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DOI: 10.4230/OASIcs.CCA.2009.2263

Computability of Homology for Compact Absolute Neighbourhood Retracts

Authors: Pieter Collins

Published in: OASIcs, Volume 11, 6th International Conference on Computability and Complexity in Analysis (CCA'09) (2009)

Abstract

In this note we discuss the information needed to compute the homology groups of a topological space. We argue that the natural class of spaces to consider are the compact absolute neighbourhood retracts, since for these spaces the homology groups are finite. We show that we need to specify both a function which defines a retraction from a neighbourhood of the space in the Hilbert cube to the space itself, and a sufficiently fine over-approximation of the set. However, neither the retraction itself, nor a description of an approximation of the set in the Hausdorff metric, is sufficient to compute the homology groups. We express the conditions in the language of computable analysis, which is a powerful framework for studying computability in topology and geometry, and use cubical homology to perform the computations.

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Pieter Collins. Computability of Homology for Compact Absolute Neighbourhood Retracts. In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, pp. 107-118, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{collins:OASIcs.CCA.2009.2263,
  author =	{Collins, Pieter},
  title =	{{Computability of Homology for Compact Absolute Neighbourhood Retracts}},
  booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
  pages =	{107--118},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-12-5},
  ISSN =	{2190-6807},
  year =	{2009},
  volume =	{11},
  editor =	{Bauer, Andrej and Hertling, Peter and Ko, Ker-I},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.CCA.2009.2263},
  URN =		{urn:nbn:de:0030-drops-22635},
  doi =		{10.4230/OASIcs.CCA.2009.2263},
  annote =	{Keywords: Computability, homology, compact absolute neighbourhood retract}
}

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Computability of Homology for Compact Absolute Neighbourhood Retracts

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