DVI FILE: bjw.dvi
TITLE: Unstable operations in generalized cohomology
AUTHORS: J. Michael Boardman, David Copeland Johnson, W. Stephen Wilson
TO APPEAR: Handbook of Algebraic Topology, ed. I.M.James,
Elsevier (Amsterdam, 1995).
We describe the structure of the unstable operations on E-cohomology
in terms of comonads, in the style of the companion paper on stable
operations. There are two variants, depending on whether we consider only
the additive operations, or all unstable operations. For practical use, we
unpack the comonad information and express it in terms of Hopf rings. Five
examples are discussed: ordinary mod p cohomology, unitary cobordism MU,
Brown-Peterson BP-cohomology, complex K-theory KU, and Morava K-theory K(n).
We give two applications to BP-cohomology. The first shows that the
presence of unstable operations imposes dimensional restrictions on the
Landweber filtration of the BP-cohomology of a finite complex. The second
constructs idempotent operations in degree k that recover the known
unstable splittings of BP-cohomology.