4. Cech closure spaces
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Eilenberg-Steenrod axioms
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
Problem 4.2.
[Nikola Milicevic] Is the Excision Axiom true for cubical homology of Cech closure spaces with respect to interior covers?-
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: $\{4, 0, 1\}$, $\{0, 1, 2\}$, $\{1, 2, 3\}$, $\{2, 3, 0\}$.
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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.This was computed by Nikola Milicevic for $n \leq 5$, but is not known for higher values of $n$.Problem 4.3.
[Nikola Milicevic] Compute $H^{sing}_*(\mathbb{Z}/n, c_k)$. Is it isomorphic to the homomology of the clique complex of $(\mathbb{Z}/n, c_k)$? -
Manifolds/cobordism
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.