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Real Computable Manifolds and Homotopy Groups

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Unconventional Computation (UC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5715))

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Abstract

Using the model of real computability developed by Blum, Cucker, Shub, and Smale, we investigate the difficulty of determining the answers to several basic topological questions about manifolds. We state definitions of real-computable manifold and of real-computable paths in such manifolds, and show that, while BSS machines cannot in general decide such questions as nullhomotopy and simple connectedness for such structures, there are nevertheless real-computable presentations of paths and homotopy equivalence classes under which such computations are possible.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Murray State University, Murray, KY, 42071, U.S.A.

    Wesley Calvert

  2. Queens College of CUNY, 65-30 Kissena Blvd., Flushing, NY, 11367, USA

    Russell Miller

  3. The CUNY Graduate Center, 365 Fifth Avenue, New York, NY, 10016, USA

    Russell Miller

Authors
  1. Wesley Calvert
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  2. Russell Miller
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Editor information

Editors and Affiliations

  1. Department of Computer Science, The University of Auckland, Science Centre, 38 Princes, 1142, St., Auckland, New Zealand

    Cristian S. Calude

  2. Centro de Matemática e Aplicações Fundamentais do Complexo Interdisciplinar, University of Lisbon, Av. Gama Pinto, 1649-003, Lisboa, Portugal

    José Félix Costa

  3. School of Computer Science, Tel Aviv University, 69978, Ramat Aviv, Israel

    Nachum Dershowitz

  4. Department of Mathematics, University of Azores, Rua da Mãe de Deus, 9501-855, Ponta Delgada, Portugal

    Elisabete Freire

  5. Leiden Center of Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333, Leiden, CA, The Netherlands

    Grzegorz Rozenberg

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Calvert, W., Miller, R. (2009). Real Computable Manifolds and Homotopy Groups. In: Calude, C.S., Costa, J.F., Dershowitz, N., Freire, E., Rozenberg, G. (eds) Unconventional Computation. UC 2009. Lecture Notes in Computer Science, vol 5715. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03745-0_16

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  • DOI: https://doi.org/10.1007/978-3-642-03745-0_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03744-3

  • Online ISBN: 978-3-642-03745-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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