Mapping Hilbert Cube Manifolds to ANR's: A Solution of a Conjecture of Borsuk

Abstract

After Borsuk introduced the notion of absolute neighborhood retracts (ANR's) and J. H. C. Whitehead demonstrated that all ANR's have the homotopy types of cell-complexes [34], the question naturally arose as to whether compact (metric) ANR's must necessarily be homotopy-equivalent to finite cell-complexes. Borsuk expressly posed this conjecture in his address to the Amsterdam Congress in 1954 15], and over the ensuing years considerable progress was made (see [261), including (a) ANR's admitting brick decompositions, by Borsuk [5], (b) the simply connected case, by de Lyra [21], (c) products with the circle, by M. Mather [22], (d) applications of Wall's obstruction to finiteness [29], (e) compact n-manifolds, by Kirby and Siebenmann [16], and (f) compact Hilbert cube manifolds and locally triangulable spaces, by Chapman [9]. The full problem remained open, however. In this paper the conjecture is settled positively by application of Hilbert cube manifold theory. Specifically, it is shown that each compact ANR is the image of some Hilbert cube manifold (Q-manifold) by a cell-like (CE) mapping. Such mappings, between ANR's, are always homotopy-equivalences [14], [17], [20], [28] (although for more general metric compacta they need not be shape-equivalences [27]), so that by appealing to (f) above, the conjecture is settled. In the process, a considerably stronger result is established, namely, that there exists a cell-like map f from a Q-manifold onto the ANR X whose mapping cylinder M(f) is itself a Q-manif old. The mapping cylinder collapse of M(f) to X provides a particularly nice CE-map of a Q-manifold to the ANR. The general outline of the proof is as follows: First, it is shown that the mapping cylinder of a CE-map from a Q-manifold to an ANR is always a Q-manifold. Second, this result is used to show that the ANR A is the image of a Q-manifold by a CE-mapping if and only if whenever A is em-