Segal operations in the algebraic $K$-theory of topological spaces

Abstract

We extend earlier work of Waldhausen which defines operations on the algebraic $K$-theory of the one-point space. For a connected simplicial abelian group $X$ and symmetric groups ${\mathrm{\Sigma }}_n$, we define operations ${\theta }^n:A\left(X\right)\to A\left(X\times B{\mathrm{\Sigma }}_n\right)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of $E_{\infty }$-maps. Let ${\varphi }_n:A\left(X\times B{\mathrm{\Sigma }}_n\right)\to A\left(X\times E{\mathrm{\Sigma }}_n\right)\simeq A\left(X\right)$ be the ${\mathrm{\Sigma }}_n$-transfer. We also develop an inductive procedure to compute the compositions ${\varphi }_n\circ {\theta }^n$, and outline some applications.