Monday, May 28th, 2012

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May 28, 2012

The Hurewicz map and the logarithm of complex cobordism

Posted by Akhil Mathew under algebra, topology | Tags: , , , |
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In earlier posts, we analyzed the Hurewicz homomorphism

\displaystyle \pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq \mathbb{Z}[b_1, b_2, \dots ]

in purely algebraic terms: {\pi_* MU} was the Lazard ring, and the Hurewicz map was the map classifying the formal group law {\exp( \exp^{-1}(x) + \exp^{-1}(y))} for {\exp(x)} the “change of coordinates”

\displaystyle \exp(x) = x + b_1 x + b_2 x^2 + \dots.

We can also express the Hurewicz map in more geometric terms. The ring {\pi_* MU} is, by the Thom-Pontryagin construction, the cobordism ring of stably almost-complex manifolds. The ring {H_*(MU; \mathbb{Z})} is isomorphic to the Pontryagin ring {H_*(BU; \mathbb{Z})} by the Thom isomorphism. In these terms, we can describe the Hurewicz map explictly:

Theorem 1 The Hurewicz map {\pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq H_*(BU; \mathbb{Z})} sends the cobordism class of a stably almost complex manifold {M} to the element {f_* [M]} where

\displaystyle f_* : M \rightarrow BU

classifies the stable normal bundle of {M} (together with its complex structure), and where {[M]} is the fundamental class of {M}.

In particular, we will be able to work out explicitly where a given complex manifold representing a cobordism class goes under this map. As an application, we’ll show using Lagrange inversion that the complex projective spaces determine the logarithm of the formal group law on {\pi_* MU}. (more…)

 

May 28, 2012

The formal group law of unoriented cobordism

Posted by Akhil Mathew under algebra, topology | Tags: , , |
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Before moving on, I’d like to work out the analog for real-oriented cohomology theories (which is tangential to the rest of the story, though). This is considerably less interesting, but perhaps it’s a toy example of the ideas explained in the last few posts without the full-blown machinery of the Adams spectral sequence and so forth.

So, let’s state what the analogous ideas are in the real-oriented context:

  1. A real-oriented ring spectrum {E} is a ring spectrum together with a functorial, multiplicative choice of Thom classes for real vector bundles; equivalently, there is a morphism of ring spectra

    \displaystyle MO \rightarrow E.

    That is, the universal real-oriented ring spectrum is unoriented cobordism, whose homotopy groups can be completely computed.

  2. If {E} is real-oriented, then {\pi_* E} is a {\mathbb{Z}/2}-vector space, and all the usual computations of {H_*( BO; \mathbb{Z}/2)} and so forth work just fine for {E}, and there is a theory of Stiefel-Whitney classes in {E}-cohomology.
  3. Given a real-oriented spectrum {E}, we thus have {E^*(\mathbb{RP}^\infty) = \pi_* E [[t]]} where {t} is the Stiefel-Whitney class of the tautological bundle. Similiarly, {E^*(\mathbb{RP}^\infty \times \mathbb{RP}^\infty) = \pi_* E [[t_1, t_2]]}.

Since {\mathbb{RP}^\infty} is the classifying space for real line bundles, there is a monoidal product

\displaystyle \mathbb{RP}^\infty \times \mathbb{RP}^\infty \rightarrow \mathbb{RP}^\infty,

classifying the tensor product of line bundles. As before, this means that we can extract a formal group law over {\pi_* E}. This formal group law {f(\cdot, \cdot) \in \pi_* E [[x, y]]} has the property that if {\mathcal{L}_1, \mathcal{L}_2} are two line bundles over a finite-dimensional space {X}, then in {E^*(X)},

\displaystyle w_1( \mathcal{L}_1 \otimes \mathcal{L}_2) = f( w_1(\mathcal{L}_1), w_1(\mathcal{L}_2)).

So far everything has been analogous to the complex-oriented case, but there is an extra feature here which changes the picture drastically. Namely, when one works with realline bundles {\mathcal{L}}, we have that {\mathcal{L}^{\otimes 2}} is always trivial. (The analog is very false for complex line bundles.) This means that the formal group law {f} must satisfy

\displaystyle f(x, x) = 0.

This is not satisfied by, say, the multiplicative formal group law. (And in fact, {KO}-theory—the natural candidate for this—is not real-oriented, only spin-oriented.)

With this in mind, the analog of Quillen’s theorem for {MU} becomes:

Theorem 1 (Quillen) The formal group law for {MO} is the universal formal group law over a satisfying {f(x, x) = 0}. (more…)

 

May 28, 2012

Quillen’s theorem on the formal group law of MU

Posted by Akhil Mathew under algebra, topology | Tags: , , |
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Our goal is now to return to topology, and in particular to study the formal group law of the universal complex-oriented theory {MU} (complex cobordism). As we computed using the Adams spectral sequence,

\displaystyle \pi_* MU \simeq \mathbb{Z}[x_1, x_2, \dots ] , \quad \deg x_i = 2i.

This is the Lazard ring, by the computations of the previous couple of posts. On the other hand, it is not at all clear that the map

\displaystyle L \rightarrow \pi_* MU

classifying the formal group law over {\pi_* MU} (arising from the complex orientation) is actually an isomorphism: in other words, that the formal group law of {MU} is the universal one. The fact that it is in fact an isomorphism is the content of Quillen’s theorem, which will be proved in this post. (more…)