What are the 8 non-compact Euclidean 3-manifolds?
I have found several sources that mention that there are eight non-compact Euclidean 3-manifolds, with four orientable and four non-orientable. There are two standard references for this, but unfortunately both are in German:
Nowacki, Werner. “Die euklidischen, dreidimensionalen, geschlossenen und offenen Raumformen.” Commentarii Mathematici Helvetici 7, no. 1 (1934): 81–93.
Hantzsche, Walter, and Hilmar Wendt. “Dreidimensionale euklidische Raumformen.” Mathematische Annalen 110, no. 1 (1935): 593–611.
So what are the eight non-compact Euclidean 3-manifolds?
The four orientable non-compact Eulcidean 3-manifolds are:
- $\mathbb{R}^3$,
- $\mathbb{R}^2\times S^1$,
- $\mathbb{R} \times S^1 \times S^1$, and
- Another manifold $X$.
The manifold $X$ can be described as the quotient of $[0,1]\times[0,1]\times\mathbb{R}$ obtained by making the identifications $$ (0,y,z) \sim (1,y,z) \qquad\text{and}\qquad (x,0,z) \sim (1-x,1,-z) $$ for all $x,y\in[0,1]$ and $z\in\mathbb{R}$. A few notes about this manifold:
The image of $[0,1]\times[0,1]\times\{0\}$ is a Klein bottle, and $X$ deformation retracts onto this Klein bottle. Thus $X$ isn't homeomorphic to any of the other three listed possibilities.
$X$ can be described as the vector bundle of $2$-forms on the Klein bottle. Note that this vector bundle is nontrivial since it has no nonzero global section (since the Klein bottle is not orientable).
$X$ can also be described as a cylinder ($S^1\times\mathbb{R}$) bundle over a circle, where going once around the circle “flips” the cylinder, i.e. rotates the cylinder $180^\circ$ around an axis perpendicular to the axis of the cylinder.
The four non-orientable non-compact Euclidean 3-manifolds are:
$M\times \mathbb{R}$,
$M\times S^1$,
$K\times \mathbb{R}$, and
Another manifold $Y$.
Note that (2) and (3) are different since $M\times S^1$ deformation retracts onto a torus and $K\times\mathbb{R}$ deformation retracts onto a Klein bottle.
The manifold $Y$ can be described as the quotient of $[0,1]\times[0,1]\times\mathbb{R}$ obtained by making the identifications $$ (0,y,z) \sim (1,y,-z) \qquad\text{and}\qquad (x,0,z) \sim (1-x,1,z) $$ or alternatively $$ (0,y,z) \sim (1,y,-z) \qquad\text{and}\qquad (x,0,z) \sim (1-x,1,-z) $$ for all $x,y\in[0,1]$ and $z\in\mathbb{R}$. A few notes about this manifold:
The image of $[0,1]\times[0,1]\times\{0\}$ is a Klein bottle, and $Y$ deformation retracts onto this Klein bottle.
However, $Y$ is not homeomorphic to $K\times\mathbb{R}$. For example, $Y$ has just one end, whereas $K\times\mathbb{R}$ has two ends.
$Y$ can also be described as a Mobius band bundle over a circle, where going once around the circle corresponds to a homeomorphism of the Mobius band that is not isotopic to the identity. (There is only one such homeomorphism up to isotopy, which acts as inversion on the fundamental group.)
$Y$ is double covered by the orientable manifold $X$ given above. This also distinguishes it from $K\times \mathbb{R}$, whose orientable double cover is $S^1\times S^1 \times \mathbb{R}$. The deck transformation of $X$ that gives $Y$ is the map $(x,y,z)\mapsto (x+\tfrac12,y,-z)$.
Any line bundle over the Klein bottle is homeomorphic to either $K\times\mathbb{R}$, $X$, or $Y$.