Computer Search for Nilpotent Complexes

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 threedimensional nilpotent complex exists was Z 2 ⊕ Z 6. The authors, together with a team of undergraduate students at Fordham University, used computers to search for threedimensional finite nilpotent complexes over groups of the form Z n ⊕ Z m. Such complexes were eventually found for Z 2 ⊕ Z 6, Z 2 ...