0.1 Cubically thin homotopy

Let u,u′ be squares in X with common vertices.

1. A cubically thin homotopy U : u ≡_T ^□u′ between u and u′ is a cube U ∈ R ₃^□(X) such that
   • U is a homotopy between u and u′,

     i.e. ∂₁⁻(U) = u, ∂ ₁⁺(U) = u′,

   • U is rel. vertices of I²,

     i.e. ∂₂⁻∂ ₂⁻(U), ∂ ₂⁻∂ ₂⁺(U), ∂ ₂⁺∂ ₂⁻(U), ∂ ₂⁺∂ ₂⁺(U) are constant,

   • the faces ∂_i^α(U) are thin for α = ±1, i = 1, 2.
2. The square u is cubically T-equivalent to u′, denoted u ≡_T ^□u′ if there is a cubically thin homotopy between u and u′.

This definition enables one to construct ρ₂^□(X) , by defining a relation of cubically thin homotopy on the set R₂^□(X) of squares.

References