RE : [ALGTOP-L] the (co)homology of \Sigma_n and O(n-1)

Joyal, André joyal.andre at uqam.ca
Thu Oct 16 00:06:14 EDT 2008


The theory of Q-rings can be used to answer your question.
See my paper with Terry Bisson

"Q-Rings and the Homology of the Symmetric Group" 
http://hopf.math.purdue.edu/cgi-bin/generate?/Bisson-Joyal/Luminy

(A Q-ring R is defined to be a commutative algebra equipped with a ring 
homomorphism Q_t:R-->R[[t]] satisfing two simple conditions)

Your map 
\phi_n: H_* B\Sigma_n \to  H_*BO(n-1) 
is closely related to the map 
\psi_n: H_* B\Sigma_n \to  H_*BO(n) 
induced by the natural representation \Sigma_n \to  BO(n) 
I will describe the latter first.
It is easier to describe the maps phi_n globally by taking a direct sum:

\psi_*:H_* B\Sigma_* \to  H_*BO(*) 

where B\Sigma_* is the disjoint union of the classifying spaces B\Sigma_n 
and BO(*) is the disjoint union of the classifying spaces BO(n).
The map psi_* is a map of Q-rings because it is induced by a map of E_\infty-spaces.

The Q-ring H_* B\Sigma_* is in fact the free Q-ring on one generator 
Q<x> where x is the non-zero class in H_0(\Sigma_1).
Thus \psi_* is completely determined by \psi_*(x)=b_0 the non-zero class of H_0BO(1).
The Q-ring structure of H_*BO(*) can be described explicitly (by using a result of Priddy).
As an algebra,  H_*BO(*) is a symmetric algebra on the homology of BO_1=RP^\infty. 
It is thus a polynomial ring in an infinite sequences of variables  b_0,b_1,b_3, ... 
where b_n\in H_nRP^\infty is the non-zero class. 
The Q-ring structure of H_*BO(*) is completely determined by the formula

Q_t(b)(x(x+t))=b(x)b(x+t)

where b(x) is the power series b_0+b_1x+b_2x^2+---
and where Q_t(b)(x)=Q_t(b_0)+Q_t(b_1)x+Q_t(b_2)x^2+---

Everything can be computed explicitly from these formulas.
For example, we have Q_t(b_0)=b_0b(t). 
Finally, we have the relation

\psi_n(x)=b_0\phi_n(x)

for every n>0, since the representation \Sigma_n-->B(n-1)
is obtained by cancelling the trivial representation from the 
natural representation \Sigma_n-->B(n).

aj


-------- Message d'origine--------
De: algtop-l-bounces+joyal.andre=uqam.ca at lists.lehigh.edu de la part de Dev Sinha
Date: mer. 15/10/2008 15:30
À: algtop-l at lists.lehigh.edu
Objet : [ALGTOP-L] the (co)homology of \Sigma_n and O(n-1)
 


I have a group cohomology question for the list.  Let R: \Sigma_n \to 
O(n-1) denote the homomorphism given by the reduced regular 
representation. What is known about the effect of BR :  B\Sigma_n \to 
BO(n-1) on cohomology or homology?

I'm more interested in explicit information, in particular how 
Stiefel-Whitney classes evaluate on the Dyer-Lashof-Kudo-Araki algebra, 
but structural theorems would also be of interest.

Thanks,
Dev

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