<font size=2> In the example below, we use a database for torsion free space groups, which is part of the HTML documents for CARAT. If your browser does not know frames, you have to go to this page after reading the example. Otherwise you should see this page in the frame on the left. </font>

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! <hr width=80%> <center> <font size=2> <a href="http://www-math.math.rwth-aachen.de/~LBFM/carat/bieberbach.html" target="_top">Go</a> to the page of torsion free space groups! <table border=0 cellpadding=5> <tr align=center> <td><font size=2><a href="./Ex10.html" target="_parent">Previous Example</a></font></td> <td><font size=2><a href="../introduction.html" target="_parent">Introduction</a></font></td> <td><font size=2><a href="./Ex12.html" target="_parent">Next Example</a></font></td> </tr> </table> </font> </center>
last change: 13.10.2000