Real Computable Manifolds and Homotopy Groups

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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