transfer argument which, combined with Chen--Zhang's theorem, establishes that $H_*(B_k (Σ_1);\mathbb{Z})$ has no $p$-torsion for $k\leq 2p-1$. Secondly, at the threshold $k=2p$, we prove that $H_{2p-2}(B_{2p}(Σ_1);\mathbb{Z})$ has $p$-torsion if and only if $p\geq 5$. For $p\geq5$, the class is the image under puncture filling of a generator of the Bianchi--Stavrou torsion group $\mathbb{Z}/p$ of the once-punctured torus; for $p=3$, Napolitano's calculation shows that this punctured class dies after filling. Finally, we also prove that $H_{2p}(B_{2p}(Σ_1);\mathbb{Z})$ and $H_{2p+1}(B_{2p}(Σ_1);\mathbb{Z})$ have no $p$-torsion for every odd $p$.
CommentsComments welcome. v2: minor exposition fixes
