Algebraic surgery X (Ex)

Let $(f,b) \colon M \rightarrow X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(f,b) \colon M \rightarrow X$ be a degree one normal map of $n$$n$-GPC. Denote by $\nu_M$$\nu_M$, $\nu_X$$\nu_X$ the respective SNFs. We form the $(n+1)$$(n+1)$-dimensional geometric (normal, Poincaré) pair

$\displaystyle \big( (W,M \sqcup X), (\nu_W,\nu_{M \sqcup X}), (\rho_W,\rho_{M \sqcup X}) \big)$ $$\displaystyle \big( (W,M \sqcup X), (\nu_W,\nu_{M \sqcup X}), (\rho_W,\rho_{M \sqcup X}) \big)$$

with $W = \textup{cyl} (f)$$W = \textup{cyl} (f)$. The symbol $\nu_W$$\nu_W$ denotes the $k$$k$-spherical fibration over $W$$W$ induced by $b$$b$ and

$\displaystyle (\rho_W,\rho_{M \sqcup X}) \colon (D^{n+1+k},S^{n+k}) \rightarrow (\textup{Th} (\nu_W), \textup{Th} (\nu_M \sqcup \nu_X))$ $$\displaystyle (\rho_W,\rho_{M \sqcup X}) \colon (D^{n+1+k},S^{n+k}) \rightarrow (\textup{Th} (\nu_W), \textup{Th} (\nu_M \sqcup \nu_X))$$

is the map induced by $\rho_M$$\rho_M$ and $\rho_X$$\rho_X$ and denote $j \colon M \sqcup X \hookrightarrow W$$j \colon M \sqcup X \hookrightarrow W$.

Let $C'$$C'$ be the underlying chain complex obtained by algebraic surgery on the $(n+1)$$(n+1)$-dimensional symmetric pair

$\displaystyle (j_\ast \colon C(M) \oplus C(X) \rightarrow C(W),(\delta \varphi,\varphi)).$ $$\displaystyle (j_\ast \colon C(M) \oplus C(X) \rightarrow C(W),(\delta \varphi,\varphi)).$$

Show that it is homotopy equivalent to the mapping cone $\mathcal{C} (f^{!})$$\mathcal{C} (f^{!})$ of the 'Umkehr' map associated to $(f,b)$$(f,b)$.