PlanetPhysics/Cubically Thin Homotopy 2

Cubically thin homotopy

Let $u,u^{\prime }$ be [[../PiecewiseLinear/|squares]] in $X$ with common vertices.

1. A {\it cubically thin homotopy} $U:u\underset{T}{\overset{◻}{\equiv }}{u}^{\prime }$

between $u$ and $u^{\prime }$ is a cube $U\in R_3^{◻}(X)$ such that

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• $U$ is a [[../ThinEquivalence/|homotopy]] between $u$ and $u^{\prime },$

  i.e. Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',}

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• $U$ is rel. vertices of $I^2,$

  i.e. Failed to parse (unknown function "\enskip"): {\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,

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• the faces $\underset{i}{\overset{\alpha }{\partial }}(U)$ are thin for $\alpha =\pm 1,i=1,2$.

1. The square $u$ is {\it cubically} $T$-{\it equivalent} to

$u^{\prime },$ denoted $u\underset{T}{\overset{◻}{\equiv }}{u}^{\prime }$ if there is a cubically thin homotopy between $u$ and $u^{\prime }.$

This definition enables one to construct ${\mathit{\rho }}_2^{◻}(X)$ , by defining a [[../Bijective/|relation]] of cubically thin homotopy on the set $R_2^{◻}(X)$ of squares.