4

I wanted to know how can I prove there are only two twisted I bundle over the Klein bottle. As I looked some topology texts, I couldn't find any solid definition of twisted I-bundle (or general definition of twisted fiber bundle) except that it is not the trivial product bundle $K\times I$(K is the Klein bottle). I was looking to finding out

(1)How can I show that $K\tilde{\times}I$ is orientable?

(2)why there are only two $K\tilde{\times}I$?

I would appreciate any help anyone can give me. Also, any references to this subject might do the job.

Edith: I have denoted the twisted I bundle over the Klein bottle by $K\tilde{\times}I$

siavash
  • 61
  • Random aside to anyone reading: is writing $K \tilde{\times} I$ common for denoting a twisted product (a bundle)? – Osama Ghani Dec 25 '17 at 20:17
  • 1
    Hint: Count homomorphisms from the Klein bottle fundamental group to $Z/2$. – Moishe Kohan Dec 25 '17 at 20:29
  • Husemoller, Fiber Bundles. It's a terrible reference, but it's a reference. Steenrod, Topology of Fiber Bundles: less terrible, but still not good. – John Hughes Dec 25 '17 at 20:53
  • @MoisheCohen can you please elaborate your hint? I was thinking about the characteristic classes because an I-bundle can be regarded as the disk bundle of rank 1 vector bundle on K. Is that correct? – siavash Dec 26 '17 at 09:30

0 Answers0