A universal addition theorem for genera

Abstract

Let M ²ⁿ be a compact, oriented, differentiable manifold and U ⊂ M an oriented, differentiable submanifold of codimension two. Let u ∈ H ²(M; ℤ) be the Poincarédual cohomology class of the fundamental cycle of U. Further, let TU and TM be the tangent bundles of U, resp. M,and let NU be the normal bundle of U in M. By choosing a Riemannian metric we get an isomorphism TU ⊕ NU ≉ TM| _U and NU gets the structure of an O(2)-bundle. Since M and U are oriented, the structure group of NU can be further reduced to SO(2) ≅ U(1).