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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...

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    $\begingroup$ If you allow finitely dominated instead of finite, it changes only $\pi_0$. Analogously, in defining $K(R)$ if you use finitely generated projective modules instead of free, it changes only $\pi_0$. 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. $\endgroup$ Oct 29, 2020 at 14:04
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    $\begingroup$ Waldhausen mentioned the variation with finitely dominated spaces on page 389 of his foundational paper math.uni-bielefeld.de/~fw/LNM1126_318-419.pdf . Getting \pi_0 right plays a bigger role in the papers by Huettemann, Klein, Vogell, Waldhausen and Williams on the Fundamental Theorem. See Lemma 1.7(3) of their 2001 JPAA paper. $\endgroup$ Oct 31, 2020 at 21:57

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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 $\pi_0$.

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  • $\begingroup$ So, is the SW category of finitely dominated retractives over $X$ a model for the $K$-theory of $\Sigma_+^\infty(\Omega X)$? I mean, $\mathbb S[\Omega X]$ is still not literally the same as $\Sigma_+^\infty(\Omega X)$, right? Or is the comparison between them trivial? $\endgroup$ Oct 30, 2020 at 9:25
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    $\begingroup$ @მამუკაჯიბლაძე $\mathbb{S}[\Omega X]$ is just the notation for $\Sigma^\infty_+(\Omega X)$ when we see it equipped with the canonical $E_1$-ring structure, in the same way $\mathbb{Z}[G]$ is the standard notation for the free abelian group on the underlying set of a group $G$ when we see it as a ring. To be clear, the answer to your first question is yes. $\endgroup$ Oct 30, 2020 at 9:32
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    $\begingroup$ Normally, I think the best practice in situations like this is to make the answer CW, so one person doesn't get reputation points for an answer written by a different person in the comments. Not that reputation points really matter in life. $\endgroup$ Oct 30, 2020 at 13:44
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    $\begingroup$ @DavidWhite I don't particularly care for the points, so I'm going to convert the answer to CW. I thought that this applies only when one just copypastes the comment and adds nothing though.. But not worth debating, they're only fake internet points :) $\endgroup$ Oct 30, 2020 at 15:48
  • $\begingroup$ @DenisNardin, wait, what? We can't redeem internet points for coffee or anything? :) $\endgroup$ Dec 11, 2021 at 1:43

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