Characterizing k-dimensional universal menger compacta

n the past few years H. Toruńczyk and R.D. Edwards-F. Quinn characterized infinite-dimensional manifolds modeled on Q or ℓ₂ and n-dimensional (n ≥ 5) manifolds modeled on Euclidean spaces, respectively. Briefly, if a space X satisfies the correct homotopy-theoretic local property, together with a certain "general positioning" property, then X is a manifold of the expected sort.

The main result of the thesis follows the same pattern: If a compact, (k-1)-connected, locally (k-1)-connected, k-dimensional metric space X has the "disjoint k-cells property" (i.e. any two maps f, g: I^k → X can be approximated by maps with disjoint images), then X is homeomorphic to the k-dimensional universal Menger space µ^k

Using this result we answer in the affirmative the question whether different constructions of the universal k-dimensional compactum appearing in the literature yield the same space.

Also, we prove a few theorems strongly resembling the well-known facts about Q-manifolds. We mention only one example: the Z-set un-knotting theorem. In particular, we show that µ^k is strongly homogeneous.

To prove these theorems, one needs to have a device for constructing homeomorphisms (with control) of µ^k onto itself. To that end, we use a "combinatorial approach": µ^k is represented as the intersection of a nice family of PL manifolds, each of which possesses a "handlebody structure" (where "handles" reflect the properties of µ^k) Using some standard PL-techniques we construct sequences of "handlebody structures" with correct properties, which ultimately yield the desired homeomorphism.

The "philosophical" output can be summarized in one sentence. The universal k-dimensional Menger compactum µ^k and the Hilbert cube Q cannot be distinguished from the point of view of k-dimensional compacta.