Characteristic power series for maps of E_{\infty} ring spectra

2010-08-22T06:16:37Z

Let me admit right at the outset that I have a very superficial outsider's knowledge of homotopy theory. Nevertheless, I was trying to gain some understanding of Hopkins' ICM lecture 'algebraic topology and modular forms.'

In section 6, he mentions two constructions. To a map

$$\phi: MSpin\rightarrow KO$$

of $E_{\infty}$ ring spectra, he associates a characteristic power series $$K_{\phi}(x)\in \mathbb{Q}[[x]].$$ Similarly, to an $E_{\infty}$-map

$$\psi: MO\langle 8\rangle \rightarrow tmf,$$

he associates a power series $$K_{\psi}(x)\in MF_{\mathbb{Q}}[[x]],$$ where $tmf$ is the topological modular form spectrum and $MF_{\mathbb{Q}}=MF\otimes _{\mathbb{Z}}\mathbb{Q}$ is the ring of modular forms with rational coefficients.

I wonder if someone could give a brief outline of how these associations are carried out. I presume it is something elementary having to do with the homotopy groups of $MSpin$ and $MO\langle 8\rangle$, but I don't quite have the resources right now to track these down.

As usual with questions of this sort, I'm sure my level of ignorance is incongruous with the words I am employing already, but thank you in advance for any tolerant answers or references.

Added:

Maybe I should summarize the point of this question for fellow number-theorists who are too busy to look into the paper. In the notation above, one associates to $\phi$ a characteristic sequence

$$b(\phi)=(b_2, b_4, b_6,\ldots)$$

via the formula

$$\log(K_{\phi}(x))=-2\sum_{n>0} b_n\frac{x^n}{n!}.$$

Incredibly, this procedure sets up a bijection:

homotopy classes of $E_{\infty}$ maps from $MSpin$ to $KO$ $\leftrightarrow$ the set of sequences of rational numbers $(b_i)$ as above that satisfy

(1) $b_n\equiv B_{n}/n \ \ \mod \mathbb{Z}$, where the $B_n$ are the Bernouilli numbers;

(2) for each odd prime $p$ and $p$-adic unit $c$,

$$m\equiv n \ \mod p^k(p-1) \Rightarrow (1-c^n)(1-p^{n-1})b_n \equiv (1-c^m)(1-p^{m-1})b_m \ \mod p^{k+1};$$

(3) for each $2$-adic unit $c$,

$$m\equiv n \ \mod 2^k \Rightarrow (1-c^n)(1-2^{n-1})b_n \equiv (1-c^m)(1-2^{m-1})b_m \ \mod 2^{k+2}.$$

In the case of the homotopy classes of maps from $MO\langle 8\rangle$ to $tmf$, one gets similar congruences involving Eisenstein series instead of their constant terms. Incidentally, perhaps these congruences imply the ones above?

Answer by Charles Rezk for Characteristic power series for maps of E_{\infty} ring spectra

2010-08-22T16:15:12Z

In short, the series $K_\phi$ is the "Hirzebruch characteristic series" which arises in the construction/calculation of genera, and in Hirzebruch-Riemann-Roch. The first few chapters of Manifolds and modular forms by Hirzebruch et al. describe the classical version of this pretty well.

If I have a one dimensional formal group law $F$ over a ring $A$, then over $A_{\mathbb{Q}}$ there is an isomorphism $\mathrm{exp}_F: G_a\to F$ with the additive formal group. Let $K(x)=x/\mathrm{exp}_F(x)$.

Now suppose $R$ is a "complex orientable cohomology theory", which means we are given a suitable isomorphism of rings $R^*(CP^\infty)\approx \pi_*R[[x]]$ . Such a theory has an associated formal group law $F$ (induced by the map $CP^\infty\times CP^\infty\to CP^\infty$ which classifies tensor product of line bundles), and thus there is an assocated series $K(x)=x/\mathrm{exp}_F(x)$ in $\pi_*R_\mathbb{Q}[[x]]$ .

