Homotopy, simple homotopy and compacta

Abstract

Question 1 is related to a series of questions which have been studied over the past 30 yr. Thus, in 1950 J.H.C. Whitehead proved that such a space is homotopy equivalent to an infinite dimensional CW complex. He also asked ([21, p. 1081) whether such a space necessarily had the homotopy type of a finite dimensional CW complex. In 1957, Milnor ([15], p. 273) asked whether an arbitrary space which is homotopy dominated by a finite CW complex must be homotopy equivalent to some finite complex. In 1%5 Whitehead’s question was answered affirmatively by Mather[l2] and by Wa11[19]. Wall went on to produce counterexamples to Milnor’s question. In more controlled geometric situations, positive results were obtained. In 1969, Kirby and Siebenmann[ 101 proved that compact TOP manifolds have the homotopy types of finite polyhedra, and in 1974 West [20], building on work of Chapman [ 11 and Miller [ 131, proved that compact ANRs have the homotopy types (actually, preferred simple homotopy types) of finite polyhedra. In 1975 Edwards and Geoghegan produced compacta shape dominated by finite CW complexes which are not shape equivalent to finite CW complexes. Siebenmann’s 1965 thesis is also relevant to this problem. We will answer Question 1 by proving the following general theorem: