CARAT Introduction / Examples / Find the fundamental group of the Handtsche-Wendt manifold!
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 11 - Find the fundamental group of the Handtsche-Wendt manifold!
Keywords: CARAT, crystallography, spacegroups, space groups, space group, spacegroup, mathematics, algorithms, tables, algorithm, table, examples, example, Bieberbach, Bieberbach group, group, torsionfree, Bieberbachgroup, centre, trivial, manifold, Handtsche, Wendt, fundamental, fundamental group, torsion, torsion free

### Example 11

Find the fundamental group of the Handtsche-Wendt manifold.

### Solution

Note that the fundamental group of the Handtsche-Wendt manifold is the only 3-dimensional torsion free space group with trivial center.

1. Select "Degree" 3.
2. Choose "Yes" for "Has trivial center".
3. Choose "all" for "Order of point group".
4. Choose "all" for "Isomorphism type of point group".
5. Click on the button "Count selectes groups". We see that there is only one torsion free space group with trivial center.
6. Click on "Show groups" to see the fundamental group.

Note that the program Torsionfree decides whether a given space group is a torsion free space group. If so, the program additionally decides whether it has trivial center. See also for Example 12!
last change: 13.10.2000