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Discrete Morse Theory for Computing Cellular Sheaf Cohomology

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Abstract

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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Notes

  1. That is, the complexity of composing two \(d \times d\) matrices with \(\mathbf{R}\)-entries is \(\text {O}(d^\omega )\).

  2. When striving for greater generality, one replaces this requirement by the following local finiteness hypothesis on the covering relation: each \(x \in X\) can have only finitely many y so that \(y \prec x\) or \(x \prec y\).

  3. In principle, any method for constructing acyclic partial matchings on graded posets will suffice, provided that it ensures sheaf compatibility by only matching cell pairs whose restriction maps are invertible.

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Acknowledgments

This work was supported in part by federal contracts FA9550-12-1-0416, FA9550-09-1-0643, and HQ0034-12-C-0027.

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Authors and Affiliations

  1. Department of Mathematics, Duke University, Durham, NC, USA

    Justin Curry

  2. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA

    Robert Ghrist & Vidit Nanda

  3. Department of Electrical/Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA

    Robert Ghrist

Authors
  1. Justin Curry
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  2. Robert Ghrist
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  3. Vidit Nanda
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Correspondence to Vidit Nanda.

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Communicated by Gunnar Carlsson.

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Curry, J., Ghrist, R. & Nanda, V. Discrete Morse Theory for Computing Cellular Sheaf Cohomology. Found Comput Math 16, 875–897 (2016). https://doi.org/10.1007/s10208-015-9266-8

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