Stochastic order on metric spaces and the ordered Kantorovich monad

In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. The spaces we consider, which we call L-ordered, are spaces where the order satisfies a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is related to the theory of Lawvere metric spaces, in which the partial order structure is induced by the zero distances. We show that the algebras of the ordered Kantorovich monad are the closed convex subsets of Banach spaces equipped with a closed positive cone, with algebra morphisms given by the short and monotone affine maps. Considering the category of L-ordered metric spaces as a locally posetal 2-category, the lax and oplax algebra morphisms are exactly the concave and convex short maps, respectively. In the unordered case, we had identified the Wasserstein space as the colimit of the spaces of empirical distributions of finite sequences. We prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the finite sequences, generalizing a recent result of Lawson. The proof holds on any metric space equipped with a closed partial order.Comment: 49 pages. Removed incorrect statement (Theorem 6.1.10 of previous version

Proper actions and proper invariant metrics

We show that if a (locally compact) group $G$ acts properly on a locally compact $\sigma$-compact space $X$ then there is a family of $G$-invariant proper continuous finite-valued pseudometrics which induces the topology of $X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and only if there exists a $G$-invariant proper compatible metric on $X$.Comment: The paper has been completely rewritten and differs essentially from "Constructing invariant Heine-Borel metrics for proper G-spaces". The main result extended to the more general case when $G$ is a topological group which acts properly on a locally compact $\sigma$-compact Hausdorff space $X$. Note that there is a gap in the proof of Theorem 2.4 of the old versio

Spectral triples for noncommutative solenoidal spaces from self-coverings

Examples of noncommutative self-coverings are described, and spectral triples on the base space are extended to spectral triples on the inductive family of coverings, in such a way that the covering projections are locally isometric. Such triples are shown to converge, in a suitable sense, to a semifinite spectral triple on the direct limit of the tower of coverings, which we call noncommutative solenoidal space. Some of the self-coverings described here are given by the inclusion of the fixed point algebra in a C$^*$-algebra acted upon by a finite abelian group. In all the examples treated here, the noncommutative solenoidal spaces have the same metric dimension and volume as on the base space, but are not quantum compact metric spaces, namely the pseudo-metric induced by the spectral triple does not produce the weak$^*$ topology on the state space.Comment: v3: the paper will appear in the Journal of Mathematical Analysis and Applications, 42 pages, no figure

End spaces and spanning trees

AbstractWe determine when the topological spaces |G| naturally associated with a graph G and its ends are metrizable or compact.In the most natural topology, |G| is metrizable if and only if G has a normal spanning tree. We give two proofs, one of them based on Stone's theorem that metric spaces are paracompact.We show that |G| is compact in the most natural topology if and only if no finite vertex separator of G leaves infinitely many components. When G is countable and connected, this is equivalent to the existence of a locally finite spanning tree. The proof uses ultrafilters and a lemma relating ends to directions

Completions, branched covers, Artin groups and singularity theory

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT(k) metric space for the completion to be CAT(k). We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT(k) if and only if all tangent spaces are CAT(0) and the base has sectional curvature bounded above by k. We also show that the branched cover is a geodesic space. Using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally CAT(k) and the complement of the branch locus to be contractible. We conjecture that the universal branched cover of complex Euclidean n-space over the mirrors of a finite Coxeter group is CAT(0). Conditionally on this conjecture, we use our machinery to prove the Arnol'd-Pham-Thom conjecture on K(pi,1) spaces for Artin groups. Also conditionally, we prove the asphericity of moduli spaces of amply lattice-polarized K3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in Arnol'd's hierarchy

All hypertopologies are hit-and-miss

[EN] We solve a long standing problem by showing that all known hypertopologies are hit-and-miss. Our solution is not merely of theoretical importance. This representation is useful in the study of comparison of the Hausdorff-Bourbaki or H-B uniform topologies and the Wijsman topologies among themselves and with others. Up to now some of these comparisons needed intricate manipulations. The H-B uniform topologies were the subject of intense activity in the 1960's in connection with the Isbell-Smith problem. We show that they are proximally locally finite topologies from which the solution to the above problem follows easily. It is known that the Wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in”nice” metric spaces including the normed linear spaces. With the introduction of a new far-miss topology we show that the Wijsman topology is hit-and-miss for all metric spaces. From this follows a natural generalization of the Wijsman topology to the hyperspace of any T1 space. Several existing results in the literature are easy consequences of our workNaimpally, S. (2002). All hypertopologies are hit-and-miss. Applied General Topology. 3(1):45-53. doi:10.4995/agt.2002.2111SWORD45533