4

I was trying to solve the exact same problem that was discussed in this question: Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle

I came up with the step to identify one-dimensional projective space with the one-sphere by myself, but that didn't make it any clearer. Even when I draw a picture I cannot see how an element of the tautological bundle should "twist around" the one-sphere (I seem to end up with the trivial bundle), let alone can I write down an explicit bundle isomorphism. Can anyone make the identifications when one passes from projective space to the circle clear? It seems that my problem lies there.

Community
  • 1
Beemer
  • 41

1 Answers1

3

Hint: Take a line in $\mathbb{R}^2$, and rotate it by 180 degrees. The line is now the same, but what can you say about what happens to the orientation?

Remember that the Grassmannian $G_1(\mathbb{R}^2)$ parameterizes lines in $\mathbb{R}^2$, but does not concern itself with a basis of that line, i.e. an orientation of the line.

Simon Rose
  • 6,793