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 {B} be a base space, and let {p: E \rightarrow B} be a real vector bundle. There are numerous constructions for the characteristic classes of {B}. Recall that these are elements in the cohomology ring {H^*(B; R)} (for {R} some ring) that measure, in some sense, the twisting or nontriviality of the bundle {B}.

Over a smooth manifold {B}, with {E} a smooth vector bundle, a construction can be made in de Rham cohomology. Namely, one chooses a connection {\nabla} on {E}, computes the curvature tensor of {E} (which is an {\hom(E,E)}-valued 2-form {\Theta} on {B}), and then applies a suitable polynomial from matrices to polynomials to the curvature {\Theta}. 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 {G}-bundles on a manifold for {G} a Lie group.

But there is something that this approach misses: torsion. By working with de Rham cohomology (or equivalently, cohomology with {\mathbb{R}}-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 {R = \mathbb{Z}/2}, and so de Rham cohomology is out. However, in return, we have cohomology operations. We can use them instead. (more…)

Suppose {B} is a topological space, and {p: E \rightarrow B} is a vector bundle over {B}. Here {B} is a “reasonable” space, for instance a manifold or a CW complex. There are many familiar examples of how such bundles arise “in nature,” for instance the tangent bundle to a manifold. It is of interest to understand how the apparatus of algebraic topology applies to the bundle.

Now, clearly there is a deformation retraction of {E} given by {(t,v) \rightarrow tv} that deformation retracts {E} to the image of the zero section. In particular, {E} is homotopy equivalent to {B}. So {E} by itself might not all be that interesting.

Nonetheless, let us suppose that {E} is a normed bundle, e.g. the tangent bundle to a manifold with a Riemannian metric. In this case, we can consider the set of elements of norm one; it is no longer a vector bundle over {B}, but it is a fiber bundle (with fibers various spheres). This is homotopically very different from {B} in general.

For a general bundle {E}, we can still consider the subset {\dot{E}} consisting of nonzero elements. Then there is a map {p: \dot{E} \rightarrow  B} whose fibers are of the form {\mathbb{R}^n -  \left\{0\right\}}. It is easy to see that if {E} is a normed bundle, then {\dot{E}} deformation retracts onto the associated sphere bundle. So we might as well see what {\dot{E}} is like. By using the long exact sequence, we might try studying the pair {(E, \dot{E})}.

Today, I want to talk about the cohomology of the pair {(E, \dot{E})}. Namely, the main result is:

Theorem 1 (Thom) For all {k}, there is an isomorphism (to be described)\displaystyle  H^k(B; \mathbb{Z}/2) \simeq H^{k+n}(E, \dot{E}; \mathbb{Z}/2)

if {E} has dimension {n}.

(To ease notation, all cohomology will henceforth be with {\mathbb{Z}/2}-coefficients.) (more…)