2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112805By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spin$^c$ manifolds. Especially, we get twisted Rokhlin congruences for $8k+4$ dimensional spin$^c$ manifolds30 pages, new contents addedDifferential GeometryAlgebraic TopologyPrimary 53C20, 57R20; Secondary 53C80, 11Z05text