# Talk:Surgery obstruction map I (Ex)

If is a manifold, then the normal map gives the base point of . An element of is given by a bundle together with a fiber homotopy trivialization . Under the isomorphism , the pair corresponds to a normal map covered by . Assume that the dimension is and that is simply connected. Then the surgery obstruction of a normal map covered by equals

by the Hirzebruch signature theorem and several properties of the -genus. In particular the surgery obstruction depends only on the bundle over . Now is the sum of and in with respect to the Whitney sum. Moreover

If this is non-zero, then the surgery obstruction is not a group homomorphism with respect to the Whitney sum.

As an example take :

There are fiber homotopically trivial bundles on corresponding to classes in which restrict to any given class in , as follows from the Puppe sequence with . From another exercise we know that on we have such vector bundles with first Pontryagin class times the generator of . This means that on we have a vector bundle with whose sphere bundle is fiber homotopically trivial, by a fiber homotopy equivalence . We compute

where the constant can be computed from the L-genus to be .

So the surgery obstruction is not a group homomorphism with respect to the Whitney sum.