### Algebraic Cycles and the Classical Groups - Part I, Real Cycles, by H. Blaine Lawson, Jr., Paulo Lima-Filho, and Marie-Louise Michelsohn

The groups of algebraic cycles on complex projective space P(V) are known to
have beautiful and surprising properties. Therefore, when V carries a real
structure, it is natural to ask for the properties of the groups of real
algebraic cycles on P(V). Similarly, if V carries a quaternionic structure,
one can define quaternionic algebraic cycles and ask the same question. In
this paper and its sequel the homotopy structure of these cycle groups is
completely determined. It turns out to be quite simple and to bear a direct
relationship to characteristic classes for the classical groups. It is
shown, moreover, that certain functors in K-theory extend directly to these
groups. It is also shown that, after taking colimits over dimension and
codimension, the groups of real and quaternionic cycles carry E_{\infty}-ring
structures, and that the maps extending the K-theory functors are
E_{\infty}-ring maps. In fact this stabilized space is a product of
(Z/2Z)-equivariant Eilenberg-MacLane spaces indexed at the representations
R^{n,n} for n \geq 0. This gives a wide generalization of the results in
[\BLLMM] on the Segal question. The ring structure on the homotopy groups of
these stabilized spaces is explicitly computed. In the real case it is a
simple quotient of a polynomial algebra on two generators corresponding to
the first Pontrjagin and first Stiefel-Whitney classes. These calculations
yield an interesting total characteristic class for real bundles. It is a
mixture of integral and mod 2 classes and has nice multiplicative properties.
The class is shown to be the (Z/2Z)-equivariant Chern class on Atiyah's
KR-theory.

H. Blaine Lawson, Jr. <blaine@math.sunysb.edu>

Paulo Lima-Filho <plfilho@math.tamu.edu>

Marie-Louise Michelsohn <mmichelsohn@math.sunysb.edu>