Our aim is to develop topological analogues of an ongoing programme in toric
geometry, which seeks to express arithmetic, elliptic, and related genera of
toric varieties as functions of their fans. In this context, we introduce
methods for computing equivariant genera of omnioriented quasitoric manifolds M
purely in terms of the combinatorial data (P,\Lambda) by which such M are
determined. We develop the theory around the universal example \Phi, which was
introduced independently by Krichever and Loeffler in 1974, albeit from
radically different viewpoints.