Characteristic power series for maps of E_{\infty} ring spectra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:53:03Z http://mathoverflow.net/feeds/question/36362 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36362/characteristic-power-series-for-maps-of-e-infty-ring-spectra Characteristic power series for maps of E_{\infty} ring spectra Minhyong Kim 2010-08-22T06:16:37Z 2010-08-22T16:26:06Z <p>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 <a href="http://arxiv.org/abs/math/0212397" rel="nofollow">'algebraic topology and modular forms.'</a> </p> <p>In section 6, he mentions two constructions. To a map </p> <p>$$\phi: MSpin\rightarrow KO$$</p> <p>of $E_{\infty}$ ring spectra, he associates a characteristic power series $$K_{\phi}(x)\in \mathbb{Q}[[x]].$$ Similarly, to an $E_{\infty}$-map </p> <p>$$\psi: MO\langle 8\rangle \rightarrow tmf,$$</p> <p>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.</p> <p>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. </p> <p>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.</p> <p>Added:</p> <p>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 </p> <p>$$b(\phi)=(b_2, b_4, b_6,\ldots)$$</p> <p>via the formula</p> <p>$$\log(K_{\phi}(x))=-2\sum_{n>0} b_n\frac{x^n}{n!}.$$</p> <p>Incredibly, this procedure sets up a bijection:</p> <p>homotopy classes of $E_{\infty}$ maps from $MSpin$ to $KO$ $\leftrightarrow$ the set of sequences of rational numbers $(b_i)$ as above that satisfy</p> <p>(1) $b_n\equiv B_{n}/n \ \ \mod \mathbb{Z}$, where the $B_n$ are the Bernouilli numbers;</p> <p>(2) for each odd prime $p$ and $p$-adic unit $c$,</p> <p>$$m\equiv n \ \mod p^k(p-1) \Rightarrow (1-c^n)(1-p^{n-1})b_n \equiv (1-c^m)(1-p^{m-1})b_m \ \mod p^{k+1};$$</p> <p>(3) for each $2$-adic unit $c$,</p> <p>$$m\equiv n \ \mod 2^k \Rightarrow (1-c^n)(1-2^{n-1})b_n \equiv (1-c^m)(1-2^{m-1})b_m \ \mod 2^{k+2}.$$</p> <p>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?</p> http://mathoverflow.net/questions/36362/characteristic-power-series-for-maps-of-e-infty-ring-spectra/36381#36381 Answer by Charles Rezk for Characteristic power series for maps of E_{\infty} ring spectra Charles Rezk 2010-08-22T16:15:12Z 2010-08-22T16:26:06Z <p>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 <em>Manifolds and modular forms</em> by Hirzebruch et al. describe the classical version of this pretty well.</p> <p>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 $K(x)=x/\mathrm{exp}_F(x)$.</p> <p>Now suppose $R$ is a "complex orientable cohomology theory", which means we are given a suitable isomorphism of rings <code>$R^*(CP^\infty)\approx \pi_*R[[x]]$</code>. 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 <code>$\pi_*R_\mathbb{Q}[[x]]$</code>.</p> <p>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.</p> <p>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.)</p> <p>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).</p> <p>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.</p> <p>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 $K_\phi(x)\in \pi_*R_{\mathbb{Q}}$.</p> <p>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.</p> <p>Finally: There is a writeup of this at <a href="http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf" rel="nofollow">http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf</a>, which may or may not be of any use to you!</p>