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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?

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1 Answer 1

up vote 13 down vote accepted

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!

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Than you very much for the detailed answer! It's quite surprising that the homotopy class of a map, say in the $MSpin$ case, is determined by the restriction to $MU$. –  Minhyong Kim Aug 23 '10 at 0:14

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