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1. Digital topology

    1. Higher homotopy groups


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      Problem 1.1.

      [Gregory Lupton] Define higher homotopy groups in digital topology. As a test case, one should verify that $\pi_2(S^2, *) = \mathbb{Z}$, where $S^2$ is the digital 2-sphere, i.e., a graph with six vertices appropriately embedded in $\mathbb{Z}^3$.
        1. Remark. [Oleg Musin] Is there a relation between homotopy groups in digital topology and the Tucker Lemma?

            • Loop spaces


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              Problem 1.2.

              [Anton Dochtermann] Is there a good notion of a loop space of a digital space?

                • Topological realization

                      This problem assumes that higher homotopy groups of digital spaces have been constructed.

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                  Problem 1.3.

                  Is there a functor $F$ from the category of digital images to that of topological spaces such that the digital invariants of a digital image $X$ agree with the corresponding classical invariants of $FX$, e.g., $\pi_n(X, *) \cong \pi_n(FX, *)$?
                      Open.

                    • Digital topology v $x$-homotopy theory


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                      Problem 1.4.

                      What is the relation between digital topology and $\times$-homotopy theory of reflexive graphs?

                          Cite this as: AimPL: Discrete and combinatorial homotopy theory, available at http://aimpl.org/combhomotop.