---
_id: '3133'
abstract:
- lang: eng
  text: 'This note contributes to the point calculus of persistent homology by extending
    Alexander duality from spaces to real-valued functions. Given a perfect Morse
    function f: S n+1 →[0, 1 and a decomposition S n+1 = U ∪ V into two (n + 1)-manifolds
    with common boundary M, we prove elementary relationships between the persistence
    diagrams of f restricted to U, to V, and to M. '
acknowledgement: "his research is partially supported by the National Science Foundation
  (NSF) under grant DBI-0820624, the European Science Foundation under the Research
  Networking Programme, and the Russian Government Project 11.G34.31.0053.\r\nThe
  authors thank an anonymous referee for suggesting the simplified proof of the Contravariant
  PE Theorem given in this paper. They also thank Frederick Cohen, Yuriy Mileyko and
  Amit Patel for helpful discussions."
article_processing_charge: No
arxiv: 1
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Michael
  full_name: Kerber, Michael
  id: 36E4574A-F248-11E8-B48F-1D18A9856A87
  last_name: Kerber
  orcid: 0000-0002-8030-9299
citation:
  ama: 'Edelsbrunner H, Kerber M. Alexander duality for functions: The persistent
    behavior of land and water and shore. In: <i>Proceedings of the Twenty-Eighth
    Annual Symposium on Computational Geometry </i>. ACM; 2012:249-258. doi:<a href="https://doi.org/10.1145/2261250.2261287">10.1145/2261250.2261287</a>'
  apa: 'Edelsbrunner, H., &#38; Kerber, M. (2012). Alexander duality for functions:
    The persistent behavior of land and water and shore. In <i>Proceedings of the
    twenty-eighth annual symposium on Computational geometry </i> (pp. 249–258). Chapel
    Hill, NC, USA: ACM. <a href="https://doi.org/10.1145/2261250.2261287">https://doi.org/10.1145/2261250.2261287</a>'
  chicago: 'Edelsbrunner, Herbert, and Michael Kerber. “Alexander Duality for Functions:
    The Persistent Behavior of Land and Water and Shore.” In <i>Proceedings of the
    Twenty-Eighth Annual Symposium on Computational Geometry </i>, 249–58. ACM, 2012.
    <a href="https://doi.org/10.1145/2261250.2261287">https://doi.org/10.1145/2261250.2261287</a>.'
  ieee: 'H. Edelsbrunner and M. Kerber, “Alexander duality for functions: The persistent
    behavior of land and water and shore,” in <i>Proceedings of the twenty-eighth
    annual symposium on Computational geometry </i>, Chapel Hill, NC, USA, 2012, pp.
    249–258.'
  ista: 'Edelsbrunner H, Kerber M. 2012. Alexander duality for functions: The persistent
    behavior of land and water and shore. Proceedings of the twenty-eighth annual
    symposium on Computational geometry . SCG: Symposium on Computational Geometry,
    249–258.'
  mla: 'Edelsbrunner, Herbert, and Michael Kerber. “Alexander Duality for Functions:
    The Persistent Behavior of Land and Water and Shore.” <i>Proceedings of the Twenty-Eighth
    Annual Symposium on Computational Geometry </i>, ACM, 2012, pp. 249–58, doi:<a
    href="https://doi.org/10.1145/2261250.2261287">10.1145/2261250.2261287</a>.'
  short: H. Edelsbrunner, M. Kerber, in:, Proceedings of the Twenty-Eighth Annual
    Symposium on Computational Geometry , ACM, 2012, pp. 249–258.
conference:
  end_date: 2012-06-20
  location: Chapel Hill, NC, USA
  name: 'SCG: Symposium on Computational Geometry'
  start_date: 2012-06-17
corr_author: '1'
date_created: 2018-12-11T12:01:35Z
date_published: 2012-06-20T00:00:00Z
date_updated: 2025-06-11T08:10:04Z
day: '20'
department:
- _id: HeEd
doi: 10.1145/2261250.2261287
external_id:
  arxiv:
  - '1109.5052'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://arxiv.org/abs/1109.5052
month: '06'
oa: 1
oa_version: Preprint
page: 249 - 258
publication: 'Proceedings of the twenty-eighth annual symposium on Computational geometry '
publication_status: published
publisher: ACM
publist_id: '3564'
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Alexander duality for functions: The persistent behavior of land and water
  and shore'
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2012'
...