Reconstruction of compacta by finite approximations and Inverse Persistence

Polyhedral expansions of compacta associated to finite approximations

This paper introduces some inverse sequences of different polyhedra all based on finite approximations of a compact metric space so they can be used to capture the shape type of the original space. It is shown that they are HPol-expansions, proving the so-called General Principle. We use these sequences to compute explicitly some inverse persistent homology groups of a space and measure its errors in the approximation process.

可数度量与超度量紧统的有限逼近

Finite approximations of countable metric and ultrametric compacta

通过改编同伦重构定理，证明可数度量或超度量紧空间可由有限 $T_0$ 空间序列的逆极限进行拓扑重构。

Gromov boundaries as Markov compacta

We prove that the Gromov boundary of every hyperbolic group is homeomorphic to some Markov compactum. Our reasoning is based on constructing a sequence of covers of $\partial G$, which is quasi-$G$-invariant wrt. the ball $N$-type (defined by Cannon) for $N$ sufficiently large. We also ensure certain additional properties for the inverse system representing $\partial G$, leading to a finite description which defines it uniquely. By defining a natural metric on the inverse limit $\lim K_n$ and proving it to be bi-Lipschitz equivalent to an accordingly chosen visual metric on $\partial G$, we pr…

Steenrod homotopy

Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. We attempt to simplify foundations of the theory and to clarify and improve some of its major results. Using geometric tools such as Milnor's telescope compactification, comanifolds (=mock bundles) and the Pontryagin-Thom Construction, we obtain n…