geometry of circle bundles over the circle, sphere, projective plane, torus, and Klein bottle

Question

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.

Also state whether or not the total space of the bundle is homogeneous (transitive action of a Lie group)

This question is of interest since $ U_1 $ principal bundles are a topic of independent mathematical interest as are Thurston geometries on 3 manifolds. Also circle bundles over surfaces are related to Seifert fibrations which are an important topic in the theory of 3 manifolds.

Also based on structure theory of homogeneous spaces due to Mostow https://math.stackexchange.com/a/4374850/758507 nearly all homogeneous 3 manifold should be fiber bundles of the sort asked about in this question.

Answer 1

Base $ S^1 $:

• 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). Total space is Riemannian homogeneous.

• Only base orientable: $ S^1 \to K^2 \to S^1 $ where $ K^2 $ is the Klein bottle. Total space is a solvmanifold.

Base $ S^2 $:

• 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. Total space is Riemannian homogeneous.

Base $ T^2 $:

• 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. Total space is a nilmanifold.

• 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. Total space of any circle bundle over a torus is solvmanifold by https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds .

Base $ \mathbb{R}P^2 $:

• 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. Total space is Riemannian homogeneous.

• 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. Total space is Riemannian homogenous see Mapping torus of the antipodal map of $ S^2 $ .

• 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. Total space is Riemannian homogeneous.

• 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. Total space is Riemannian homogeneous.

• 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. Total space has transitive action by $ SE(3,\mathbb{R}) $ see Connected sum of two copies of $ RP^3 $

Base $ K^2 $:

• 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. Again the total space of the bundle is always a solvmanifold since the total space of any circle bundle over a torus is a solvmanifold by https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds .

• 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. Both have total space which are not homogeneous see the argument given here Is a mapping torus of a solvmanifold always a solvmanifold? and the arguments given here https://math.stackexchange.com/a/4374850/758507

• 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. The mapping torus of $ T^2 $ is the total space of a bundle $ T^2 \rtimes S^1 $ and thus must be a solvmanifold by https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds .