What are $E_{\infty}$ ring spaces good for?

TL;DR

This paper explores the significance of $E_{inity}$ ring spaces in connecting infinite loop space theory with geometric topology and algebraic K-theory, highlighting their roles in solving calculational problems and understanding algebraic structures.

Contribution

It clarifies the central role of $E_{inity}$ ring spaces in linking infinite loop space theory with topology and algebraic K-theory, and discusses their applications.

Findings

Infinite loop space theory aids in calculating characteristic classes.

$E_{inity}$ ring spaces are crucial for understanding algebraic K-theory.

The approach simplifies complex calculations in geometric topology.

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 E infinity ring spaces.