# PlanetPhysics/Cubically Thin Homotopy 2

### Cubically thin homotopy

Let $u,u'$ be squares in $X$ with common vertices.

1. A {\it cubically thin homotopy} $U:u\equiv _{T}^{\square }u'$ between $u$ and $u'$ is a cube $U\in R_{3}^{\square }(X)$ such that

 #

• $U$ is a homotopy between $u$ and $u',$ \begin{center} i.e. $\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',$ \end{center} #
• $U$ is rel. vertices of $I^{2},$ \begin{center} i.e. $\displaystyle \partial^{-}_2\partial^{-}_2 (U),\enskip\partial^{-}_2 \partial^{+}_2 (U),\enskip \partial^{+}_2\partial^{-}_2 (U),\enskip\partial^{+}_2 \partial^{+}_2 (U)$ are constant,\end{center} #
• the faces $\partial _{i}^{\alpha }(U)$ are thin for $\alpha =\pm 1,\ i=1,2$ .
1. The square $u$ is {\it cubically} $T$ -{\it equivalent} to

$u',$ denoted $u\equiv _{T}^{\square }u'$ if there is a cubically thin homotopy between $u$ and $u'.$ This definition enables one to construct ${\boldsymbol {\rho }}_{2}^{\square }(X)$ , by defining a relation of cubically thin homotopy on the set $R_{2}^{\square }(X)$ of squares.