On CW-complexes over groups with periodic cohomology
On CW-complexes over groups with periodic cohomology
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DOI：
10.1090/tran/8411
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发表时间：
2019-05
影响因子：
1.3
通讯作者：
John Nicholson
中科院分区：
文献类型：
--
作者：
John Nicholson
如果$G$有$4$-周期上同调，那么$G$上的D2复形由它们的欧拉特征确定为极化同伦当且仅当$G$至多有两个一维四元数表示。我们用它来解决非交换群的几个无限族的Wall的D2问题，在这些情况下，还表明，任何有限的Poincar\'{e} $3$-复$X$与$\pi_1（X）=G$允许一个细胞结构与一个单一的$3$-细胞。证明涉及消除定理的$\mathbb{Z} G$模块，其中$G$具有周期上同调。
If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Wall's D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincar\'{e} $3$-complex $X$ with $\pi_1(X)=G$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $\mathbb{Z} G$ modules where $G$ has periodic cohomology.