Minimal supports of geometric cycle codes

For an incidence geometry Δ, its geometric cycle code H(Δ) is the top-dimensional homology group over F2 recognized as a subspace of the chain group [5]. Minimal supports, the subchamber systems corresponding to minimal-weight vectors, of H(Δ) are worthy of study, because those for buildings Δ of spherical type coincide with apartments of Δ (Theorem 1). The minimal supports of H(Δ) for the sporadic A7-geometry are also determined (Theorem 2).