Applications of Modularity in Topology

generalization was given by Atiyah and Hirzebruch [AH], which states that ∧ Agenus of a closed oriented smooth spin manifold is an even integer. Landweber [La] shows that we can use the elliptic genus to get the Ochanine result directly from the divisibility results of Atiyah and Hizebruch [AH]. In 1972, Rokhlin [R2] established a congruence formula of the type φ(B) ≡ Sign(M)−Sign(B·B) 8 mod 2Z, where B ·B is the self-intersection of B in M , and B is an orientable characteristic submanifold of M , which is the Poincaré dual of the scond class of tangent bundle. Ochanine [O] generalized this congruence formula to the case of 8k + 4 dimensional closed spin manifolds. Guillon and Marin [GM] generalizes the result when B might be non-orientable. On the physics aspect, when Alvarez-Gaumé and Witten [AW] directly computed gravitational anomaly, they discoverd a so called ”Miraculous Cancella-