List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle. State the orientability of the total space, the base and the bundle (orientability of a circle bundle is equivalent to whether it is $ U_1 $ principal see https://mathoverflow.net/questions/144092/is-every-orientable-circle-bundle-principal ). For circle bundles over a surface state which Thurston geometry the total space admits, if any.
 A: Base $ S^1 $:

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*All three:  $ S^1 \to T^2 \to S^1 $ (in general any trivial bundle $ S^1 \times M $ for any orientable manifold $ M $ has base, bundle and total space all orientable)


*Only base orientable: $ S^1 \to K^2 \to S^1 $ where $ K^2 $ is the Klein bottle.
Base $ S^2 $:

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*All three: The lens spaces $ S^1 \to L_{n,1} \to S^2 $, where $ n $ is the Euler class of the bundle (for small values of $ n $ we have $ L_{0,1}= S^1 \times S^2, L_{1,1}\cong S^3, L_{2,1}\cong \mathbb{R}P^3 $). $ E^1 \times S^2 $ geometry for $ n=0 $, $ S^3 $ geometry otherwise.

Base $ T^2 $:

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*All three: The circle bundles $ S^1 \to MT(\begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix}) \to T^2 $ where $ MT(\begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix}) $ denotes the mapping torus of $ T^2 $ corresponding to the mapping class $ \begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix} $, which is the $ r $th power of the Dehn twist $ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $. For $ r=0 $ this is $ T^3 $ and admits $ E^3 $ (flat) geometry while for $ r \neq 0 
 $ these are the nilmanifolds $ N_r $ described in Is every Nil manifold a nilmanifold? and they admit Nil geometry. $ E^3 $ geometry for $ r=0 $ otherwise Nil geometry.


*Only base orientable: $ S^1 \rtimes_b T^2 $ two of the four flat compact non orientable three manifolds. For $ b=0 $ this is $ S^1 \times K^2 $ with first homology $ \mathbb{Z}^2 \times C_2 $, for $ b=1 $ this is the mapping torus of the Dehn twist diffeomorphism of $ K^2 $ with first homology $ \mathbb{Z}^2 $. The total space is not orientable. These coincide with the two $ U_1 $ principal bundles over $ K^2 $. $ E^3 $ geometry.
Base $ \mathbb{R}P^2 $:

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*Only bundle orientable: $ S^1 \to S^1 \times \mathbb{R}P^2 \to \mathbb{R}P^2 $. (in general any trivial bundle $ S^1 \times M $ for any non orientable manifold $ M $ has only the bundle orientable). $ E^1 \times S^2 $ geometry.


*Only bundle orientable: $ S^1 \to (S^2 \times S^1)/(-1,-1) \to \mathbb{R}P^2 $. This is the mapping torus of the antipodal map of $ S^2 $. It is the unique nontrivial $ U_1 $ principal bundle over $ \mathbb{R}P^2 $. $ E^1 \times S^2 $ geometry.


*Only total space orientable: $ S^1 \to P_{4n,1} \to \mathbb{R}P^2 $ where $ P_{4n,1} $ is the standard prism manifold with $ 4n $ element dicyclic fundamental group. $ S^3 $ geometry.


*Only total space orientable: $ S^1 \to UT(\mathbb{R}P^2) \cong L_{4,1} \to \mathbb{R}P^2 $, the unit tangent bundle of $ \mathbb{R}P^2 $. $ S^3 $ geometry.


*Only total space orientable: $ S^1 \to \mathbb{R}P^3 \# \mathbb{R}P^3 \to \mathbb{R}P^2 $. $ E^1 \times S^2 $ geometry.
Base $ K^2 $:

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*Only the bundle is orientable: The two principal $ U_1 $ bundles over $ K^2 $ coincide with the two non principal $ S^1 $ bundles over $ T^2 $. These are two of the four non orientable compact flat three manifolds they can also be viewed as two of the four mapping tori of $ K^2 $. $ E^3 $ geometry.


*none of 3: The other two of the four non orientable compact flat three manifolds, they can also be viewed as the other two of the four mapping tori of $ K^2 $. They are non principal bundles $ S^1 \to S^1 \rtimes_b K^2 \to K^2 $ both with non orientable total space. For $ b=0 $ this is the mapping torus of the Y homoeomorphism of $ K^2 $, it has first homology $ \mathbb{Z}^2 \times C_2 \times C_2 $. For $ b=1 $  this is the mapping torus of $ K^2 $ for the mapping class corresponding to the combination of a Dehn twist and a Y homoemorphism. It has first homology $ \mathbb{Z}^2 \times C_4 $. $ E^3 $ geometry.


*Only total space orientable: The circle bundles $ S^1 \to MT(\begin{bmatrix} -1 & -r \\ 0 & -1 \end{bmatrix}) \to K^2 $ where $ MT(\begin{bmatrix} -1 & -r \\ 0 & -1 \end{bmatrix}) $ denotes the mapping torus of $ T^2 $ corresponding to the mapping class $ \begin{bmatrix} -1 & -r \\ 0 & -1 \end{bmatrix} $. These manifolds are double covered by $ MT(\begin{bmatrix} 1 & 2r \\ 0 & 1 \end{bmatrix}) $. For $ r=0 $ this is the unit tangent bundle of the Klein $ UT(K^2) $, which admits $ E^3 $(flat) geometry, while for $ r \neq 0 
 $ these admit Nil geometry. $ E^3 $ geometry for $ r=0 $, Nil geometry otherwise.
