Talk:Surgery obstruction map I (Ex)

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We take X=\mathbb H P^2 and consider various bundle reductions of the normal bundle: The normal map id_X gives the base point of \mathcal N (X). The surgery obstruction of a normal map M\to X covered by \nu_M\to \xi equals

\displaystyle  sign(M)-sign(X)=\langle L(-\xi),[X]\rangle -1,

so it depends only on the bundle over X. There are fiber homotopically trivial bundles on X corresponding to classes in [X,G/TOP] which restrict to any given class in [S^4,G/Top], since the corresponding Atiyah-Hirzebruch spectral sequence collapses. From another exercise we know that on S^4 we have such vector bundles with first Pontryagin class 48k times the generator of H^4(S^4). This means that on X we have vector bundles \xi_1,\xi_2 whose sphere bundles are fiber homotopically trivial, by fiber homotopy equivalences \phi_i. Then (\xi_1\oplus\xi_2,\phi_i * \phi_2) is the sum of (\xi_1,\phi_1) and (\xi_2,\phi_2) in \mathcal N (X)=[X,G/O] with respect to the Whitney sum. Now we compute

\displaystyle \sigma(\xi_1,\phi_1)+\sigma(\xi_2,\phi_2) - \sigma(\xi_1\oplus\xi_2,\phi_i * \phi_2) = \langle L(TX\oplus\xi_1), [X] \rangle +  \langle L(TX\oplus\xi_2), [X] \rangle - \langle L(TX\oplus\xi_1\oplus\xi_2), [X] \rangle -\langle L(TX), [X] \rangle.

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