On the Semi-Tensor Product of the Dyer-Lashof and Steenrod Algebras

1989 ◽  
Vol 41 (4) ◽  
pp. 676-701
Author(s):  
H. E. A. Campbell ◽  
P. S. Selick
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Vector Space ◽  
Steenrod Algebra ◽  
Joint Work ◽  
Loop Spaces ◽  
Joint Structure ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Adem Relations

This paper arises out of joint work with F. R. Cohen and F. P. Peterson [5, 2, 3] on the joint structure of infinite loop spaces QX. The homology of such a space is operated on by both the Dyer-Lashof algebra, R, and the opposite of the Steenrod algebra A∗. We describe a convenient summary of these actions; let M be the algebra which is R ⊗ A∗ as a vector space and where multiplication Q1 ⊗ PJ. Q1’ ⊗ PJ’∗ is given by applying the Nishida relations in the middle and then the appropriate Adem relations on the ends. Then M is a Hopf algebra which acts on the homology of infinite loop spaces.

scholarly journals Homology operations and power series

1983 ◽  
Vol 24 (2) ◽  
pp. 161-168
Author(s):  
Richard Steiner
Keyword(s):  
Power Series ◽  
Loop Spaces ◽  
Similar Treatment ◽  
Ring Spectra ◽  
Homology Operations ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Cohomology Operations ◽  
Adem Relations

Bullett and Macdonald [1] have used power series to simplify the statement and proof of the Adem relations for Steenrod cohomology operations. In this paper I give a similar treatment of May's generalized Adem relations [4, §4] and of the Nishida relations ([6], [2, 1.1.1(9)], [5, 3.1(7)]). Both sets of relations apply to Dyer-Lashof operations in E∞, spaces such as infinite loop spaces ([3], [2, I.I]) and in H^ ring spectra ([5, §3]).

scholarly journals ¡+-structures—II: a recognition principle for infinite loop spaces

Topology ◽  
1974 ◽  
Vol 13 (2) ◽  
pp. 113-126 ◽  
Author(s):  
M.G. Barratt ◽  
Peter J. Eccles
Keyword(s):  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop

On the quotients of mapping class groups of surfaces by the Johnson subgroups

2019 ◽  
pp. 1-23
Author(s):  
TOMÁŠ ZEMAN
Keyword(s):  
Boundary Component ◽  
Loop Space ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Classifying Spaces ◽  
Class Groups ◽  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

Abstract We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

INFINITE LOOP SPACES

1979 ◽  
Vol 11 (3) ◽  
pp. 363-364
Author(s):  
John Hubbuck
Keyword(s):  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop
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