University of Georgia
Mathematics Department Colloquium
John H. Palmieri
University of Notre Dame
February 2th 1999
Stable homotopy theory and the Steenrod algebra
Abstract:
Over the past half-century, topologists have developed many powerful
tools for studying stable homotopy theory. One such tool is the Adams
spectral sequence; together with mod p homology, it provides a strong
link between stable homotopy theory and the representation theory of a
certain Hopf algebra, the mod p Steenrod algebra. It turns out that
one can use many stable homotopy theoretic tools in other settings:
Hovey, Strickland and I have developed axioms extending them to, for
example, the representation theory of Hopf algebras. This provides a
second, more formal, connection between stable homotopy and the
Steenrod algebra.
Around thirty years ago, Quillen proved an important result in group
cohomology; this result has revolutionized modular representation
theory. Recently, I have proved the analogous result for the Steenrod
algebra by using the homotopy theoretic methods provided by the
axioms. In this talk, I will discuss the some of the basic features
of stable homotopy theory; I will describe the Steenrod algebra
analogue of Quillen's result; and I will explain how to use the
axiomatic approach to prove it.