# On transfer maps in the algebraic $K$-theory of spaces

## TL;DR

This paper investigates the naturality properties of the Waldhausen trace map in algebraic K-theory of spaces, establishing its compatibility with transfer maps up to weak homotopy, thus linking algebraic K-theory and stable homotopy.

## Contribution

It proves that the Waldhausen trace map is natural up to weak homotopy with respect to transfer maps, clarifying its functorial behavior in algebraic K-theory.

## Key findings

- Waldhausen trace map is natural up to weak homotopy
- Compatibility with transfer maps established
- Links algebraic K-theory to stable homotopy through transfer maps

## Abstract

We show that the Waldhausen trace map $\mathrm{Tr}_X \colon A(X) \to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to \emph{weak} homotopy with respect to transfer maps in algebraic $K$-theory and Becker-Gottlieb transfer maps respectively.