Sphere Triangulations and their Double Homology

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.