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…)