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