# Discrete Morse Theory for Computing Cellular Sheaf Cohomology

@article{Curry2016DiscreteMT, title={Discrete Morse Theory for Computing Cellular Sheaf Cohomology}, author={Justin Curry and Robert Ghrist and Vidit Nanda}, journal={Foundations of Computational Mathematics}, year={2016}, volume={16}, pages={875-897} }

Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.

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## References

SHOWING 1-10 OF 99 REFERENCES

### Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

- Mathematics, Computer ScienceFound. Comput. Math.
- 2014

A new Morse theoretic preprocessing framework for deriving chain maps from set-valued maps is introduced, and hence an effective scheme for computing the morphism induced on homology by the approximated continuous function is provided.

### The Nyquist theorem for cellular sheaves

- Mathematics
- 2013

We develop a unified sampling theory based on sheaves and show that the Shannon-Nyquist theorem is a cohomological consequence of an exact sequence of sheaves. Our theory indicates that there are…

### Morse Theory for Filtrations and Efficient Computation of Persistent Homology

- MathematicsDiscret. Comput. Geom.
- 2013

An efficient preprocessing algorithm is introduced to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups through an extension of combinatorial Morse theory from complexes to filtrations.

### Morse theory from an algebraic viewpoint

- Mathematics
- 2005

Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the…

### Elements of algebraic topology

- Mathematics
- 1984

Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in…

### On the sheaf theory

- Mathematics
- 2006

The“Gel’fand sheaf” of a topological algebra is endowed with auniform structure, this being complete if and only if, the spectrum of the algebra considered is complete. Examples are also provided.

### An introduction to homological algebra

- Mathematics
- 1960

Preface 1. Generalities concerning modules 2. Tensor products and groups of homomorphisms 3. Categories and functors 4. Homology functors 5. Projective and injective modules 6. Derived functors 7.…

### Morse Theory for Cell Complexes

- Mathematics
- 1998

In this paper we will present a very simple discrete Morse theory for CW complexes. In addition to proving analogues of the main theorems of Morse theory, we also present discrete analogues of such…

### On discrete Morse functions and combinatorial decompositions

- Mathematics, Computer ScienceDiscret. Math.
- 2000