# Surgery obstruction map I (Ex)

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Show that the surgery obstruction map

$\displaystyle \theta \colon \mathcal{N} (X) \rightarrow L_{n} (\Zz [\pi_1 (X)])$

is not in general a homomorphism of abelian groups, when the normal invariants are viewed as an abelian group with the group structure coming from the Whitney sum of vector bundles.

Hint: in the simply connected case and $n = 4k$$; Show that the surgery obstruction map \theta \colon \mathcal{N} (X) \rightarrow L_{n} (\Zz [\pi_1 (X)]) is not in general a homomorphism of abelian groups, when the normal invariants are viewed as an abelian group with the group structure coming from the Whitney sum of vector bundles. Hint: in the simply connected case and n = 4k, find a formula for \theta in terms of the \mathcal{L}-class. See Exercise 13.3 in {{cite|Ranicki2002}}. == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises with solution]]n = 4k$, find a formula for $\theta$$\theta$ in terms of the $\mathcal{L}$$\mathcal{L}$-class. See Exercise 13.3 in [Ranicki2002].