Chromotopy

August 24, 2011, by Eric Peterson

The summer is coming to a close, and so I’ve been trying to summarize some of the things that have happened. For one, I did manage to successfully perform the computation of $(E_n)^\vee_* K(\mathbb{Z}/p^j, q)$ for all conceivable values of n, p, j, and q, and the whole thing did culminate, as expected, in an isomorphism of graded ring schemes

$\mathbb{C}\mathrm{P}^\infty_E[p^\infty]^{\wedge *} \cong K(\mathbb{Z}/p^\infty, *)_E.$

So, that’s pretty exciting. But some other things happened too, and here’s one I’ve been meaning to complain about for a while. For the most part, this idea of associating to a compactly generated space X the formal scheme $X_E := \mathrm{colim}_\alpha E^* X_\alpha$ , where $X_\alpha$ ranges over the compact subspaces of X, is a mechanism for introducing the language and ideas of algebraic geometry into algebraic topology. The first nontrivial example that I learned in this framework is the Ando-Hopkins-Strickland result (or, rather, the Cartier dual of what I’m about to say) that the formal scheme $BU\langle 2k \rangle_E$ corresponds to the kth symmetric power of the augmentation ideal of a “group ring” type object $C_0 \mathbb{C}\mathrm{P}^\infty_E$ , for all kinds of theories E and at least in the range $k \le 3$ . A big part of their proof was an explicit analysis of the schemes $C_k$ , especially in the case of k = 3, where chasing out the definition of $C^3 := \mathrm{Hom}(C_3 \mathbb{C}\mathrm{P}^\infty_E, \mathbb{G}_m)$ shows that it is the scheme of “cubical structures”.

In topology, there is a fiber sequence $K(\mathbb{Z}, 3) \to BU \langle 6 \rangle \to BSU$ , which for a p-local cohomology theory E gives at the very least a map

$\mathrm{Hom}(K(\mathbb{Z}/p^\infty, 2)_E, \mathbb{G}_m) = \mathrm{Hom}(K(\mathbb{Z}, 3)_E, \mathbb{G}_m) \leftarrow \mathrm{Hom}(BU\langle 6 \rangle_E, \mathbb{G}_m).$

But, at least for specific choices of E like the one in my opening unrelated paragraph, we know that $K(\mathbb{Z}/p^\infty, 2)_E \cong (B\mathbb{Z}/p^\infty_E)^{\wedge 2}$ , and so we expect the map above to mimic a map in algebraic geometry of the form

$\mathrm{Hom}((B\mathbb{Z}/p^\infty_E)^{\wedge 2}, \mathbb{G}_m) \leftarrow C^3(\mathbb{C}\mathrm{P}^\infty_E, \mathbb{G}_m),$

i.e., something that swallows cubical structures and outputs alternating biexponential maps. Such an object is known to the enterprising algebraic geometer: the Weil pairing. This object was crucially used by Mumford in his study of biextensions, and has received all kinds of press before and since.

To give you the dimmest sense of what this is, if you’re unfamiliar with it already, I’ll outline its construction. A biextension over a pair of group schemes G and H is a $\mathbb{G}_m$ -torsor over $G \times H$ , such that if you pick any point $g \in G$ or $h \in H$ , the pullback of the torsor along the inclusions $\{g\} \times H \hookrightarrow G \times H$ and $G \times \{h\} \hookrightarrow G \times H$ respectively is an honest split central extension of the base by $\mathbb{G}_m$ . Each of these central extensions has a controlling 2-cocycle determining the isomorphism type of the extension, and a “cubical structure” is a biextension for which $G = H$ , all these 2-cocycles are equal, and the 2-cocycle is $\Sigma_3$ -invariant. Beginning with a cubical structure $\mathcal{L}$ , we can consider the multiplication-by-p maps $p_L$ and $p_R$ in the left- and right-hand factors of the base respectively. These maps factor as

$\mathcal{L} \xrightarrow{i} \mathcal{L}^{\otimes p} \xrightarrow{\mu_L} \mathcal{L},$

and similarly for $p_R$ . These two maps $i$ and $\mu_L$ themselves factor as $\mathcal{L} \to p_* \mathcal{L} \to \mathcal{L}^{\otimes p}$ and $\mathcal{L}^{\otimes p} \to (p \times 1)^* \mathcal{L} \to \mathcal{L}$ . The two factoring maps $p_* \mathcal{L} \to \mathcal{L}^{\otimes p}$ and $\mathcal{L}^{\otimes p} \to (p \times 1)^* \mathcal{L}$ can be checked to be isomorphisms, and finally the composite

$(p \times 1)^* \mathcal{L} \to p_* \mathcal{L} \to (1 \times p)^* \mathcal{L}$

is the thing called the “Weil pairing”. Over the p-torsion in the base, at least, the two pullback torsors trivialize canonically, and so we can write down a formula for the action of the composite map in the $\mathbb{G}_m$ factor:

$e_n = \prod_{i=1}^{p-1} \frac{u(x_1, ix_1, x_2)}{u(x_1, ix_2, x_2)},$

where u is the controlling 2-cocycle. And, finally, the real punchline: this is exactly the map induced by the topological fiber $K(\mathbb{Z}, 3) \to BU\langle 6 \rangle$ .

Of course, the topological story doesn’t stop there. At the next stage in the Whitehead-Postnikov decomposition there’s another fiber sequence $K(\mathbb{Z}, 5) \to BU\langle 8 \rangle \to BU\langle 6 \rangle$ , and we can attempt to play a similar game by pplying the functor $\mathrm{Hom}(-_E, \mathbb{G}_m)$ to the fiber map. Again, assuming that E is a nice cohomology theory, the object $\mathrm{Hom}(K(\mathbb{Z}, 5), \mathbb{G}_m)$ is known: it’s the scheme of alternating, multiexponential functions on the p-divisible subgroup $\mathbb{C}\mathrm{P}^\infty_E[p^\infty]$ of the formal group associated to the complex orientability of E-theory. Hughes, Lau, and I have calculated the ring of functions on the affine scheme $C^k(\mathbb{G}_a, \mathbb{G}_m) \times \mathrm{Spec}(\mathbb{Z}_{(2)})$ , and computational experiment suggests that, further pulling back to $\mathrm{Spec}(\mathbb{F}_2)$ , the Ando-Hopkins-Strickland map $\mathcal{O}_{C^k} \otimes \mathbb{F}_2 \to H_*(BU \langle 2k \rangle; \mathbb{F}_2)$ is very close to being an isomorphism, and the amount it misses it by is exactly the closure of the odd-dimensional parts of the homology under the coaction of the dual Steenrod algebra. This is a bit of a mess of a conjecture, but by hook or by crook we seem to end up with a map from the $(k-1)$ -dimensional analogues of cubical structures to $(2k-4)$ -variate alternating, multiexponential maps.

The real question is: where is this guy in the algebraic geometry literature? The construction used above has no obvious higher dimensional analogue. An equivalent phrasing of the question is: given two line bundles $L_1$ and $L_2$ , we can consider their virtual differences $(1-L_1)$ and $(1 - L_2)$ , which we can put together in creative ways to get a virtual bundle whose classifying map lifts to $BU\langle 6 \rangle$ . These creative ways specifically look rather like the formula for $e_n$ , inexorably leading you toward the Weil pairing story. But, given four line bundles, how do we build a bundle with good properties whose classifying map lifts to $BU\langle 8 \rangle$ ? I, for one, have no idea.