  We introduce a homological approach to exhibiting instances of Stembridge's
q=-1 phenomenon. This approach is shown to explain two important instances of
the phenomenon, namely that of partitions whose Ferrers diagrams fit in a
rectangle of fixed size and that of plane partitions fitting in a box of fixed
size. A more general framework of invariant and coinvariant complexes with
coefficients taken mod 2 is developed, and as a part of this story an analogous
homological result for necklaces is conjectured.
