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 $\Sigma_n$, we define operations $\theta^n \colon A(X) \rightarrow A(X{\times}B\Sigma_n)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of $E_{\infty}$-maps. Let $\phi_n \colon A(X{\times}B\Sigma_n) \rightarrow A(X{\times}E\Sigma_n) \simeq A(X)$ be the $\Sigma_n$-transfer. We also develop an inductive procedure to compute the compositions $\phi_n \circ \theta^n$, and outline some applications.