  Fast and efficient homology algorithms are in demand in the applied sciences
for analyzing solid materials and proteins, processing digital imaging data, or
pattern classification among others. Recent advances employ discrete Morse
theory as a preprocessor. Research in this area has lead to the need to find
complicated test examples. We present an infinite series of examples that have
been constructed to test some of the latest algorithms under development. This
family of 4-spheres (known as the Akbulut-Kirby spheres) is based on a
handlebody construction via finitely presented groups.
