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 $f:M\rightarrow X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}f:M\rightarrow X$ be a degree 1 normal map from a $2k$$2k$-dimensional (resp. $(2k+1)$$(2k+1)$-dimensional) manifold to a geometric Poincaré complex, inducing the isomorphism $f_\ast:\pi_1(M)\cong\pi_1(X)=:\pi$$f_\ast:\pi_1(M)\cong\pi_1(X)=:\pi$. Denote by $K_i(M)=K_i(\widetilde{M})$$K_i(M)=K_i(\widetilde{M})$ the homology surgery kernel $\mathbb{Z}[\pi]$$\mathbb{Z}[\pi]$-module. If $f$$f$ is $k$$k$-connected the kernel module $K_k(M)$$K_k(M)$ 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 $R$$R$ be a ring with involution and $C=C_\ast$$C=C_\ast$ a finite chain complex of finitely generated projective (left) $R$$R$-modules.

1) If for $i<n$$i<n$, $H_i(C)=0$$H_i(C)=0$ for some integer $n$$n$ then the $R$$R$-module $H_n(C)$$H_n(C)$ is finitely generated and

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

is an isomorphism.

2) If in addition for $j>n$$j>n$, $H^j(C)=0$$H^j(C)=0$ for the same integer $n$$n$ then there are isomorphisms

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

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

where under further assumption that $C$$C$ is a chain complex of free modules, the latter isomorphism implies that $H_n(C)$$H_n(C)$ and $H^n(C)$$H^n(C)$ are stably free and hence $H_n(C)$$H_n(C)$ and $H^n(C)$$H^n(C)$ are dual.

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].