On the group-theoretical approach to the study of interpenetrating nets

Using group–subgroup and group–supergroup relations, a general theoretical framework is developed to describe and derive interpenetrating 3-periodic nets. The generation of interpenetration patterns is readily accomplished by replicating a single net with a supergroupGof its space groupHunder the condition that site symmetries of vertices and edges are the same in bothHandG. It is shown that interpenetrating nets cannot be mapped onto each other by mirror reflections because otherwise edge crossings would necessarily occur in the embedding. For the same reason any other rotation or roto-inversion axes fromG \ Hare not allowed to intersect vertices or edges of the nets. This property significantly narrows the set of supergroups to be included in the derivation of interpenetrating nets. A procedure is described based on the automorphism group of aHopf ring net[Alexandrovet al.(2012).Acta Cryst.A68, 484–493] to determine maximal symmetries compatible with interpenetration patterns. The proposed approach is illustrated by examples of twofold interpenetratedutp,diaandpcunets, as well as multiple copies of enantiomorphic quartz (qtz) networks. Some applications to polycatenated 2-periodic layers are also discussed.

k(n)-Torsion-Free H-Spaces and P(n)-Cohomology

The H-space that represents Brown–Peterson cohomology BPk(–) was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum P(n), by constructing idempotent operations in P(n)–cohomology P(n)k(–) in the style of Boardman–Johnson–Wilson; this relies heavily on the Ravenel–Wilson determination of the relevant Hopf ring. The resulting (i – 1)-connected H-spaces Yi have free connective Morava K-homology k(n)*(Yi), and may be built from the spaces in the Ω-spectrum for k(n) using only vn-torsion invariants.We also extend Quillen's theorem on complex cobordism to show that for any space X, the P(n)*-module P(n)*(X) is generated by elements of P(n)i(X) for i ≥ 0. This result is essential for the work of Ravenel–Wilson–Yagita, which in many cases allows one to compute BP–cohomology from Morava K-theory.