# Is there a Wall finiteness obstruction in other settings?

Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW complexes (i.e., the smallest subcategory of $\mathcal{S}$ containing $\ast$ and closed under finite homotopy colimits). Namely, given $X$ (let's say connected), the homology of the universal cover defines a perfect complex of $\mathbb{Z}[\pi_1 X]$-modules, which gives a class in $\widetilde{K}_0( \mathbb{Z}[\pi_1 X])$, which vanishes when $X$ is a finite CW complex. This class is the Wall finiteness obstruction; it is a classical result that this is the only obstruction for $X$ to be a finite CW complex. The reason for its existence is that taking the "image" of an idempotent is not a finite homotopy colimit (though it would be if one worked with $n$-categories for some $n$).

I'm curious about other examples of this type of finiteness obstruction. For instance, what happens if we do this in (for example) the $(\infty, 1)$-category of $E_\infty$-rings? The analogs of the finite CW complexes are the $E_\infty$-rings obtained from the free ones (free on a generator in some degree) via finite colimits, and the compact objects are these and the retracts. Is there an analog of the finiteness obstruction here? Other examples: algebras over some other operad, simplicial commutative rings, (pre)sheaves of spaces (resp. spectra), and equivariant spaces (resp. spectra). The only example that I am aware of considers the derived category of modules over a ring, where $\widetilde{K_0}$ measures the difference between "perfect complexes" and "those representable via finite complexes of frees."

• That's an intriguing characterization of the finite CW complexes! Do you have a reference for it? Aug 27, 2013 at 6:49
• @Qiaochu: Here's a link to Wall's paper: math.uchicago.edu/~shmuel/tom-readings/…. Andrew Ranicki gives more references below. I also blogged about it at one point, see amathew.wordpress.com/2012/09/22/…. Aug 27, 2013 at 13:50
• Ah, sorry, by "that" I meant "the smallest subcategory of $S$ containing $\bullet$ and closed under finite homotopy colimits." Aug 27, 2013 at 21:55
• @QiaochuYuan: well, the operation of building up a CW complex via attaching cells is essentially that of forming iterated (finite) homotopy pushouts, and vice versa. (This argument works for the derived category of modules over a ring, too.) Aug 27, 2013 at 23:13
• Have you looked at Andy Baker's papers on TAQ? He approaches TAQ by thinking of it as a sort of cellular homology theory (which is an awesome perspective). See the paper by Baker Gilmour and Reinhard first, then the computational approach ones on the arxiv. Sep 4, 2013 at 9:34