Novikov additivity II (Ex)

This is an exercise in Novikov additivity for normal complexes.

Let $(Y,X)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(Y,X)$ be a $2n$$2n$-dimensional Poincare pair with the fundametal class $[Y]$$[Y]$ which maps to the fundamental class $[X]$$[X]$ under the boundary homomorphism, $n = 2k$$n = 2k$.

Suppose that we have a $2n$$2n$-dimensional normal pair $(Y',X)$$(Y',X)$ whose fundamental class $[Y']$$[Y']$ maps to $-[X]$$-[X]$ under the boundary map. Form the normal space $Z = Y' \cup_{X} Y$$Z = Y' \cup_{X} Y$. Show that

$\displaystyle \partial \textup{sign}^{\mathbf{NL}^{\bullet}} (Z) = \partial \textup{sign}^{(\mathbf{NL}^{\bullet},\mathbf{L}^{\bullet})} (Y',X)$ $$\displaystyle \partial \textup{sign}^{\mathbf{NL}^{\bullet}} (Z) = \partial \textup{sign}^{(\mathbf{NL}^{\bullet},\mathbf{L}^{\bullet})} (Y',X)$$

Hint: check the Novikov additivity exercise and the definition of the quadratic boundary of a (normal,Poincare) pair.