Structured chain complexes III (Ex)

Let $(\bar \Zz,\nu)$$<wikitex>; Let $(\bar \Zz,\nu)$ be the <script type="math/tex">(\bar \Zz,\nu)$ be the $0$$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

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

References

$-dimensional symmetric complex where $$ \bar \Zz_i := \begin{cases} \Zz & i = 0 \ 0 & i \neq 0, \end{cases} $$ and the symmetric structure is given by $$ \nu = 1 \in (\bar \Zz \otimes \bar \Zz)_0. $$ Let $I$ be the cellular chain complex of the unit interval and let $i_0$, $i_1$ be the inclusions of the two ends. Find a (\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

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

References

$-dimensional chain $\omega \in W^{\%}(I)$ such that $$ d_{W^{\%}{I}} (\omega) = i_1^{\%} (\nu) - i_0^{\%} (\nu). $$ == References== {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]](\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

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