2012/11/09 by Daniel Farley, Farley, Daniel, Ivonne Johanna Ortiz +2
Mathematics · #19A31 #19B28 #19D35 #20H15 #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #math.AT #math.GT #math.KT #msc:19A31 #msc:19B28 #msc:19D35 #msc:20H15
paper · pdf · doi:10.48550/arxiv.1211.2024
116 pages, 22 Figures, 24 Tables
arxiv created 2012/11/09 · openalex publication_date 2012/11/09 · arxiv updated 2012/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.