Middle-dimensional surgery kernel (Ex)
The goal of the following 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 is stated here as an exercise.
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
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]. In fact the exercise is borrowed from the latter where it is stated verbatim as a lemma.