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The following general decomposition result is obtained: Suppose K* (pel) is a finite collapsible ^-complex topologically embedded as a subset of a separable metric space X" where, for some r=ï 1, X" x Er is homeomorphic to Euclidean ((3> + /")-space Eq + r. Then the Cartesian product of the quotient space XqjKv with E' is topologically Eq*r provided that <7ä3 and, for each simplex Afc e K", (X'xE', A*x([0, l]r_1 x 0)) is homeomorphic, as pairs, to (£«*%[<U]*+'-lx(0,...,0)). It is known that this condition is satisfied iîq— p^2andq + r^5. This result implies that if K" is a finite collapsible Ar-complex topologically embedded as a subset of Euclidean «-space En, then the Cartesian product of the quotient space En/Kk with E1 is topologically E"*1 provided either (i) nt¿3, (ii) n — k~i2, or (iii) each simplex of A:fc is flat in£" + 1. 0. Introduction and definitions. In [1], by using a result of Klee [14] and some of the techniques developed by Bing in [2], Andrews and Curtis proved that if A is an arc in Euclidean «-space En, then the Cartesian product of the quotient space En/A with E1 is topologically En + 1. The significant aspects of this result appear when «=g3, for in this case En/A is not necessarily homeomorphic to En. In [3], Bryant proves the following result: Theorem 0.1. Suppose that Dk is a k-cell topologically embedded in En. Then (En/Dk) xE1 is homeomorphic to En + 1 provided I. Dk is flat in En + 1, or II. n-k^2. In proving this result under Condition II, Bryant made use of Homma's Approximation Theorem ([12], which later was found to contain a gap) to homeomorphically approximate Dk in En + 1 by a piecewise-linear k-ce\\. However, by recent results obtained in [8] or [16], this approximation may now be obtained, Received by the editors May 1, 1970 and, in revised form, August 17, 1970. AMS 1969 subject classifications. Primary 5478, 5705; Secondary 5425, 5525.

Euclidean $(q+r)$-space modulo an $r$-plane of collapsible $p$-complexes