Surgery obstruction map I (Ex)

Revision as of 17:16, 28 May 2012

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 $<wikitex>; 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}}. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]n = 4k$, find a formula for $\theta$ in terms of the $\mathcal{L}$-class. See Exercise 13.3 in [Ranicki2002].

References

• [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001