I am currently going through Hirsch's differential topology and am currently learning about the amazing world of vector bundles!!
I was hoping to start a thread where some experts could help me develop some intuition on why two homotopic maps into the infinite dimensional grassmanian will have isomorphic pullbacks. This result is amazing, yet somehow makes a lot of sense. Could somebody help me to understand why this is true? Thanks!
I like examples, I think they clarify.
Here is the example I like to tell to explain this. Consider an embedded $k$-dimensional submanifold $M$ in $\mathbb{R}^n$. Then each tangent space $T_xM$ is a $k$ dimensional subspace in $\mathbb{R}^n$. Sending $x\mapsto T_xM$ defines a map $M\rightarrow \mathrm{Gr}_k(\mathbb{R}^n)$ into the Grassmannian. Pulling back the tautological bundle along this map reproduces the tangent bundle. This is the essence of the classifying map!
Check that an isotopy of embeddings defines a homotopy class of maps. You can then stabilize: If you have an embedding to $\mathbb{R}^n$ you automatically get an embedding into $\mathbb{R}^{n+1}$, by mapping it to $\mathbb R^n\times\{0\}$. You can take a limit $n\mapsto \infty$ and get a map $M\rightarrow \mathrm{Gr}_k(\mathbb R^\infty)$. What is true is that if $n$ is sufficiently large there is basically one embedding into $\mathbb{R}^n$ up to isotopy. This means that the tangent bundle itself (so not the embedding) defines a homotopy class of maps into the infinite Grassmannian.
Now it is a bit more work to show that you can do something like this for arbitrary vector bundles. What you need to show is that for any vector bundle can be seen as a subbundle of a sufficiently high dimensional trivial vector bundle.
This is nicely worked out in books on characteristic classes (e.g. Milnor, or Hatcher, or Husemoller)