nLab equivariant cell structure on 2-tori -- section

Equivariant cell structure on 2-tori

The following shows, for the symmorphic 2D crystallographic groups (wallpaper groups) $G \ltimes \mathbb{Z}^2 \subset Iso(2)$, the $G$-CW complex structure on the resulting tori $T^2 \coloneqq \mathbb{R}^2/\mathbb{Z}^2$ (graphics from SS25).

The point groups $G$ arising are the cyclic groups $\mathbb{Z}_{/1} =$$1$, $\mathbb{Z}_{/2}$, $\mathbb{Z}_{/3}$, $\mathbb{Z}_{/4}$, $\mathbb{Z}_{/6}$ and the dihedral groups $Dih_1$, $Dih_2$, $Dih_3$, $Dih_4$, and $Dih_6$ (making 10 distinct point groups $G$, but the first three of the latter come with two inequivalent actions each).

Example

(p1) For completeness, here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with trivial group-action, corresponding to the $p 1$ wallpaper group:

Example

(pm) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $Dih_1$ $\simeq$ $\mathbb{Z}_{/2}$-action which reflects one of the two coordinate axes, corresponding to the $p m$ wallpaper group:

Example

(cm) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $Dih_1$ $\simeq$ $\mathbb{Z}_{/2}$-action which reflects along the coordinate diagonal, corresponding to the $c m$ wallpaper group:

Example

(p2) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $\mathbb{Z}_{/2}$-action which rotates by multiples of $\pi$ around the origin, corresponding to the wallpaper group $p2$:

Example

(pmm) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with $Dih_{2}$-action according to the wallpaper group pmm:

Example

(cmm) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with $Dih_{2}$-action according to the wallpaper group cmm:

Example

(p4) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $\mathbb{Z}_{/4}$-action which rotates by multiples of $\pi/2$ around the origin:

Example

(p4m) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with $Dih_{4}$-action:

Example

(p3) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $\mathbb{Z}_{/3}$-action which rotates by multiples of $2\pi/3$ around the origin:

Example

(p31m) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with $Dih_{3}$-action corresponding to the $p31m$ wallpaper group:

Example

(p3m1) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $Dih_{3}$-action corresponding to the $p3m1$ wallpaper group:

Example

(p6) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with the $\mathbb{Z}_{/6}$-action which rotates by multiples of $\pi/3$ around the origin:

Example

(p6m) Here is a $G$-CW complex structure for the torus $T^2 \,\equiv\, \mathbb{R}^2/\mathbb{Z}^2$ equipped with $Dih_{6}$-action: