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4. Cech closure spaces

    1. Eilenberg-Steenrod axioms


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      Problem 4.1.

      [Antonio Rieser] Do the Eilenberg-Steenrod axioms or a variant thereof determine the homology theory uniquely for closure spaces (or graphs)?

        • Cubical homology


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          Problem 4.2.

          [Nikola Milicevic] Is the Excision Axiom true for cubical homology of Cech closure spaces with respect to interior covers?
            1. Remark. Here is an example of an interior cover of the 4-cycle C_4, a graph with vertices: 0, 1, 2, 3, and edges between i and i+1 mod 4. The minimal interior cover is given by: \{4, 0, 1\}, \{0, 1, 2\}, \{1, 2, 3\}, \{2, 3, 0\}.

                • Singular homology of graphs

                      Every simple graph can be viewed as a closure space as follows: given a graph X = (V, E), we define a closure space by taking its underlying set to be V and c(A) = \bigcup_{v \in A} c(v), where c(v) = \{ w \in V \ | \ \{ v, w \} \in E\}. For n , k \in \mathbb{Z}, define the graph (\mathbb{Z}/n, c_k) to have the set of vertices \mathbb{Z}/n = \{ 0, 1, \ldots, n-1\} and an edge between i and j whenever i and j are no more than k away.

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                  Problem 4.3.

                  [Nikola Milicevic] Compute H^{sing}_*(\mathbb{Z}/n, c_k). Is it isomorphic to the homomology of the clique complex of (\mathbb{Z}/n, c_k)?
                      This was computed by Nikola Milicevic for n \leq 5, but is not known for higher values of n.

                    • Manifolds/cobordism


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                      Problem 4.4.

                      [Samira Jamil] What are manifolds in Cech closure spaces? What is the notion of cobordism in Cech closure spaces?

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