Equivalences of classifying spaces completed at odd primes
Equivalences of classifying spaces completed at odd primes
复制标题
在奇素数处完成的分类空间的等价
DOI：
10.1017/s0305004104007728
复制
发表时间：
2004
影响因子：
0.8
通讯作者：
B. Oliver
中科院分区：
文献类型：
--
作者：
B. Oliver
证明了奇素数$p$的Martino-Priddy猜想：两个群$G和$G^\素数的分类空间的$p$-完备性是同伦等价的当且仅当它们的Sylow$p$-子群之间存在保持融合的同构.第二个定理是用在$G$中保持融合的Sylow$p$-子群的自同构来刻画$p$-完成$BG$的同伦类群的奇数$p$。这两个结果都是技术代数结果的结果，即对于奇素数$p$和有限群$G$，对于$G$的$p$-子群轨道范畴上的某个函子$\calz_G$，逆极限的所有高阶派生函子都为零。
We prove the Martino–Priddy conjecture for an odd prime $p$: the $p$-completions of the classifying spaces of two groups $G$ and $G^\prime$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $p$-subgroups which preserves fusion. A second theorem is a description for odd $p$ of the group of homotopy classes of self homotopy equivalences of the $p$-completion of $BG$, in terms of automorphisms of a Sylow $p$-subgroup of $G$ which preserve fusion in $G$. These are both consequences of a technical algebraic result, which says that for an odd prime $p$ and a finite group $G$, all higher derived functors of the inverse limit vanish for a certain functor $\calz_G$ on the $p$-subgroup orbit category of $G$.