Genus of A Multiplicative Sequence - L Genus and The Hirzebruch Signature Theorem

L Genus and The Hirzebruch Signature Theorem

The L genus is the genus of the formal power series

{\sqrt{z}\over \tanh(\sqrt z)} = \sum_{k\ge 0} {2^{2k}B_{2k}z^k\over (2k)!} = 1 + {z \over 3} - {z^2 \over 45} +\cdots

where the numbers B2k are the Bernoulli numbers. The first few values are

  • L0 = 1
  • L1 = p1/3
  • L2 = (7p2 − p12)/45.

Now let M be a closed smooth oriented manifold of dimension 4n with Pontrjagin classes . Friedrich Hirzebruch showed that the L genus of M in dimension 4n evaluated on the fundamental class of M, is equal to, the signature of M (i.e. the signature of the intersection form on the 2nth cohomology group of M ):

This is now known as the Hirzebruch signature theorem (or sometimes the Hirzebruch index theorem). René Thom had earlier proved that the signature was given by some linear combination of Pontryagin numbers, and Hirzebruch found the exact formula for this linear combination given above.

The fact that L2 is always integral for a smooth manifold was used by John Milnor to give an example of an 8-dimensional PL manifold with no smooth structure. Pontryagin numbers can also be defined for PL manifolds, and Milnor showed that his PL manifold had a non-integral value of p2, and so was not smoothable.

Read more about this topic:  Genus Of A Multiplicative Sequence

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