The concept of nilpotency for a topological space is a generalization
of simple connectivity. That it is a fruitful generalization was
shown by Dror, Kan, Bousfield, Hilton, and others. In 1977 Brown and
Kahn proved that the dimension of a nilpotent complex can be read from
the ordinary homology groups, just as in the case of a simply
connected complex. They also showed that if a nilpotent complex has
finite and nontrivial fundamental group, its dimension must be at
least 3.
¶ In 1985 Lewis showed that for any finite nilpotent group there is a
(not necessarily finite) three-dimensional nilpotent complex with that
fundamental group. The smallest finite nilpotent group for which it
was unknown whether a finite three-dimensional nilpotent
complex exists was $\bold Z_2 \oplus \bold Z_6$.
¶ The authors, together with a team of undergraduate students at Fordham
University, used computers to search for three-dimensional finite
nilpotent complexes over groups of the form $\bold Z_n \oplus \bold Z_m$. Such
complexes were eventually found for $\bold Z_2 \oplus \bold Z_6$, $\bold Z_2 \oplus
\bold Z_{10}$, and $\bold Z_3 \oplus \bold Z_6$.
¶ This article describes the strategy for constructing nilpotent
complexes of dimension three, and some of the issues in implementing
the computer search. The main computational issues are
"normalizing'' matrices, especially to the Smith normal form, and
mapping matrices over $\bold Z$ to matrices over $\bold Z_p$ for various primes
p. We conclude with a summary of the complexes discovered and open
questions.