Real line bundles on the Klein bottle

Question

I am trying to better understand real line bundles. The real line bundles on a compact manifold are classified by $ H^1(M,C_2) $. See

https://mathoverflow.net/a/113944/387190

So, for example, every real line bundle on a simply connected manifold is trivial.

Here is a list of some low dimensional base spaces together with all possible real line bundles up to homeomorphism of the total space

• $ S^1 $: trivial $ \mathbb{R} \times S^1 $, and tautological bundle on $ \mathbb{R}P^1 $ (coincides with the Moebius strip $ \text{Mo} $ )

• $ \mathbb{R} $: trivial $ \mathbb{R}^2 $

• $ S^2 $: trivial $ \mathbb{R} \times S^2 $

• $ \mathbb{R}P^2 $: trivial $ \mathbb{R} \times \mathbb{R}P^2 $, and nontrivial coincides with the tautological bundle on $ \mathbb{R}P^2 $

• $ \mathbb{R}^2 $: trivial $ \mathbb{R}^3 $

• $ T^2 $: trivial $ T^2 \times \mathbb{R} $ and nontrivial which coincides with $ \text{Mo} \times S^1 $

• $ \text{Mo} $: trivial $ \text{Mo} \times \mathbb{R} $ and nontrivial which coincides with $ \mathbb{R}^2 \times S^1 $

• $ \mathbb{R} \times S^1 $: trivial $ \mathbb{R}^2 \times S^1 $ and nontrivial which coincides with $ \text{Mo} \times \mathbb{R} $

Now I'm trying to understand the real line bundles on the Klein bottle $ K $. $$ H^1(K,C_2) \cong C_2 \times C_2 $$ So there are at most four real line bundles over $ K $ up to homeomorphism of the total space. One is the trivial bundle $ K \times \mathbb{R} $. One nontrivial line bundle over $ K $ has total space $ M_0^3 $ as described in table 1 of

https://www.researchgate.net/publication/242949481_Three-dimensional_homogeneous_spaces

What about the other two real line bundles over the Klein bottle?

Answer 1

I haven't checked all the details but arguing by analogy with the complex case one should have a picture like this.

Consider $K$ as an $S^1$-bundle over $S^1$, pulling back the trivial bundle and tautological bundle of the base appears to give two bundles. I presume that this $S^1$ -bundle as a projectivisation of a rank 2 real bundle, Then I guess tensoring with the tautologival bundle of the projective bundle should give all 4.