Chromatic Homotopy Theory

3 MU

Let \(E\) be a complex oriented associative ring spectrum (COCT for short), i.e it is equipped with a unital map from \(\MU (1)\). Equivalently it is a Thom class for the universal line bundle. This implies the degeneration of the \(E\)-based AHSSs for \(\BU (1) = \Sigma ^2\MU (1)\), and consequently for \(\BU (n)\) and \(\MU (n)\) by the splitting principle. Moreover the degeneration shows that for these spaces there is an isomorphism \(E^*= H^*\otimes E^*(pt)\) respecting the ring structure, because in \(E^*(\CC \PP ^n)\) a class \(x\) in the kernel of the augementation can be taken relative to a basepoint, and \((n+1)\) contractible sets cover \(\CC \PP ^n\). Thus we really get a Thom class for every virtual bundle and a theory of \(E\)-Chern classes satisfying a Whitney sum formula. One can also prove projective bundle formulas and the like.

We get a one-dimensional commutative formal group law on \(E_*\) by looking at the map classifying the tensor product of line bundles on \(E^*(\CC \PP ^\infty \otimes \CC \PP ^\infty ) \to E^*(\CC \PP ^\infty )\). Moreover the computation of \(E^*\MU (n)\) gives compatible maps \(\MU (n) \hookrightarrow \MU (n+1) \to E\) given by summing with a trivial line bundle. In particular, we get a map of COCTs \(\MU \to E\), which is a ring map essentially by additivity of Chern classes, so \(\MU \) is the universal complex oriented cohomology theory.

Our hope is that a lot of the information of \(E\) is contained in its formal group law, and the first evidence in this direction is the computation of \(\MU _*\) Namely, the universal complex oriented cohomology theory should have the universal formal group law, defined on \(L\), Lazard’s ring. This is true, and in fact the polynomial generators \(t_i\) in \(\MU _*\) can be identified with \(\CC \PP ^{i}\) if \(\MU \) is identified with complex bordism, and the logarithm for \(\MU _*\) becomes \(\sum _i \frac {[\CC \PP ^i]}{i+1}t^{i+1}\)

First one can get a more conceptual description of \(E_*\MU \). It is the homotopy groups of the spectrum \(E\otimes \MU \) which has two complex orientations, one coming from \(E\), \(t_E\)and one from \(MU\), \(t_{\MU }\). Thus \(E\otimes \MU ^*\CC \PP ^\infty \) has two formal group laws, which must be strictly isomorphic, since they are orientations on the same ring spectrum. In fact, from definition of the orientations, we see that \(t_{\MU } = t_E + t_E^2b_1 + t_E^3b_2 + \dots \) so that we get

Proposition 3.1. \(E\otimes \MU _*\) is the universal strict isomorphism of the formal group law on \(E\).

Now we can try to prove:

Theorem 3.2 (Quillen’s Theorem). \(\MU _* = L\) with its formal group law.

Rationally, we have already computed \(\MU _*\otimes \QQ = H\QQ _*(\MU ) = \QQ [b_1,b_2,\dots ]\). We will use the \(H\FF _p\)-based Adams spectral sequence to compute \(\MU _*\) p-adically. In particular it will degenerate at the \(E_2\) page.

First we need to understand the homology of \(\MU \) even better, namely as a \(H\FF _p\otimes H\FF _p\) comodule. To do this, we can consider the quotient \(B\ZZ /p\ZZ \to \CC \PP ^\infty \), and look at the induced map in cohomology. This map is injective, and the generator \(b_1\) in cohomology of \(\CC \PP ^\infty \) gets sent to the polynomial generator in degree \(2\) of \(B\ZZ /p\ZZ \). Now let \(\beta _i\) be the elements of degree \(2i\) in homology. The coaction is then \(\beta _i \mapsto \beta _i \otimes 1 + \dots \) when \(i \neq 2p^j\) and \(\beta _{p^i}\mapsto \beta _{p^i} \otimes 1 + \beta _1 \otimes \zeta _i \dots \) where \(\zeta _i\) are the polynomial generators of the dual Steenrod algebra (when \(p=2\) they are the squares of those generators). Now the homology of \(MU\) is the polynomial algebra on the \(\beta _i\) so this essentially determines the coaction by multiplicativity. The coaction is really then determined by the subalgebra generated by the \(\zeta _j\). In otherwords, the action of the super algebraic group \(\Spec H\FF _{p*}H\FF _p = \Spec A\) factors through a quotient group \(\Spec P\), which is the free polynomial algebra on the \(\zeta _i\). In fact this group is really the group of automorphisms of the additive formal group. \(H\FF _{p*}\MU \) is the universal strict isomorphism of the additive formal group on \(H\FF _{p}\) and with this description, the action of \(\Spec A\) is the inclusion.

Given this description, one easily sees that the action of \(\Spec P\) is free, namely, every strict automorphism \(t \to t+t^2b_1 + t^3b_2 + \dots \) can be written uniquely as the composite of a strict automorphism of the additive formal group (i.e where only \(b_{p^i-1} \neq 0\)) and an automorphism where \(b_{p^i-1}\) are \(0\). The quotient by the action is \(\Spec \FF _p[b_i, i \neq p^j-1]\).

The kernel of \(\Spec {A} \to \Spec {P}\) is a tensor product of exterior algebras in degrees \(2p^i-1, i\geq 0\). Since it acts trivially, the \(E_2\)-term of the Adams SS is \(\FF _p[b_i, i \neq p^j-1] \otimes _{\FF _p} \Ext _{\ker }(\FF _p,\FF _p)\). The cohomology of an exterior algebra is a polynomial algebra in degree \(1\), which will be called \(r_i\). Thus we get:

Proposition 3.3. The \(E_2\)-term of the Adams SS for \(\pi _*\MU \) is a polynomial algebra of generators \(b_i, i > 0, i \neq p^{i}-1\) in bidegree \((0,2i)\), and \(r_i, i\geq 0\) in bidegree \((1,2p^i-1)\).

With this we can finish Quillen’s Theorem.

Proof. Since the \(E_2\) term of the Adams SS is concentrated in even degrees, it degenerates at that page. Moreover, \(r_0\) is an element of \(\pi _0\) in homotopy degree \(0\) and Adams filtration \(1\), so it must be \(p\) (up to a unit). Thus we get that there is no \(p\)-torsion in the homotopy since \(r_0\) is free. Since the rational Hurewicz map is an isomorphism, The Hurewicz map is thus injective. We can look at indecomposables (i.e \(I/I^2\) of the augmentation ideals). The \(b_i\) are in Adams filtration \(0\) and so are not killed by Hurewicz, and are really the same classes \(b_i\) from before (up to a unit). \(r_i\) is in Adams filtration \(1\), so is sent to \(p\) times \(b_{p^i-1}\). This is the same injection as the one of \(L\) into the universal change of variables, and since that is exactly what the formal group law on homology is, \(\MU _* = L\). □