What are E∞ring spaces good for?

Abstract

Infinite loop space theory, both additive and multiplicative, arose largely from two basic motivations. One was to solve calculational questions in geometric topology. The other was to better understand algebraic K–theory. The Adams conjecture is intrinsic to the first motivation, and Quillen’s proof of that led directly to his original, calculationally accessible, definition of algebraic K–theory. In turn, the infinite loop understanding of algebraic K–theory feeds back into the calculational questions in geometric topology. For example, use of infinite loop space theory leads to a method for determining the characteristic classes for topological bundles (at odd primes) in terms of the cohomology of finite groups. We explain just a little about how all that works, focusing on the central role played by E1 ring spaces.