Structured chain complexes II (Ex)

Let

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$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}f,g \co C \ra D$ be two chain maps between bounded chain complexes over~$R$$R$. Show that the following hold:

1.)

$\displaystyle ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_0 = (1+T) (f \otimes g) \varphi_0, \\ ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_s = \; 0 \; \textup{for} \; s \geq 1,$ $$\displaystyle ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_0 = (1+T) (f \otimes g) \varphi_0, \\ ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_s = \; 0 \; \textup{for} \; s \geq 1,$$

2.)

$\displaystyle ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_0 = (f \otimes g) (1+T) \psi_0, \\ ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_s = 0 \; \textup{for} \; s \geq 1,$ $$\displaystyle ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_0 = (f \otimes g) (1+T) \psi_0, \\ ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_s = 0 \; \textup{for} \; s \geq 1,$$

3.)

$\displaystyle ((\widehat{f + g})^{\%} (\varphi) - \widehat{f}^{\%} (\varphi) - \widehat{g}^{\%} (\varphi))_s = 0 \; \textup{for} \; s \geq 0.$ $$\displaystyle ((\widehat{f + g})^{\%} (\varphi) - \widehat{f}^{\%} (\varphi) - \widehat{g}^{\%} (\varphi))_s = 0 \; \textup{for} \; s \geq 0.$$