# Discrete Homotopy Theory and Critical Values of Metric Spaces

@article{Conant2012DiscreteHT, title={Discrete Homotopy Theory and Critical Values of Metric Spaces}, author={James F. Conant and Victoria Curnutte and Corey Jones and Conrad Plaut and Kristen Pueschel and Maria Walpole and Jay Wilkins}, journal={arXiv: Metric Geometry}, year={2012} }

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3/2. The latter two spectra are known to be discrete for compact geodesic…

## 13 Citations

### Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces

- Mathematics
- 2011

Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines…

### Essential circles and Gromov–Hausdorff convergence of covers

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The δ-covers of Sormani–Wei ([20]) are known not to be “closed” with respect to Gromov–Hausdorff convergence. In this paper we use the essential circles introduced in [19] to define a larger class of…

### Various Covering Spectra for Complete Metric Spaces

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- 2012

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with…

### Spectra related to the length spectrum

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We show how to extend the Covering Spectrum (CS) of Sormani-Wei to two spectra, called the Extended Covering Spectrum (ECS) and Entourage Spectrum (ES) that are new for Riemannian manifolds but…

### Weakly Chained Spaces

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We introduce "weakly chained spaces", which need not be locally connected or path connected, but for which one has a reasonable notion of generalized fundamental group and associated generalized…

### The Persistent Topology of Optimal Transport Based Metric Thickenings

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A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X . Thickenings have found use in applications of topology to data analysis, where one may…

### Cofibration and Model Category Structures for Discrete and Continuous Homotopy

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- 2022

. We show that the categories PsTop and Lim of pseudotopological spaces and limit spaces, respectively, admit coﬁbration category structures, and that PsTop admits a model category structure, giving…

### Metric constructions of topological invariants

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- 2017

We present a general mechanism for obtaining topological invari-ants from metric constructs. In more detail, we describe a process which, undervery mild conditions, produces topological invariants…

## References

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### Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces

- Mathematics
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Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines…

### Essential circles and Gromov–Hausdorff convergence of covers

- Mathematics
- 2013

The δ-covers of Sormani–Wei ([20]) are known not to be “closed” with respect to Gromov–Hausdorff convergence. In this paper we use the essential circles introduced in [19] to define a larger class of…

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We introduce the R cut-off covering spectrum and the cut-off covering spectrum of a metric space or a Riemannian manifold. The spectra measure the sizes of localized holes in the space and are…

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We define a new spectrum for compact length spaces and Riemannian manifolds called the “covering spectrum” which roughly measures the size of the one dimensional holes in the space. More…