I am trying to understand isomorphism classes of bundles
$$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$
These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$.

**ATTEMPT 1.**
$Aut(\mathbb{S}^1)$ are the diffeomorphisms $\mathbb{S}^1\to \mathbb{S}^1$, and it is homotopically equivalent to $O(2).$ Therefore the universal classifying space $BAut(\mathbb{S}^1)$ is homotopically equivalent to $BO(2)$. Thus we are interested in homotopy classes in$[\mathbb{RP}^2,BO(2)]$.

Now topologically $O(2) \simeq SO(2) \sqcup SO(2)\simeq \mathbb{S}^1 \sqcup \mathbb{S}^1,$ (it has two connected component that are diffeomorphic to $\mathbb{S}^1$). Therefore
$$BO(2)\simeq B\mathbb{S}^1\sqcup B\mathbb{S}^1 \simeq \mathbb{CP}^\infty \sqcup \mathbb{CP}^\infty.$$

Consequently
$$[\mathbb{RP}^2,BO(2)] \simeq [\mathbb{RP}^2,\mathbb{CP}^\infty \sqcup \mathbb{CP}^\infty] \simeq [\mathbb{RP}^2,\mathbb{CP}^\infty]\times[\mathbb{RP}^2,\mathbb{CP}^\infty]$$
Using that $\mathbb{CP}^\infty$ is a $K(\mathbb{Z},2)$ we get that
$[\mathbb{RP}^2,\mathbb{CP}^\infty]\simeq H^2(\mathbb{RP}^2,\mathbb{Z})\simeq \mathbb{Z}/2\mathbb{Z}$.
This would imply that there are $4$ isomorphism classes of circle bundles over $\mathbb{RP}^2$.

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> Question 1: which characteristic classes give this classification?
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> Question 2: what are these 4 manifolds $E$?
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**ATTEMPT 2.** This is inspired by [Classification of $O(2)$-bundles in terms of characteristic classes.](https://math.stackexchange.com/questions/1712414/classification-of-o2-bundles-in-terms-of-characteristic-classes)
The first Steifel-Whitney class $w\_1 \in H^1(\mathbb{RP}^2,\mathbb{Z}/2\mathbb{Z})$ is $0$ iff the bundle is orientable. In this case we have a reduction of the structure group to $SO(2)$ and $SO(2)$ principal bundles are classified by the Euler class $e\in H^2(\mathbb{RP}^2,\mathbb{Z})\simeq [\mathbb{RP}^2, \mathbb{CP}^\infty]\simeq \mathbb{Z}/2\mathbb{Z}.$ Therefore we have $2$ orientable bundles and an unknown number of non-orientable ones when $w\_1 = 1\in \mathbb{Z}/2\mathbb{Z} \simeq H^1(\mathbb{RP}^2,\mathbb{Z}/2\mathbb{Z})$.

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> Question 3: how to find out how many non-orientable bundles we have?
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> Please tell me if I am correct and if you know, the answer to the above questions.
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