On the trace of KZ (after John Rognes (after Marcel Böckstedt))
Part 0: Proof skeleton
While reading the Ausoni-Rognes computation of “algebraic $K$-theory of topological $K$-theory”, one of the first steps they take is define a certain class in algebraic $K$-theory by pulling it through the trace map, which relies on the surjectivity of that trace map in some degree. My goal in these posts is to detail a proof by Rognes for this innocent-looking theorem of Böckstedt.
Theorem: $\tr : K_{2p-1}(\Z) \to \THH_{2p-1}(\Z)$ is surjective on $p$-torsion, for any prime $p$.
The theorem may seem a bit “dry” or arbitrary, and without any context, I guess it sort of is. One way in which it’s interesting is that $2p-1$ is the first degree in which $\THH(\Z)$ has nontrivial torsion. In general we’d be interested in the behavior of the trace map on higher degrees too, and nowadays there’s a lot more information known about it (largly, but not only, due to the later works of Rognes), but this is the first & easiest step. More interestingly to me is the geometric method behind its proof – this fairly algebraic result builds on some deep facts from manifold topology.
I will assume here that we know how to compute the codomain. In particular, I will assume $\THH_{2p-1}(\Z) \iso \Z/p$ and this is the first $p$-torsion in $\pi_*\THH(\Z)$. Henceforth all homotopy & homology groups will be implicitly $p$-localized. All throughout, unless mentioned otherwise, whenever we say that a spectrum is “trivial” or a morphism of spectra is “trivial” we really mean “trivial on bottom-most $p$-torsion”, which will be in degree $2p-2$ for most (though not all) spaces and spectra considered below. Likewise for other adjectives, such as “injective”, “surjective”, “isomorphism”, “cyclic”, and so on.
The proof skeleton
If $F$ is a functor, we will generally denote $F(X \to Y) := \hofib(F(X) \to F(Y))$. This applies in particular to the linearlization map $\ell : \S \to \Z$,1 which gives fiber sequences both in $K$-theory and $\THH$. By naturality, their connecting homomorphisms (vertical faces in the diagram) commute with the trace map:
Our claim, surjectivity of $\tr$ on $\pi_{2p-1}$, is equivalent to the top face $\Omega\tr$ being surjective on $\pi_{2p-2}$. Recall $\pi_{2p-2}$ of the top-right corner is $\Z/p$. We will see later that $\pi_{2p-2}$ of the top-left corner is also nontrivial, so to prove surjectivity of $\Omega\tr$ it’d suffice to show that the diagonal composite is nonzero.
This is done using the lower triangle in the diagram; namely it’d suffice to show that both the bottom face is nonzero and left face is surjective – we’ll do this in Part 1 and Part 2, respectively.
I briefly outline how to prove each part. For Part 1,
- We will approximate the relative $K$-theory via a certain map $j : B\SL_1\S \to K(\S\to\Z)$2. The domain is more well-understood, and it has nontrivial $p$-torsion in degree $2p-2$, so if we can show the whole composite $\tr\circ j$ is nonzero, that would imply $\tr$ is nonzero.
- For this, we will use the fact that $\tr$ factors (basically by definition) through a variant of $K$-theory, namely the cyclic $K^{cyc}$. Consequently we obtain an analogous approximate factorization $B\SL_1\S\to B^{cyc}\SL_1\S \to \THH(\S\to\Z)$3. This will be easier to study – the first map has an easily-constructed retraction, so it’s (split) injective through any functor.
- As for the second map, both the domain and codomain will have fairly computable homotopy in degree $2p-2$ and both turn out to be just $\Z/p$. To prove that the map in question is an isomorphism between those, we will exploit its equivariance with respect to an appropriate circle action.
The proof of Part 2 is more involved. We start from the long exact sequence in homotopy (whose connecting map is exactly the left face of our square)
\[\cdots \xto{} \pi_{2p-1} K(\S\to\Z) \xto{} \pi_{2p-1}K(\S) \xto{} \pi_{2p-1}K(\Z) \xto{\del} \pi_{2p-2}K(\S\to\Z) \xto{} \pi_{2p-2}K(\S) \xto{} \pi_{2p-2}K(\Z) \xto{} \cdots\]To prove surjectivity of $\del$, an obvious idea thanks to exactness is to prove the map after it is zero, or equivalently, the map $\pi_{2p-2}K(\S) \to \pi_{2p-2}K(\Z)$ is injective. However it is not entirely clear, at least to me, how such argument can be made. Following Waldhausen, we will show vanishing of $K(\S\to\Z) \to K(\S)$ by a direct computation. As before, this computation is carried out using the approximation map $j : B\SL_1\S \to K(\S\to\Z)$.
- Our first task will be to study the map $j$ in more detail, and in particular, we will show that it induces a surjection on bottom-most $p$-torsion. Therefore it’d suffice to show that the composition $B\SL_1\S \to K(\S\to\Z) \to K(\S)$ is trivial.
- Recall the equivalence $K(\S) \equiv A(\ast)$ between the algerbaic $K$-theory of $\S$ and Waldhausen’s algebraic $A$-theory of the one-point space. Waldhausen proved some remarkable properties of his $A$ functor, a fundamental one among them being the splitting \(A(X) \equiv \S_*(X) \times \mathrm{Wh}^{DIFF}(X)\). Thus vanishing of the map $B\SL_1\S \to A(\ast)$ is reduced to the vanishing of each factor $B\SL_1\S \to \S_*(\ast)$ and $B\SL_1\S \to \mathrm{Wh}^{DIFF}(\ast)$ separately.
- The first factor is trivial (on $p$-torsion, in degree $2p-2$) by degree considerations. To deal with the second factor we need yet another big result of Waldhausen: vanishing of the composite $BO \xto{J} B\SL_1\S \xto{} A(\ast) \to \mathrm{Wh}^{DIFF}(\ast)$ where $J$ is the usual $J$-homomorphism. With this in mind, it just remains to apply the well known fact, that the $J$ homomorphism is surjective on bottom-most $p$-torsion.
Footnotes
-
Super technically, I should write here $\S_{(p)} \to H\Z_{(p)}$. ↩
-
I should be writing here the underlying infinite loop space of the $K$-theory spectrum. Throughout the article I’ll be a bit uncareful and conflate the two, hopefully without causing too much confusion. ↩
-
Again, I should really be writing the underlying loopspace of the $\THH$ spectrum. ↩