Segal operations in the algebraic $K$-theory of topological spaces
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 $Σ_n$, we define operations $θ^n \colon A(X) \rightarrow A(X{\times}BΣ_n)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of $E_{\infty}$-maps. Let $ϕ_n \colon A(X{\times}BΣ_n) \rightarrow A(X{\times}EΣ_n) \simeq A(X)$ be the $Σ_n$-transfer. We also develop an inductive procedure to compute the compositions $ϕ_n \circ θ^n$, and outline some applications.
math.AT↗