scholarly journals The Hopf ring for bo and its connective covers

2007 ◽  
Vol 210 (1) ◽  
pp. 219-247 ◽  
Author(s):  
Dena S. Cowen Morton
Keyword(s):  
Hopf Ring

scholarly journals The Hopf ring for complex cobordism

1977 ◽  
Vol 9 (2-3) ◽  
pp. 241-280 ◽  
Author(s):  
Douglas C. Ravenel ◽  
W.Stephen Wilson
Keyword(s):  
Complex Cobordism ◽  
Hopf Ring

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

2007 ◽  
Vol 59 (6) ◽  
pp. 1154-1206
Author(s):  
J. Michael Boardman ◽  
W. Stephen Wilson
Keyword(s):  
Complex Cobordism ◽  
Hopf Ring ◽  
Torsion Free

AbstractThe 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.

The periodic Hopf ring of connective Morava K-theory

1999 ◽  
Vol 11 (6) ◽  
Author(s):  
J. M. BOARDMAN ◽  
R. L. KRAMER ◽  
W. S. WILSON
Keyword(s):  
Hopf Ring ◽  
K Theory

Dyer–Lashof operations in the Hopf ring for complex cobordism

1993 ◽  
Vol 114 (3) ◽  
pp. 453-460 ◽  
Author(s):  
Paul R. Turner
Keyword(s):  
The Other ◽  
Complex Cobordism ◽  
Hopf Ring

AbstractWe give a complete description of the additive Dyer-Lashof operations on the Hopf ringwith coefficients in. We re-prove a result of Ravenel and Wilson [9] giving the operations on the elementand give formulae for the operations on the other generators.

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

2016 ◽  
Vol 72 (3) ◽  
pp. 366-375 ◽  
Author(s):  
Igor A. Baburin
Keyword(s):  
Automorphism Group ◽  
Theoretical Approach ◽  
Theoretical Framework ◽  
Acta Cryst ◽  
Space Group ◽  
Multiple Copies ◽  
Hopf Ring ◽  
Ring Net ◽  
Edge Crossings

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.

The Hopf Ring for P(n)

1996 ◽  
Vol 48 (5) ◽  
pp. 1044-1063 ◽  
Author(s):  
Douglas C. Ravenel ◽  
W. Stephen Wilson
Keyword(s):  
Cohomology Operations ◽  
Hopf Ring ◽  
Set Up ◽  
Stable Result

AbstractWe show that , the E-homology of the Ω-spectrum for P(n), is an E* free Hopf ring for E a complex oriented theory with In sent to 0. This covers the cases and . The generators of the Hopf ring are those necessary for the stable result. The motivation for this paper is to show that P(n) satisfies all of the conditions for the machinery of unstable cohomology operations set up in [BJW95]. This can then be used to produce splittings analogous to those for BP done in [Wil75]

scholarly journals The mod 2 Hopf ring for connective Morava K-theory

2015 ◽  
Vol 11 (3) ◽  
pp. 469-491
Author(s):  
Paul Thomas Pearson
Keyword(s):  
Hopf Ring ◽  
K Theory

scholarly journals The Hopf ring for Bockstein-nil homology of QSn

2012 ◽  
Vol 216 (2) ◽  
pp. 267-275 ◽  
Author(s):  
Takuji Kashiwabara
Keyword(s):  
Hopf Ring

On the Hopf ring for the sphere

1997 ◽  
Vol 224 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Peter J. Eccles ◽  
Paul R. Turner ◽  
W. Stephen Wilson
Keyword(s):  
Hopf Ring
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