Title: Algebra over the Steenrod algebra, lambda-ring, and Kuhn's Realization Conjecture
Author: Donald Yau
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street
Urbana, IL 61801
dyau@math.uiuc.edu
In this paper we study the relationships between operations in
$K$-theory and ordinary mod $p$ cohomology. In particular, conditions
are given under which the mod $p$ associated graded ring of a filtered
$\lambda$-ring is an unstable algebra over the Steenrod algebra. This
result partially extends to the algebraic setting a topological result
of Atiyah about operations on $K$-theory and mod $p$ cohomology for
torsionfree spaces. It is also shown that any polynomial algebra that
is an algebra over the Steenrod algebra can be realized as the mod $p$
associated graded of a filtered $\lambda$-ring. Another observation is
that Atiyah's result gives rise to a $K$-theoretic analogue of Kuhn's
Realization Conjecture concerning the size of spaces in cohomology.