scholarly journals HIGHER CONJUGATION COHOMOLOGY IN COMMUTATIVE HOPF ALGEBRAS

2001 ◽  
Vol 44 (1) ◽  
pp. 19-26 ◽  
Author(s):  
M. D. Crossley ◽  
Sarah Whitehouse
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Symmetric Group ◽  
Group Action ◽  
Hopf Algebras ◽  
Homotopy Theory ◽  
Cohomology Ring ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Primary 16W30

AbstractLet $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

The transfer and stable homotopy theory

1978 ◽  
Vol 83 (1) ◽  
pp. 103-111 ◽  
Author(s):  
Daniel S. Kahn ◽  
Stewart B. Priddy
Keyword(s):  
Symmetric Group ◽  
Homotopy Theory ◽  
Stable Homotopy ◽  
Splitting Theorem ◽  
Stable Homotopy Theory

The purpose of this paper is to give a proof of the following splitting theorem in stable homotopy theory. We assume all spaces are localized at a fixed prime p. Let k be the symmetric group on {1, …, k}, Q(.) = lim ΩnΣn(.), and QkS0, k ∈ , denote the components of QS0.

On the transfer in the homology of symmetric groups

1978 ◽  
Vol 83 (1) ◽  
pp. 91-101 ◽  
Author(s):  
Daniel S. Kahn ◽  
Stewart B. Priddy
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Homotopy Theory ◽  
Symmetric Groups ◽  
Characterization Theorem ◽  
Group Cohomology ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Transfer Homomorphism ◽  
Image Position

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphismin group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND TENSOR PRODUCTS OF SYMMETRIC GROUP REPRESENTATIONS

1991 ◽  
Vol 01 (02) ◽  
pp. 207-221 ◽  
Author(s):  
JEAN-YVES THIBON
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Tensor Products ◽  
Ring Structure ◽  
Algebra Structure ◽  
Group Representations ◽  
Symmetric Group Representations

The Hopf algebra structure of the ring of symmetric functions is used to prove a new identity for the internal product, i.e., the operation corresponding to the tensor product of symmetric group representations. From this identity, or by similar techniques which can also involve the λ-ring structure, we derive easy proofs of most known results about this operation. Some of these results are generalized.

v 1 - and v 2 -Periodicity in Stable Homotopy Theory

1981 ◽  
Vol 103 (4) ◽  
pp. 615 ◽  
Author(s):  
Donald M. Davis ◽  
Mark Mahowald
Keyword(s):  
Homotopy Theory ◽  
Stable Homotopy ◽  
Stable Homotopy Theory

Spectra of derived module homomorphisms

1987 ◽  
Vol 101 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Alan Robinson
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Theory ◽  
Ring Spectrum ◽  
Homotopy Invariant ◽  
New Construction

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

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