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

TL;DR

This paper extends Waldhausen's work by defining new operations in the algebraic K-theory of spaces for connected simplicial abelian groups, establishing their structure as E-infinity maps, and providing computational methods and applications.

Contribution

It introduces new operations in algebraic K-theory for spaces, proves their E-infinity structure, and develops an inductive procedure for their composition with transfer maps.

Findings

Operations $ heta^n$ are defined on $A(X)$ for connected simplicial abelian groups.

These operations can be structured as $E_{ olinebreak} _{ olinebreak} olinebreak ext{infinity}$-maps.

An inductive method for computing compositions with transfer maps is developed.

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.