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 Poincare-dual 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). The bundle NU can therefore be considered as a complex line bundle which has the first Chern class i*(u) (i: U → M inclusion), i.e.: $$ \begin{array}{l} c(NU) = 1 + {c_1}(NU) = 1 + i*(u) \Rightarrow p(NU) = 1 + i*({u^2}) \end{array} $$ .