Consider a smooth surface {M \subset \mathop{\mathbb P}^3(\mathbb{C})} of degree {d}. We are interested in determining its cohomology.

1. A fibration argument

A key observation is that all such {M}‘s are diffeomorphic. (When {\mathop{\mathbb P}^3} is replaced by {\mathop{\mathbb P}^2}, then this is just the observation that the genus is determined by the degree, in the case of a plane curve.) In fact, consider the space {V} of all degree {d} homogeneous equations, so that {\mathop{\mathbb P}(V)} is the space of all smooth surfaces of degree {d}. There is a universal hypersurface {H \subset \mathop{\mathbb P}^3 \times \mathop{\mathbb P}(V)} consisting of pairs {(p, M)} where {p} is a point lying on the hypersurface {M}. This admits a map

\displaystyle \pi: H \rightarrow \mathop{\mathbb P}(V)

which is (at least intuitively) a fiber bundle over the locus of smooth hypersurfaces. Consequently, if {U \subset \mathop{\mathbb P}(V)} corresponds to smooth hypersurfaces, we get an honest fiber bundle

\displaystyle \pi^{-1}(U) \rightarrow U .

But {U} is connected, since we have thrown away a complex codimension {\geq 1} subset to get {U} from {\mathop{\mathbb P}(V)}; this means that the fibers are all diffeomorphic.

This argument fails when one considers only the real points of a variety, because a codimension one subset of a real variety may disconnect the variety. (more…)

Today I would like to blog about a result of Atiyah from the 1950s, from his paper “Bott periodicity and the parallelizability of the spheres.” Namely:

Theorem 1 (Atiyah) On a nine-fold suspension {Y = \Sigma^9 X} of a finite complex, the Stiefel-Whitney classes of any real vector bundle vanish.

In particular, this means that any real vector bundle on a sphere S^n, n \geq 9 cannot be distinguished using Stiefel-Whitney classes from the trivial bundle. The argument relies on the Bott periodicity theorem and some calculations with Stiefel-Whitney classes. There is also an analog for the Chern classes of complex vector bundles on spheres; they don’t necessarily vanish but are highly divisible.

These sorts of integrality theorems often have surprising geometric consequences. In this post, I’ll discuss the classical problem of when spheres admit almost-complex structures, a problem one can solve using the second of the integrality theorems mentioned above. Atiyah was originally motivated by the question of parallelizability of the spheres. (more…)

Today I would like to take a break from the index theorem, and blog about a result of Wu, that the Stiefel-Whitney classes of a compact manifold (i.e. those of the tangent bundle) are homotopy invariant. It is not even a priori obvious that the Stiefel-Whitney classes are homeomorphism invariant; note that “homeomorphic” is a strictly weaker relation than “diffeomorphic” for compact manifolds, a result first due to Milnor. But in fact the argument shows even that the Stiefel-Whitney classes (of the tangent bundle) can be worked out solely in terms of the structure of the cohomology ring as a module over the Steenrod algebra.

Here is the idea. When A \subset M is a closed submanifold of a manifold, there is a lower shriek (Gysin) homomorphism from the cohomology of A to that of M; this is Poincaré dual to the restriction map in the other direction. We will see that the “fundamental class” of A (that is,  the image of 1 under this lower shriek map) corresponds to the mod 2 Euler (or top Stiefel-Whitney) class of the normal bundle. In the case of M \subset M \times M, the corresponding normal bundle is just the tangent bundle of M. But by other means we’ll be able to work out the Gysin map easily. Once we have this, the Steenrod operations determine the rest of the Stiefel-Whitney classes.

(more…)

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…)

I’d like to finish the series I started a while back on Chern-Weil theory (and then get back to exponential sums).

So, in the discussion of the Cartan formalism a few days back, we showed that given a vector bundle E with a connection on a smooth manifold, we can associate with it a curvature form, which is an \hom(E, E)-valued 2-form; this is a generalization of the Riemann curvature tensor (as some computations that I don’t feel like posting here will show). In the case of a line bundle, we saw that since \hom(E, E) was canonically trivialized, we could interpret the curvature form as a plain old 2-form, and in fact it turned out to be a representative — in de Rham cohomology — of the first Chern class of the line bundle. Now we want to see what to do for a vector bundle, where there are going to be a whole bunch of Chern classes.

For a general vector bundle, the curvature {\Theta} (of a connection) will not in itself be a form, but rather a differential form with coefficients in {\hom(E, E)}, which is generally not a trivial bundle. In order to get a differential form from this, we shall have to apply an invariant polynomial. In this post, I’ll describe the proof that one indeed gets well-defined characteristic classes (that are actually independent of the connection), and that they coincide with the usually defined topological Chern classes. (more…)

So, now with the preliminaries on connections and curvature established, and the Chern classes summarized, it’s time to see how they connect with one another. Namely, we want to say that, given a complex vector bundle, we can compute the Chern classes in de Rham cohomology by picking a connection — any connection — on it,  computing the curvature, and then applying various polynomials.

We shall start by warming up with a special case, of a line bundle, where the algebra needed is easier. Let {M} be a smooth manifold, {L \rightarrow M} a complex line bundle. Let {\nabla} be a connection on {L}, and let {\Theta} be the curvature.

Thus, {\Theta} is a global section of {\mathcal{A}^2 \otimes \hom(L, L)}; but since {L} is a line bundle, this bundle is canonically identified with {\mathcal{A}^2}. (Recall the notation that \mathcal{A}^k is the bundle (or sheaf) of smooth k-forms on the manifold M.)

Proposition 1 (Chern-Weil for line bundles) {\Theta} is a closed form, and the image in { H^2(M; \mathbb{C})} is {2\pi i} times the first Chern class of the line bundle {L}. (more…)

So, I’m in a tutorial this summer, planning to write my final paper on the Kodaira embedding theorem, and I’ve been finding my total ignorance of complex algebraic geometry to be something of a problem. One of my goals next year is, coincidentally, to acquire a solid understanding of most of the topics in Griffiths-Harris. To start with, I’d like to spend a few posts on Chern-Weil theory. This gives an analytic method of computing the Chern classes of a complex vector bundle, and more generally a framework for the characteristic classes of a principal bundle over a Lie group. In fact, it tells you what the cohomology of the classifying space of a Lie group is (it’s a certain algebra of invariant polynomials on the Lie algebra), from which — by Yoneda’s lemma — you can associate cohomology classes to a principal bundle on any space.

Today, I’d like to review what Chern classes are like.

1. Introduction

To start with, we will need to describe what the Chern classes really are. These are going to be natural maps

\displaystyle \mathrm{Vect}_{\mathbb{C}}(X) \rightarrow H^*(X; \mathbb{Z}),

from the complex vector bundles on a space {X} to the cohomology ring. In other words, to each vector bundle {E \rightarrow X}, we will have an element {c(E) \in H^*(X; \mathbb{Z})}. In order for this to be natural, we are going to want that, for any map {f: Y \rightarrow X} of topological spaces,

\displaystyle c(f^*E) = f^* c(E) \in H^*(Y; \mathbb{Z}).

In other words, we are going to want the map {\mathrm{Vect}_{\mathbb{C}}(X) \rightarrow H^*(X; \mathbb{Z})} to be functorial in {X}, when both are considered as contravariant functors in {X}. It turns out that each functor {\mathrm{Vect}_{n, \mathbb{C}}} (of {n}-dimensional complex vector bundles) and {H^k(X; \mathbb{Z})} is representable on the appropriate homotopy category. (more…)