# Talk:Surgery obstruction map I (Ex)

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Revision as of 21:28, 29 May 2012 by Martin Olbermann (Talk | contribs)

We take and consider various bundle reductions of the normal bundle: The normal map gives the base point of . 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 vector bundles whose sphere bundles are fiber homotopically trivial, by fiber homotopy equivalences . Then is the sum of and in with respect to the Whitney sum. Now we compute