Trivially extending decompositions of eⁿ

Let G be a monotone decomposition of Eⁿ, then G can be extended in a trivial way, to the monotone decomposition G¹ of E^{n + 1}, where Eⁿ = {{x₁ … x_n, 0) ∈ E^{n + 1}}, by adding to G all points of E^{n + 1} − Eⁿ. If the decomposition space Eⁿ/G of G is homeomorphic to Eⁿ, Eⁿ\G is said to be obtained by a pseudoisotopy if there exists a map F: Eⁿ × I→ Eⁿ × I, such that F _t(=f/Eⁿ × t) is homeomorphism onto Eⁿ × 1, for all 0 ≦; 1 < 1, F₀ is the identity and F₁ is equivalent to the projection The purpose of this paper is to present a relation between these two notions. It will then follow, that if G is the decomposition of E³ to points, circles and figure-eights, due to R. H. Bing, for which E³/G is homeomorphic to E³ then E⁴/G¹ is not homeomorphic to E⁴.