Vol. 29, No. 3, 1969

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Trivially extending decompositions of En

Joseph Zaks

Vol. 29 (1969), No. 3, 727–729

Let G be a monotone decomposition of En, then G can be extended in a trivial way, to the monotone decomposition G1 of En+1, where En = {(x1,,xn,0) En+1}, by adding to G all points of En+1 En. If the decomposition space En∕G of G is homeomorphic to En, En∕G is said to be obtained by a pseudo-isotopy if there exists a map F : En × I En × I, such that Ft(= F|En × t) is homeomorphism onto En × t, for all 0 t < 1, F0 is the identity and F1 is equivalent to the projection En En∕G.

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.

Mathematical Subject Classification
Primary: 54.78
Received: 21 May 1968
Published: 1 June 1969
Joseph Zaks