TRANSFER AND COMPLEX ORIENTED COHOMOLOGY RINGS

           MALKHAZ BAKURADZE AND STEWART PRIDDY
  

Keywords:  transfer, Chern class, classifying space, complex cobor-

dism, Morava K-theory 55N22, 55R12.

                         1.  Introduction


  Let p be a prime and let G be a subgroup of the symmetric
group S_p. In this paper we use the transfer to study homotopy orbit spaces
X^p_hG= EG x_G X^p in complex oriented cohomology. We are particularly
interested in computing the ring structure. Thus we are led to consider
the relation between cup products and transfer known as Fröbenius
reciprocity by analogy with representation theory

                       Tr*(x)y = Tr*(x rho*(y))

(formula (i) of Section 2) where rho : EG x X^p --> X^p_hG is the covering
projection and
                      Tr* : E*(X^p) ---> E*(X^p_hG)

is the associated transfer homomorphism. It is worth noting that the
multiplicative structure of the cohomology groups we consider is com-
pletely determined by this formula.

  In case E = K(s) is Morava K-theory, G is cyclic of order p, and
X is the classifying space of a finite group, Hopkins-Kuhn-Ravenel [11 ]
have studied these cohomology groups as modules over the coefficient
ring.  Our paper builds on their approach by extending their notion
of a good group to spaces.  For X = CP^infty we determine the algebra
K(s)*(X^p_hS_p) for Morava K-theory; for complex cobordism we compute
the ring MU*(X^p_hS_p) making additional use of the formal group law.
This enables us to make explicit computations of the transfer in both
cases. In an analogous fashion we compute the algebra BP *(X^p_hS_p).

  The starting point and original motivation for our work comes from
Quillen's famous formula for Tr*(1), the stable Euler class, for the uni-
versal Z/p covering.  As explained in Section 2, our results for CP^infty
provide a universal example which enable us to compute the stable Eu-
ler classes and the transfer in general for many other cases. For example
universal coverings for some nonabelian p-groups, namely those with
cyclic subgroups of index p and those which are semi-direct products
of elementary abelian p-groups with Z/p.