PlanetPhysics/Cubically Thin Homotopy 2

Cubically thin homotopy

Let ${\displaystyle u,u'}$ be squares in ${\displaystyle X}$ with common vertices.

1. A {\it cubically thin homotopy} ${\displaystyle U:u\equiv _{T}^{\square }u'}$

between ${\displaystyle u}$ and ${\displaystyle u'}$ is a cube ${\displaystyle U\in R_{3}^{\square }(X)}$ such that

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• ${\displaystyle U}$ is a homotopy between ${\displaystyle u}$ and ${\displaystyle u',}$

i.e. $\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',$

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• ${\displaystyle U}$ is rel. vertices of ${\displaystyle I^{2},}$

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,

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• the faces ${\displaystyle \partial _{i}^{\alpha }(U)}$ are thin for ${\displaystyle \alpha =\pm 1,\ i=1,2}$.
1. The square ${\displaystyle u}$ is {\it cubically} ${\displaystyle T}$-{\it equivalent} to

${\displaystyle u',}$ denoted ${\displaystyle u\equiv _{T}^{\square }u'}$ if there is a cubically thin homotopy between ${\displaystyle u}$ and ${\displaystyle u'.}$

This definition enables one to construct ${\displaystyle {\boldsymbol {\rho }}_{2}^{\square }(X)}$ , by defining a relation of cubically thin homotopy on the set ${\displaystyle R_{2}^{\square }(X)}$ of squares.

References

1. K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Applied Cat. Structures , 8 (2000): 209-234.
2. R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, {\it Theory and Applications of Categories} 10 ,(2002): 71-93.