# Local homology of abstract simplicial complexes

@article{Robinson2018LocalHO, title={Local homology of abstract simplicial complexes}, author={Michael Robinson and Christopher Capraro and Cliff Joslyn and Emilie Purvine and Brenda Praggastis and Stephen Ranshous and Arun V. Sathanur}, journal={arXiv: Algebraic Topology}, year={2018} }

This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated to the study of manifolds, submanifolds, or samplings thereof. While this is a vital perspective, the focus of this survey is squarely on the local homology of abstract simplicial complexes. Our motivation comes from the needs of the analysis of hypergraphs and graphs. In addition to presenting many…

## 3 Citations

### A (co)homology theory for some preordered topological spaces

- Mathematics
- 2020

The aim of this short note is to develop a (co)homology theory for topological spaces together with the specialisation preorder. A known way to construct such a (co)homology is to define a partial…

### Outlier-robust subsampling techniques for persistent homology

- Computer Science
- 2021

This work proposes a novel approach to select landmarks specifically for PH that preserves topological properties of the original data set and requires only local PH computation thus enabling efficient computation.

### Activation Landscapes as a Topological Summary of Neural Network Performance

- Computer Science2021 IEEE International Conference on Big Data (Big Data)
- 2021

This work compute the persistent homology of the activation data for each layer of a deep neural network and summarizes this information using persistence landscapes to provide both an informative visualization of the network and a kernel for statistical analysis and machine learning.

## References

SHOWING 1-10 OF 65 REFERENCES

### Homology stratifications and intersection homology

- Mathematics
- 1999

A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [Intersection homology theory: II, Inventiones Mathematicae,…

### Persistent Homology of Filtered Covers

- Mathematics
- 2012

We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now…

### Persistent Intersection Homology

- MathematicsFound. Comput. Math.
- 2011

This paper gives an algorithm for the computation of the persistent intersection homology groups of a filtered simplicial complex equipped with a stratification by subcomplexes, and it is proved its correctness.

### Local Cohomology: an algebraic introduction with geometric applications: The Lichtenbaum-Hartshorne Theorem

- Mathematics
- 1998

This book provides a careful and detailed algebraic introduction to Grothendieck’s local cohomology theory, and provides many illustrations of applications of the theory in commutative algebra and in…

### Topology of Manifolds

- Mathematics
- 1949

Elementary concepts characterizations of $\overline E^1$ and $S^1$ Locally connected spaces fundamental properties of the euclidean $n$-sphere Peano spaces characterizations of $S^2$ and the…

### Computing Topological Persistence for Simplicial Maps

- MathematicsSoCG
- 2014

This paper proposes a practical algorithm for computing persistence under Z2 coefficients for a (monotone) sequence of general simplicial maps and shows how these maps arise naturally in some applications of topological data analysis.

### Wilder manifolds are locally orientable.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1969

A proof is given for the long-standing conjecture of R. L. Wilder that every generalized manifold is locally orientable. Roughly speaking, a generalized n-manifold is a locally compact space whose…

### HOMOLOGY GROUPS OF RELATIONS

- Mathematics
- 1952

Any relation between the elements of a set X and the elements of a set Y is associated with two simplicial complexes K and L. A simplex of K is a finite set of elements of X related to a common…

### Inferring Local Homology from Sampled Stratified Spaces

- Mathematics48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
- 2007

This work uses the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point and proves the correctness of this assessment under the assumption of a sufficiently dense sample.

### Inverse limits of finite topological spaces

- Mathematics
- 2009

Extending a result of McCord, we prove that every finite simplicial complex is homotopy equivalent to the inverse limit of a sequence of finite spaces. In addition to generalizing McCord’s theorem,…