It turns out that a map of ring spectra $\phi:MU\to R$ corresponds exactly to giving a complex orientation of $R$. By Thom, elements of $\pi_*MU$ correspond to cobordism classes of stably-almost-complex manifolds, and there is a standard calculus due to Hirzebruch of calculating the effect of the map $\pi_*MU \to \pi_*R_{\mathbb{Q}}$ using $K(x)$, which I might as well call $K_\phi(x)$, since it depends on $\phi$. The formula (if I remember correctly), is that, if $[M]\in \pi_*MU$ is the class corresponding to a manifold of dimension $2n$, then $$ \phi(M) = \langle K_\phi(x_1)\dots K_\phi(x_n), [M] \rangle, $$ where the $x_i$ are the "chern roots" of the tangent bundle of $M$, and $[M]\in H_{2n}M$ is the fundamental class.

There is a "universal example" of a $K_\phi$, corresponding to the identity map $\phi\colon MU\to MU$. It turns out that $\pi_*MU_{\mathbb{Q}}$ is a polynomial ring on the coefficients of $K_\phi$, so that $K_\phi(x)=\sum a_{i-1}x^i$ (with $a_0=1$) and $\pi_*MU_{\mathbb{Q}}=\mathbb{Q}[a_1,a_2,\dots]$. (I'll need this later.)

In his talk, Mike isn't talking about complex orientations, but rather orientations with respect to $MSpin$ or $MO\langle 8\rangle$ (instead of $MO\langle 8\rangle$, we call it $MString$ these days, for some reason).

There is a map of ring spectrum $MU\to MSO$, induced by the apparent homomorphisms $U(n)\to SO(2n)$ of Lie groups. There is also a map $MSpin\to MSO$, induced by the double cover of lie groups. Although $MSpin\neq MSO$, we have that $\pi_*MSpin_{\mathbb{Q}}\to \pi_*MSO_{\mathbb{Q}}$ is an isomorphsism. Thus, a map $\phi\colon MSpin\to R$ induces $$ \pi_*MU_{\mathbb{Q}}\to \pi_*MSO_{\mathbb{Q}}\approx \pi_*MSpin_{\mathbb{Q}}\to \pi_*R_{\mathbb{Q}},$$ and we can get $K_\phi(x)$ from this.

The $MO\langle 8\rangle$ case is a little trickier. There is a map $MU\langle 6\rangle \to MO\langle 8\rangle$, so a ring spectrum map $\phi\colon MO\langle 8\rangle\to R$ gives rise to a map $$ MU\langle 6\rangle_{\mathbb{Q}} \to R_{\mathbb{Q}}.$$ On the other hand, the effect of the map $MU\langle 6\rangle\to MU$, on homotopy groups tensored with $\mathbb{Q}$, is $$ \mathbb{Q}[a_3,a_4,\dots]\to \mathbb{Q}[a_1,a_2,a_3,\dots].$$ So a map $\phi\colon MO\langle 8\rangle\to R$ gives us elements $\phi(a_i)\in \pi_{2i}R$ for $i\geq3$, which we can use as the coefficients of a series $K_\phi(x)\in \pi_*R_{\mathbb{Q}}$.

I must point out: there is actually an error in the statement of (2) and (3) given in Mike's talk. What he writes down are the "Kummer congruences"; but what one really needs to require are the "generalized Kummer congruences", which are basically the collection of all possible $p$-adic congruences involving Bernoulli numbers, not just the ones listed in (2) and (3). This comes from the theory of the "Mazur measure": the generalized Kummer congruences imply that the sequence $b_n(1-p^{n-1})(1-c^n)$ can be interpolated to a function $f$, so that $f(n)$ for $n$ an integer is the moment of a measure on $\mathbb{Z}_p^\times$. With (2) and (3) replaced by "interpolates to the moments of a measure on $\mathbb{Z}_p^\times$", the result is correct.

Finally: There is a writeup of this at http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf, which may or may not be of any use to you!

Characteristic power series for maps of E_{\infty} ring spectra - MathOverflow

Characteristic power series for maps of E_{\infty} ring spectra

Answer by Charles Rezk for Characteristic power series for maps of E_{\infty} ring spectra