# Middle-dimensional surgery kernel (Ex)

The goal of this exercise is to prove the following statement which will be indispensible in defining the surgery obstruction.

**Proposition 0.1.** Let be a degree 1 normal map from a -dimensional (resp. -dimensional) manifold to a geometric Poincaré complex, inducing the isomorphism . Denote by the homology surgery kernel -module. If is -connected the kernel module is finitely generated and stably free.

The statement essentially follows from the technical lemma which you are asked to prove.

**Lemma 0.2** [Ranicki2002, Lemma 10.26]**.**
Let be a ring with involution and a finite chain complex of finitely generated projective (left) -modules.

**1)**If for , for some integer then the -module is finitely generated and

**2)**If in addition for , for the same integer then there are isomorphisms

The proposition is given as lemma 4.19 in [Lück2001], however the proof is incomplete. Alternatively a good proof can be found in [Wall1999] and a more detailed one in [Ranicki2002].