Middle-dimensional surgery kernel (Ex)

Proposition 0.1. Let $<wikitex>; The goal of this exercise is to prove the following statement which will be indispensible in defining the surgery obstruction. {{beginthm|Proposition}} Let $f:M\rightarrow X$ be a degree 1 normal map from a k$-dimensional (resp. $(2k+1)$-dimensional) manifold to a geometric Poincaré complex, inducing the isomorphism $f_\ast:\pi_1(M)\cong\pi_1(X)=:\pi$. Denote by $K_i(M)=K_i(\widetilde{M})$ the homology surgery kernel $\mathbb{Z}[\pi]$-module. If $f$ is $k$-connected the kernel module $K_k(M)$ is finitely generated and stably free. {{endthm|Proposition}} The statement essentially follows from the technical lemma which you are asked to prove. {{beginthm|Lemma|{{citeD|Ranicki2002|Lemma 10.26}}}} Let $R$ be a ring with involution and $C=C_\ast$ a finite chain complex of finitely generated projective (left) $R$-modules. '''1)''' If for $i<n$, $H_i(C)=0$ for some integer $n$ then the $R$-module $H_n(C)$ is finitely generated and$$H^n(C)\rightarrow H_n(C)^\ast, \quad f\mapsto (x\mapsto f(x)).$$is an isomorphism. '''2)''' If in addition for $j>n$, $H^j(C)=0$ for the same integer $n$ then $H_n(C)$ and $H^n(C)$ are dual, and there are isomorphisms$$H_n(C)\oplus\sum_{i\in\mathbb{Z}}C_{n+2i+1}\cong\sum_{j\in\mathbb{Z}}C_{n+2j}, $$$$H^n(C)\oplus\sum_{i\in\mathbb{Z}}C^{n+2i+1}\cong\sum_{j\in\mathbb{Z}}C^{n+2j},$$where under further assumption that $C$ is a chain complex of free modules, the latter isomorphism implies that $H_n(C)$ and $H^n(C)$ are stably free. {{endthm|Lemma}} The proposition is given as lemma 4.19 in {{citeD|Lück2001}}, however the proof is incomplete. Alternatively a good proof can be found in {{citeD|Wall1999}} and a more detailed one in {{citeD|Ranicki2002}}. </wikitex> [[Category:Exercises]]f:M\rightarrow X$ be a degree 1 normal map from a $2k$-dimensional (resp. $(2k+1)$-dimensional) manifold to a geometric Poincaré complex, inducing the isomorphism $f_\ast:\pi_1(M)\cong\pi_1(X)=:\pi$. Denote by $K_i(M)=K_i(\widetilde{M})$ the homology surgery kernel $\mathbb{Z}[\pi]$-module. If $f$ is $k$-connected the kernel module $K_k(M)$ is finitely generated and stably free.

Lemma 0.2 [Ranicki2002, Lemma 10.26]. Let $R$ be a ring with involution and $C=C_\ast$ a finite chain complex of finitely generated projective (left) $R$-modules.

$$\displaystyle H^n(C)\rightarrow H_n(C)^\ast, \quad f\mapsto (x\mapsto f(x)).$$

$$\displaystyle H_n(C)\oplus\sum_{i\in\mathbb{Z}}C_{n+2i+1}\cong\sum_{j\in\mathbb{Z}}C_{n+2j},$$

$$\displaystyle H^n(C)\oplus\sum_{i\in\mathbb{Z}}C^{n+2i+1}\cong\sum_{j\in\mathbb{Z}}C^{n+2j},$$