Stable completed homology without Quillen-Lichtenbaum

Having just made (hopefully) the final revisions on my paper on stable completed cohomology groups, I wanted to record here a few remarks which didn’t otherwise make it into the paper.

The first is that, in addition to the result that \(\widetilde{H}_2(\mathrm{SL},\mathbf{Z}_p) = \mathbf{Z}_p\) for \(p > 2,\) one may also compute \(\widetilde{H}_3(\mathrm{SL},\mathbf{Z}_p)\) for \(p > 3.\) Namely:

\(\displaystyle{\widetilde{H}_3(\mathrm{SL},\mathbf{Z}_p) = 0.}\)

This result is proved in the paper up to a finite group, so the point here is the integral refinement. The computation of \(\widetilde{H}_2\) comes from the Hurewicz isomorphism

\(\displaystyle{\pi_2(SK(\mathbf{Z},\mathbf{Z}_p);\mathbf{Z}_p) \simeq \widetilde{H}^{\mathrm{cont}}_2(\mathbf{Z}_p)}.\)

However, the Hurewicz theorem also gives an epimorphism

\(\displaystyle{\pi_3(SK(\mathbf{Z},\mathbf{Z}_p);\mathbf{Z}_p) \rightarrow \widetilde{H}^{\mathrm{cont}}_3(\mathbf{Z}_p)},\)

and one finds that the first group lives in an exact sequence

\(H^2(\mathbf{Q}_p,\mathbf{Z}_p(2)) \rightarrow \pi_3(SK(\mathbf{Z},\mathbf{Z}_p);\mathbf{Z}_p) \rightarrow K_3(\mathbf{Z}) \otimes \mathbf{Z}_p\)

Since both flanking groups vanish for \(p > 3\), the middle group is zero, and the claim follows.

The second remark is that, throughout the paper, I assume the Quillen-Lichtenbaum conjecture, which is now a theorem due do Voevodsky and others. However, I must confess, I do not have the fine details of the argument at my fingertips. How much can one say without it? The answer is quite a lot. Due to work of Borel, Soulé, and Quillen (all of which is much more familiar to me, at least relatively speaking), we know that the \(K\)-groups of number fields are finitely generated abelian groups, we know their ranks, and we know that the Chern class maps to the appropriate Galois cohomology groups are surjective. Moreover, we understand \(K_2(\mathscr{O}_F)\) completely in terms of Galois cohomology by work of Tate. (In this game, I am also giving up the results of Hesselholt and Masden on the \(K\)-theory of local fields, and instead using the results of Wagoner, which similarly give everything in very small degree and up to a finite group in higher degrees.)

In particular:

1. The computation of \(H_2(\Gamma_N(p),\mathbf{F}_p)\) for large \(N\), where \(\Gamma_N(p)\) is the principal congruence subgroup of \(\mathrm{GL}_N(\mathbf{Z})\), is unaffected. This also uses the computation of \(K_3(\mathbf{Z})\) by Lee and Szczarba.
2. The identification of the completed \(K\)-groups with Galois cohomology groups still holds up to a finite group.
3. The computation of the rational stable completed homology groups\(\widetilde{H}_*(\mathrm{SL},\mathbf{Z}_p) \otimes \mathbf{Q} = \mathbf{Q}[x_2,x_6,x_{10},x_{14}, \ldots ]\)under the assumption that either \(p\) is regular or \(\zeta_p(3), \zeta_p(5), \zeta_p(7)\) etc. are all non-vanishing still holds.

Something that does require Quillen-Lichenbaum is the vanishing of the partially completed \(K\)-group for very regular primes.

Regarding the computation of the rational stable completed homology groups, the referee made a very interesting point (I will come back in a later post to the refeering of this paper and some other of my recent papers in a post on “what a great referee report should be”). I prove that the rational stable completed homology groups are the continuous homology of the homotopy fibre

\(SK(\mathbf{Z},\mathbf{Z}_p) \rightarrow SK(\mathbf{Z}) \rightarrow SK(\mathbf{Z}_p)\)

(The definition of \(SK(\mathbf{Z},\mathbf{Z}_p)\) is just homotopy fibre of this map.) Now \(SK(\mathbf{Z},\mathbf{Z}_p)\) is an infinite loop space, which under the assumption that \(p\) is regular or on the non-vanishing of the \(p\)-adic zeta function at integral arguments, has the property that the homotopy groups with coefficients \(\pi_n(SK(\mathbf{Z},\mathbf{Z}_p);\mathbf{Z}_p)\) are rationally non-zero in exactly degrees \(2,6,10\), etc. The referee noted that the computation of rational stable completed homology should follow precisely from this description using the Milnor–Moore theorem, which shows that (for simply connected \(H\)-spaces) that the homology is (rationally) the universal enveloping algebra of the rational homotopy classes (and so, in particular, the Hurewicz map is rationally injective). One consequence is that the rational homotopy groups are precisely the primitive classes in rational homology. To orient the reader, this is exactly the theorem which allowed Borel to compute the rational \(K\)-groups of (rings of integers) of number fields from his computation of stable homology over \(\mathbf{Q}\). Now I was a little worried about this, because the Milnor–Moore theorem does not literally apply, since one is comparing here homotopy groups with coefficients in \(\mathbf{Z}_p\) and continuous homology (the latter is just the inverse limit of homology groups modulo \(p^n\)). However, having looked at the argument in Milnor–Moore and then having Paul Goerss explain it to me, the argument does indeed seem to simply work in this case. (Warning, this is a weaker statement than saying I checked the details.)

To be more precise, suppose that \(G\) is a simply connected infinite loop space, and suppose that \(G\) has the property that the groups \(\pi_n(G;\mathbf{Z}/p^k)\) are finite for all \(n\) and \(k\), so \(\pi_n(G;\mathbf{Z}_p)\) is the inverse limit of these groups. There is a pairing

\([,]: \quad \pi_r(G,\mathbf{Z}/p^k) \otimes \pi_s(G,\mathbf{Z}/p^k)\rightarrow \pi_{r+s}(G,\mathbf{Z}/p^k),\)

which, after taking inverse limits in \(k\) and tensoring with \(\mathbf{Q}\), makes \(\pi_*(G,\mathbf{Z}_p) \otimes \mathbf{Q}\) into a Lie algebra over \(\mathbf{Q}_p\), then the Hurewicz map will induce an isomorphism

\(U(\pi_*(G,\mathbf{Z}_p) \otimes \mathbf{Q}) \rightarrow H^{\mathrm{cont}}_*(G,\mathbf{Q}_p):= \lim H_*(G,\mathbf{Z}/p^k) \otimes \mathbf{Q}\)

of Hopf algebras. The key technical point required here is to define the appropriate pairing on homotopy groups with coefficients, which is done by Neisendorfer. (If \(G\) is simply connected infinite loop space, one doesn’t have to worry about the issue of homotopy groups with coefficients in very low degree exhibiting certain pathologies.)

As another example of this, one can take \(G = SK(\mathbf{Z}_p)\). In this case, the rational continuous homology reduces, by work of Lazard, to lie algebra cohomology, and gives an exterior algebra in odd degrees \(> 1\). So \(SK_n(\mathbf{Z}_p;\mathbf{Z}_p) \otimes \mathbf{Q}\) has dimension one in odd degrees \(> 1\) and is zero for all even positive degrees. This is a result of Wagoner. In fact, Wagoner proves something slightly stronger, also capturing some information away from \(p\). To do this, he also proves a version of the Milnor–Moore theorem, but his assumptions are more stringent than what we discuss above.