# Talk:Surgery obstruction map I (Ex)

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$$\sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi)) | $$\sigma(-(\xi,-\phi))+\sigma(-(\xi,\phi)) - \sigma(-(\xi\oplus\xi,\phi * \phi)) | ||

= 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1 | = 2 \langle L(TX\oplus\xi), [X] \rangle - \langle L(TX\oplus\xi\oplus\xi), [X] \rangle -1 | ||

− | = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0 | + | = c \langle p_1(\xi)^2 ,[X] \rangle \ne 0, $$ |

and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum. | and so the surgery obstruction is not a group homomorphism with respect to the Whitney sum. | ||

</wikitex> | </wikitex> |

## Revision as of 21:44, 29 May 2012

We take and consider various bundle reductions of the normal bundle: 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 . The surgery obstruction of a normal map covered by equals

so it depends only on the bundle over . There are fiber homotopically trivial bundles on corresponding to classes in which restrict to any given class in , since the corresponding Atiyah-Hirzebruch spectral sequence collapses. 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 . Now is the sum of and in with respect to the Whitney sum. We compute

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