The First Differential of the Functor "Algebraic K-Theory of Spaces"

TL;DR

This paper computes the first differential of Waldhausen's algebraic K-theory functor for spaces, extending previous splitting results to a calculus framework, providing new insights into its local behavior.

Contribution

It adapts Waldhausen's splitting proof to calculate the differential of A(X) in Goodwillie's calculus, a novel application in algebraic K-theory of spaces.

Findings

Calculated the first differential of A(X) at any path-connected space

Extended Waldhausen's splitting to a differential calculus context

Provided explicit methods for local analysis of algebraic K-theory functor

Abstract

In his "Algebraic K-theory of topological spaces II" Waldhausen proved that his functor A(X) splits: There is a canonical map from the stable homotopy of X which has a retraction up to weak equivalence. We adapt Waldhausen's proof to obtain a calculation of the Differential (in the sense of Goodwillie's "Calculus I") of A(X) at any path-connected base space.