Miraculous Cancellation and Pick's Theorem

We show that the Cappell–Shaneson version of Pick’s theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula of Alvarez-Gaume and Witten, and is imposed by the action of the Landweber–Novikov algebra in the complex cobordism ring of a point. Introduction Studying gravitational anomalies Alvarez-Gaume and Witten [1] discovered a remarkable cancellation which they called the miraculous cancellation formula. The formula is a consequence of the following relation between Pontryagin characteristic numbers of 12-dimensional oriented manifolds: (1) L(M) = 8A(M,T )− 32Â(M), where L(M) is the L-genus (the signature) of M corresponding to the power series L(x) = x/ tanh(x), Â(M) is the Â-genus of M corresponding to the power series A(x) = x/2 sinh(x/2), and A(M,T ) is the twisted Â-genus defined by cohomology Kronecker product 〈 ∏