# PlanetPhysics/Cubically Thin Homotopy 2

### Cubically thin homotopyEdit

Let be squares in with common vertices.

- A {\it cubically thin homotopy}

between and is a cube such that

#

- is a homotopy between and

#i.e.

**Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',}** - is rel. vertices of

#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, - the faces are thin for .

- The square is {\it cubically} -{\it equivalent} to

denoted if there is a cubically thin homotopy between and

This definition enables one to construct , by defining a relation of cubically thin homotopy on the set of squares.

## All SourcesEdit

^{[1]}^{[2]}