Structured chain complexes III (Ex)

Let $(\bar \Zz,\nu)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(\bar \Zz,\nu)$ be the $0$$0$-dimensional symmetric complex where

$\displaystyle \bar \Zz_i := \begin{cases} \Zz & i = 0 \\ 0 & i \neq 0, \end{cases}$ $$\displaystyle \bar \Zz_i := \begin{cases} \Zz & i = 0 \\ 0 & i \neq 0, \end{cases}$$

and the symmetric structure is given by

$\displaystyle \nu = 1 \in (\bar \Zz \otimes \bar \Zz)_0.$ $$\displaystyle \nu = 1 \in (\bar \Zz \otimes \bar \Zz)_0.$$

Let $I$$I$ be the cellular chain complex of the unit interval and let $i_0$$i_0$, $i_1$$i_1$ be the inclusions of the two ends. Find a $1$$1$-dimensional chain $\omega \in W^{\%}(I)$$\omega \in W^{\%}(I)$ such that

$\displaystyle d_{W^{\%}{I}} (\omega) = i_1^{\%} (\nu) - i_0^{\%} (\nu).$ $$\displaystyle d_{W^{\%}{I}} (\omega) = i_1^{\%} (\nu) - i_0^{\%} (\nu).$$