Let $ M $ denote the Moebius band and $ K^2 $ denote the Klein bottle.
Mapping tori of the cylinder (thanks to comment from Michael Albanese):
- $ (t,z) \mapsto (t,z) $ is $ \mathbb{R} \times T^2 $ orientable
- $ (t,z) \mapsto (-t,z) $ is $ S^1 \times M $ nonorientable
- $ (t,z) \mapsto (t,\overline{z}) $ is $ \mathbb{R} \times K^2 $ nonorientable
- $ (t,z) \mapsto (-t,\overline{z}) $ is the orientable manifold $ X $ of two forms on the Klein bottle (so it is a nontrivial real line bundle over $ K^2 $) it is described in What are the 8 non-compact Euclidean 3-manifolds?
Mapping tori of the plane:
- $ (x,y) \mapsto (x,y) $ is $ S^1 \times \mathbb{R}^2 $ orientable
- $ (x,y) \mapsto (-x,y) $ is $ \mathbb{R} \times M $ nonorientable
Mapping tori of the Mobius strip:
- The trivial mapping torus $ S^1 \times M $ which already occurs above as a nontrivial mapping torus of the cylinder
- The nontrivial mapping torus $ Y $
Mapping tori of the Klein bottle (listed with first integer Homology):
- mapping torus for trivial map ($ S^1 \times K^2 $), $ H_1=\mathbb{Z}^2 \times C_2 $
- mapping torus for Dehn twist, $ H_1=\mathbb{Z}^2 $
- mapping torus for Y-homeomorphism ( defined in https://arxiv.org/pdf/1410.1123.pdf ), $ H_1=\mathbb{Z}^2 \times C^4 $
- mapping torus for Dehn twist plus Y-homeomorphism, $ H_1= \mathbb{Z}^2 \times C_2 \times C_2 $
Mapping tori of $ T^2 $ (for details see Bianchi classification of solvable Lie groups and cocompact subgroups )
- trivial mapping torus $ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $ (Trace $ =2 $) is $ T^3 $
- mapping torus of $ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} $ (Trace $ =-2 $) is unit tangent bundle of Klein bottle, made by gluing opposite faces of a cube with $ 1/2 $ twist on one pair
- mapping torus of $ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $ (Trace $ =0 $) made by gluing opposite faces of a cube with $ 1/4 $ twist on one pair
- mapping torus of $ \begin{bmatrix} 0 & -1 \\ 1 & 1 \end{bmatrix} $ (Trace $ =1 $) made by gluing opposite faces of a hexagonal prism with $ 1/6 $ twist on the hexagonal faces
- mapping torus of $ \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix} $ (Trace $ =-1 $) made by gluing opposite faces of a hexagonal prism with $ 1/3 $ twist on the hexagonal faces
- mapping torus of $ \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} $ (mapping class not diagonalizable, Trace$ = 2 $, $ n \neq 0 $) mapping torus of $ n $ many Dehn twists, has Nil geomtry
- mapping torus of $ \begin{bmatrix} -1 & n \\ 0 & - 1 \end{bmatrix} $ (mapping class not diagonalizable, Trace$ = -2 $, $ n \neq 0 $) double covered by mapping torus of $ 2n $ many Dehn twists, has Nil geomtry
- mapping torus of $ \begin{bmatrix} 0 & -1 \\ 1 & n \end{bmatrix} $ (Trace $ =\pm3, \pm 4 \dots $) has Sol geometry
The above is a full list of orientable mapping tori of $ T^2 $ i.e. mapping tori for orientation preserving mapping class. The next two bullets exhaust all the nonorientable mapping tori/ mapping tori for orientation reversing mapping class. For more details see Which mapping tori are Seifert manifolds?
- mapping torus of $ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $ is $ S^1 \times K^2 $ it is the only nonorientable mapping torus of $ T^2 $ with Euclidean geometry
- mapping torus of $ \begin{bmatrix} 0 & 1 \\ 1 & n \end{bmatrix} $ has Sol geometry for any nonzero integer $ n $.
Flat three manifolds:
There are 18 flat 3 manifolds (without boundary): 6 compact orientable, 4 compact non orientable, 4 non compact orientable, 4 non compact non orientable
The six compact flat orientable three manifolds are the Hantzsche-Wendt manifold and the mapping tori of $ T^2 $ for the five finite order mapping classes (diagonalizable and trace $ =0,\pm1,\pm2 $, first five listed above)
The four compact flat non orientable three manifolds are exactly the four mapping tori of the Klein bottle
The four noncompact orientable flat three manifolds are $ \mathbb{R}^3 , \mathbb{R}^2 \times S^1 , \mathbb{R} \times T^2 , Y $
The four noncompact non orientable flat three manifolds are $ S^1 \times M , \mathbb{R} \times M , \mathbb{R} \times K^2 , X $
My questions are: Did I correctly list all mapping tori of the Mobius strip? and Did I correctly list the first homology for all the mapping tori of the Klein bottle?
