I’ve been trying to fix the (many) gaps in my knowledge of classical algebraic topology as of late, and will probably do a few posts in the near future on vector bundles, K-theory, and characteristic classes.

Let be a base space, and let be a real vector bundle. There are numerous constructions for the *characteristic classes* of . Recall that these are elements in the cohomology ring (for some ring) that measure, in some sense, the twisting or nontriviality of the bundle .

Over a smooth manifold , with a smooth vector bundle, a construction can be made in de Rham cohomology. Namely, one chooses a connection on , computes the curvature tensor of (which is an -valued 2-form on ), and then applies a suitable polynomial from matrices to polynomials to the curvature . One can show that this gives closed forms, whose de Rham cohomology class does not depend on the choice of connection. This is the subject of Chern-Weil theory, and it applies more generally to principal -bundles on a manifold for a Lie group.

But there is something that this approach misses: torsion. By working with de Rham cohomology (or equivalently, cohomology with -coefficients), the very interesting torsion phenomena that algebraic topologists care about is lost. For the purposes of this post, we’re interested in cohomology classes where the ground ring is , and so de Rham cohomology is out. However, in return, we have cohomology operations. We can use them instead. (more…)