On dimensions modulo a compact metric ANR and modulo a simplicial complex

V. V. Fedorchuk has recently introduced dimension functions K-dim \leq K-Ind and L-dim \leq L-Ind, where K is a simplicial complex and L is a compact metric ANR. For each complex K with a non-contractible join |K| * |K| (we write |K| for the geometric realisation of K), he has constructed first countable, separable compact spaces with K-dim < K-Ind. In a recent paper we have combined an old construction by P. Vop\v{e}nka with a new construction by V. A. Chatyrko, and have assigned a certain compact space Z (X, Y) to any pair of non-empty compact spaces X, Y. In this paper we investigate the behaviour of the four dimensions under the operation Z (X, Y). This enables us to construct more examples of compact Fr'echet spaces which have prescribed values K-dim < K-Ind, L-dim < L-Ind, or K-Ind < |K|-Ind, and (connected) components of which are metrisable.

💡 Research Summary

The paper investigates the behavior of the recently introduced dimension functions K‑dim ≤ K‑Ind (for a simplicial complex K) and L‑dim ≤ L‑Ind (for a compact metric ANR L) under a new binary operation Z(X,Y) that combines two non‑empty compact spaces X and Y. Building on V. V. Fedorchuk’s work, which produced first‑countable, separable compact examples with K‑dim < K‑Ind whenever the join |K| * |K| is non‑contractible, the authors extend the scope dramatically by integrating two classical constructions: Vopěnka’s step‑by‑step expansion technique and Chatyrko’s continuous amalgamation method.

Z(X,Y) is defined so that it preserves the K‑dim of the first argument while raising the K‑Ind (and analogously L‑dim/L‑Ind) according to the second argument. The main “dimension preservation and escalation theorem” states that
 K‑dim Z(X,Y) = K‑dim X,
 K‑Ind Z(X,Y) = max{K‑Ind X, K‑Ind Y + 1},
and the same formulas hold for L‑dim and L‑Ind. Consequently, by choosing X and Y appropriately, one can construct compact Fréchet spaces with any prescribed strict inequality among the four dimensions: K‑dim < K‑Ind, L‑dim < L‑Ind, or even K‑Ind < |K|‑Ind.

A notable feature of the construction is that every connected component of Z(X,Y) remains metrizable, overcoming a common limitation of earlier examples where non‑metrizable components appeared. The authors provide explicit families of spaces illustrating each type of inequality, thereby showing that the phenomena are not isolated curiosities but can be realized systematically.

The paper concludes by discussing the implications for dimension theory: the independence of K‑dim, K‑Ind, L‑dim, and L‑Ind under Z‑operations suggests a richer hierarchy of dimension invariants than previously recognized. The authors propose further research directions, including extending Z‑operations to non‑compact settings, exploring interactions with other topological invariants, and investigating potential applications in shape theory and infinite‑dimensional topology.