May 31, 2012

The other direction of Ochanine’s theorem

Posted by Akhil Mathew under topology | Tags: , , , |
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In a previous post, I began discussing a theorem of Ochanine:

Theorem 1 (Ochanine) A genus {\phi: \Omega_{SO} \rightarrow \Lambda} annihilates the projectivization {\mathbb{P}(E)} of every even-dimensional complex bundle {E \rightarrow M} if and only if the logarithm of {\phi} is an elliptic integral

\displaystyle g(x) = \int_0^x (1 - 2\delta u^2 + \epsilon u^4)^{-1/2} du.

In the previous post, we described Ochanine’s proof that a genus whose logarithm is an elliptic integral (a so-called elliptic genus) annihilated any such projectivization. The proof relied on some computations in the projectivization and then some trickery with elliptic functions. The purpose of this post is to prove the converse: a genus with a suitably large kernel comes from an elliptic integral. (more…)

 

May 30, 2012

The Milnor hypersurfaces

Posted by Akhil Mathew under topology | Tags: , |
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In a previous post, we studied the formal group law of {\pi_* MU} in geometric terms: that is, using the interpretation of {\pi_* MU} as the cobordism ring of stably almost-complex manifolds. We found that the logarithm for this formal group law was given by the power series

\displaystyle \log x = \sum_{i \geq 0 } \frac{[\mathbb{CP}^i]}{i+1} x^{i+1}.

In particular, we saw that the complex projective spaces provided a set of independent generators for the rationalization {\pi_* MU \otimes \mathbb{Q}}: that is,

\displaystyle \pi_* MU \otimes \mathbb{Q} \simeq \mathbb{Q}[\{ [\mathbb{CP}^i] \}_{i >0}].

This is analogous to the theorem of Hirzebruch which calculates the oriented cobordism ring {\pi_* MSO \otimes \mathbb{Q}}, and could also have been established directly by arguing that {\pi_* MU \otimes \mathbb{Q} \simeq H_*(MU; \mathbb{Q})}. The structure of the latter ring can be worked out directly, and in fact was.

We might be interested, though, in a set of honest generators for {\pi_* MU} (not generators mod torsion). Such a set is provided by the Milnor hypersurfaces which I would like to discuss in this post. (more…)