I’ve been reading about spectra and stable homotopy theory lately, but don’t feel ready to start talking about them here. Instead, I shall say a few words on characteristic classes. The present post will be quite general and preparatory — the more difficult matter is to actually construct such characteristic classes. Our goal is to see that characteristic classes essentially boil down to computing the cohomology of the infinite Grassmannian.

A lot of problems in mathematics involve the existence of sections to vector bundles. For instance, there is the old question of when the sphere is parallelizable. A quick Euler characteristic argument shows that even-dimensional spheres can’t be—then there would be an everywhere nonzero vector field, whose infinitesimal flows would be homotopic to the identity (and consequently having nonzero Lefschetz number by the even-dimensionality) while having no fixed points. In fact, much more is known. Using the group or group-like structures on {S^1, S^3, S^7} (coming from the complex numbers, quarternions, and octonions), it is easy to see that these manifolds are parallelizable. But in fact no other sphere is.

A characteristic class is a means of assigning some invariant to a vector bundle. Ideally, it should be trivial on trivial bundles, so the characteristic class can be thought of as an “obstruction” to finding large numbers of linearly independent sections.

More formally, let {p: E \rightarrow B} be a vector bundle. A characteristic class assigns to this bundle (of some fixed dimension, say {n}) an element of the cohomology ring {H^*(B)} (with coefficients in some ring). To be interesting, the characteristic class has to be natural. That is, if {f: B' \rightarrow B} is a map, then the characteristic class of the pull-back bundle {f^*E \rightarrow  B'} should be the pull-back of the characteristic class of {E \rightarrow B}. (more…)

I’ve been reading a lot of algebraic topology as of late, but I still don’t think I understand most of it properly. I will try to blog about some of the ideas as a means to understand the ideas better.

The homotopy groups are essentially homotopy classes of maps from the sphere into a space {X}. For instance, the first homotopy group {\pi_1} (or fundamental group) is the group of homotopy classes of loops at a fixed base-point. But why should they be a group? There is a categorical reason for this, and while it’s not immensely deep, I’d like to explain it.

1. Definition

Fix a topological space {X} with a basepoint {x_0 \in X}. Let {n \geq 1}. We consider the set {[I^n : X]} of homotopy classes of maps {I^n \rightarrow X} from the unit {n}-cube into {X}. This, however, is not a very interesting set, because {I^n} is contractible; there is thus only one element. Instead, we consider the set {[I^n, \partial I^n: X, x_0]} of homotopy classes of maps {I^n \rightarrow X} that send the boundary {\partial I^n} into {x_0}. When I say “homotopy classes,” I mean that homotopies are required to send the boundary {\partial I^n} into {x_0}. (more…)