Grassmannian - Motivation

Motivation

By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace. Though such concepts may seem strangely out of place they can coincide with things that one is interested in, and can describe ideas that could not be considered otherwise—or at least describe them more economically.

A natural example comes from tangent bundles of smooth manifolds embedded in Euclidean space. Suppose we have a manifold M of dimension r embedded in Rn. At each point x in M, the tangent space to M can be considered as a subspace of the tangent space of Rn, which is just Rr. The map assigning to x its tangent space defines a map from M to Gr(r, n). (In order to do this, we have to translate the geometrical tangent space to M so that it passes through the origin rather than x, and hence defines a r-dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)

This idea can with some effort be extended to all vector bundles over a manifold M, so that every vector bundle generates a continuous map from M to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps viewed as continuous maps. In particular we find that vector bundles with maps that are homotopic are isomorphic. But the definition of homotopic relies on a notion of continuity, and hence a topology.