On the definition of A-theory

Question

Waldhausen's A-theory is a version of algebraic K-theory of spaces. Concretely, for a (pointed) space $X$, he considers the 'Waldhausen category' $\mathcal R_f(X)$ of finite retractive CW-complexes over $X$, applies his $S_{\bullet}$ construction to it, and obtains an infinite loop space, $A(X)$. The functor $A$ is extremely important in high-dimensional geometric topology, for instance it prominently features in the definition of $Wh^{\text{Diff}}(X)$, and in the parametrized $h$-cobordism theorem by Waldhausen, Jahren, and Rognes.

For some time, I believed that $A(X)$ could equivalently be described (using modern machinery that wasn't available when Waldhausen developed his theory) as the $K$-theory of the ring spectrum $\sum^{\infty}_+ \Omega X$, suitable interpreted.

This week I learned that while this might work to understand the connected components, it does not give the right description on $\pi_0$: for any connected space $X$, $\pi_0A(X)$ is simply $\mathbb Z$, given by the relative Euler characteristic of the relative cell complex. Moreover, the canonical map $A(X) \to K(\mathbb Z\pi_1(X))$ induces the canonical map $\mathbb Z \to K_0(\mathbb Z\pi_1(X))$ on connected components, and the cokernel of this map is $\tilde{K}_0(\mathbb Z \pi_1)$, which is often non-trivial. (See Wall's finiteness obstruction).

So my question is the following: could $A$-theory have been defined in terms of finitely dominated as opposed to finite relative CW-complexes, and then what I believed actually holds? And we just have to keep in mind that there is this difference on $\pi_0$, but besides that all is fine? Or is there something more substantial going on?

Sorry if this question is maybe rather vague, but it is my feeling that I am not the only one who might be puzzled about this...

Answer 1

Since the question remains unanswered, let me copy Tom Goodwillie's comment:

If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely generated projective modules instead of free, it changes only π0. I believe that this is discussed somewhere in Waldhausen's big foundational paper. And in the EKMM book the corresponding issue for connective ring spectra is discussed.

In more detail, for $X$ a connected space the (∞-)category of perfect $\mathbb{S}[\Omega X]$-modules is the Spanier-Whitehead category of the category of finitely dominated retractive CW-complexes over $X$, and therefore it has the same algebraic K-theory. Restricting to finite retractive CW-complexes over $X$ corresponds to taking the stable subcategory of perfect $\mathbb{S}[\Omega X]$-modules generated by the free ones under colimit, and so by Waldhausen's cofinality theorem it just replaces the $\pi_0$ with $\mathbb{Z}$.

This is worked out in detail in Lecture 21 of Jacob Lurie's course Algebraic K-theory and manifold topology. Note in particular Warning~9 there, where Lurie remarks that his definition of A-theory differs from the "traditional" one only on $\pi_0$.