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 E∞ ring
spaces.
Keywords
algebraic K–theory spectrum,
bipermutative category, classifying space,
E∞ ring space, E∞ ring
spectrum, orientation theory, Thom spectrum
