  We study the double homology associated to triangulated spheres and present
two results. First, we explicitly compute the double homology for minimum
degree sphere triangulations. Using a spectral sequence argument, we compute
the effect of removing a maximal simplex of a non-neighborly sphere
triangulation. Using these results and computational aid we generate complexes
with exotic double homology rank. We also relate the double homology of a
complex with how neighborly it is.
