10.4230/OASICS.CCA.2009.2263
Collins, Pieter
Pieter
Collins
Computability of Homology for Compact Absolute Neighbourhood Retracts
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2009
Article
Computability
homology
compact absolute neighbourhood retract
Bauer, Andrej
Andrej
Bauer
Hertling, Peter
Peter
Hertling
Ko, Ker-I
Ker-I
Ko
2009
2009-11-25
2009-11-25
2009-11-25
en
urn:nbn:de:0030-drops-22635
10.4230/OASIcs.CCA.2009
978-3-939897-12-5
2190-6807
10.4230/OASIcs.CCA.2009
OASIcs, Volume 11, CCA 2009
6th International Conference on Computability and Complexity in Analysis (CCA'09)
2012
11
15
107
118
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Bauer, Andrej
Andrej
Bauer
Hertling, Peter
Peter
Hertling
Ko, Ker-I
Ker-I
Ko
2190-6807
Open Access Series in Informatics (OASIcs)
2009
11
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
11 pages
322683 bytes
application/pdf
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
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.
OASIcs, Vol. 11, 6th International Conference on Computability and Complexity in Analysis (CCA'09), pages 107-118