= Binary family of nonhomeomorphic Sunada quotients
{title2=$2^n\text{ examples in dimension }4$}
= Binary families of nonhomeomorphic Sunada quotients
{synonym}
Choose distinct odd primes and take products of the <nonisomorphic Gassmann equivalent regular subgroups>, selecting one of the two groups at each prime. Product conjugacy-class counts give $2^n$ <Gassmann equivalent> <subgroups> of the same finite product of symmetric groups. Their <fundamental groups> are distinguished by whether the unique Sylow <subgroup> at each chosen prime is abelian. The <closed-manifold realization of a finitely presented fundamental group> and <Sunada theorem> therefore produce $2^n$ pairwise nonhomeomorphic, and hence nonisometric, isospectral closed four-manifolds. Gassmann equivalence alone, without a topology distinction, does not imply nonhomeomorphism.
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