Property A and large scale paracompactness

Jerzy Dydak and his collaborators have published a very interesting series of papers recently, whose overall theme is that property A is the large scale analog of paracompactness.  The papers that I have seen on the arXiv are these:

Cencelj, M., J. Dydak, and A. Vavpetič. “Asymptotic Dimension, Property A, and Lipschitz Maps.” Revista Matemática Complutense 26, no. 2 (July 1, 2013): 561–571. doi:10.1007/s13163-012-0102-2.

Cencelj, M., J. Dydak, and A. Vavpetič. Coarse Amenability Versus Paracompactness. ArXiv e-print, August 13, 2012.

Dydak, Jerzy. Coarse Amenability and Discreteness. ArXiv e-print, July 15, 2013.

Dydak, Jerzy, and Atish Mitra. “Large Scale Absolute Extensors.” arXiv:1304.5987 (April 22, 2013).

Recall that a Hausdorff topological space \(X\) is paracompact if every open cover of \(X\) has a refinement to a locally finite open cover.  (It is metacompact, or weakly paracompact, if every open cover has a refinement to a pointwise finite open cover.)  Most often though one applies paracompactness via the existence of partitions of unity: \(X\) is paracompact if and only if there exists a (locally finite) partition of unity subordinate to any open cover.  The papers above elucidate what should be the “coarse version” of paracompactness both in the “covering” and the “partition of unity” interpretations, and in both cases they relate it to property A.

According to the first paper above, the “optimal” version of the definition of paracompactness is this: a Hausdorff space \(X\) is paracompact if for each open cover \(\mathscr U\) of \(X\) there exist a simplicial complex \(K\) and a continuous \(f\colon X\to K\) such that the open cover \(f^{-1}(\star K)\) refines \(\mathscr U\).  Here \(\star K\) denotes the star cover of \(K\), that is the open cover of \(K\) by stars of vertices.

A natural “large scale” rendition of this definition would be to say that a metric space \(X\) is large scale paracompact if for each uniformly bounded cover \(\mathscr U\) and each pair \(\lambda,C>0 \) there exist a simplicial complex \(K\) and a \((\lambda,C)\)-Lipschitz map \(f\colon X\to K\) such that \(f^*(\star K) \) is uniformly bounded and coarsens \(\mathscr U\).

Dydak et. al. then prove that this definition is equivalent (at least for bounded geometry spaces) to property A.  Moreover, in the presence of large scale paracompactness, there is a related characterization of asymptotic dimension in terms of retracting maps to simplicial complexes onto their skeleta.

Personally, I don’t know exactly what to make of this.  I don’t have much intuition for failures of paracompactness except that the standard examples have an “uncountable” flavor to them.  (Here’s a discussion of the standard example of \(\Omega\), the first uncountable ordinal.)  But that’s a failing in me, not in this very interesting connection with property A.

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