Characteristic Classes

The next section of the book, available here, gives a very quick introduction to what the Pontrjagin classes are.  This is really just a refresher – I am sure it is too terse for someone who has never seen characteristic classes before.  What such a person should take away, I think, is simply the idea of calculable invariants (in cohomology) which measure how non-trivial a bundle is.

Chapter 3 of Hatcher’s Vector bundles and K-theory gives a nice presentation of characteristic class theory.  The most current draft version of that book is available from Hatcher’s website.  In particular, this contains a fairly detailed presentation of the 2-torsion aspect of real characteristic classes (i.e. Stiefel-Whitney classes), which of course we are going to need later in the book.  One learns from the exercises, for instance, the exact formula for the odd Chern classes of the complexified bundle in terms of the Stiefel-Whitney classes, namely

$c_{2k+1}(E\otimes{\mathbb C}) = \beta (w_{2k}(E)w_{2k+1}(E) ),$

where $$\beta$$ is the Bockstein homomorphism associated to the exact sequence of coefficient groups

$0 \to {\mathbb Z} \to {\mathbb Z} \to {\mathbb Z}/2 \to 0$

given by multiplication by 2.

Part of my “philosophy” for the book, if that is not too grand a word, is to present things on an “as needed” basis.  So, right here we need the Pontrjagin classes and nothing else, so that’s what I sketch.  later on we’ll have to come back and talk about Euler and Stiefel-Whitney as well.  This means some backtracking, but I hope it makes the material more digestible.

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