The Lower Algebraic K-Theory of Split Three-Dimensional Crystallographic Groups
The Lower Algebraic K-Theory of Split Three-Dimensional Crystallographic Groups
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分裂三维晶体群的低代数K理论
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
I. J. Ortiz
中科院分区：
文献类型：
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作者：
Daniel S. Farley;I. J. Ortiz
我们明确计算分裂三维晶体群的低代数K理论;即，群G在三维欧氏空间上通过等距适当地作用和余紧，使得从G到O（3）的自然映射是到其像上的分裂注入。在总共219种同构类型的三维晶体群中，有73种分裂的三维晶体群。
我们还提供了一个一般的分裂公式的低代数K-理论，是有效的所有三维晶体群。这一结果推广了Alves和Ontaneda的早期工作。
沿着这条路，我们给出了所有73个分裂的三维晶体群的明确描述，并完整地给出了它们的分类。分裂的晶体学基团有时被称为“分裂基团”。晶体群的一个定理说，任何晶体群都是它的分裂群的有限指数子群，所以每个三维晶体群都是我们列表中的一个的有限指数子群。
We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimensional crystallographic groups in all, out of a total of 219 isomorphism types of three-dimensional crystallographic groups.
We also provide a general splitting formula for the lower algebraic K-theory that is valid for all three-dimensional crystallographic groups. This result generalizes earlier work of Alves and Ontaneda.
Along the way, we give explicit descriptions of all 73 split three-dimensional crystallographic groups, and completely work out their classification. The split crystallographic groups are sometimes called "splitting groups". A theorem of crystallographic groups says that any crystallographic group is a finite-index subgroup of its splitting group, so each three-dimensional crystallographic group is a finite-index subgroup of one from our list.