Vertex scheme of an abstract simplicial complex/Definition

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There's a lot of work required here. Like what does it mean for "vertices to span a simplex" and so forth? Also the notation for the definition of [ilmath]\mathcal{K} [/ilmath] leaves a lot to be desired! It's implied [ilmath]n[/ilmath] varies but it is not in the spirit of this site!


Let [ilmath]K[/ilmath] be a simplicial complex and let [ilmath]V_K[/ilmath] be the vertex set of [ilmath]K[/ilmath] (not to be confused with the vertex set of an abstract simplicial complex), then we may define [ilmath]\mathcal{K} [/ilmath] - an abstract simplicial complex - as follows[1]:

  • [math]\mathcal{K}:\eq\left\{\{a_0,\ldots,a_n\}\in \mathcal{P}(V_K)\ \big\vert\ \text{Span}(a_0,\ldots,a_n)\in K\right\} [/math]Warning:[Note 1] - that is to say [ilmath]\mathcal{K} [/ilmath] is the set containing all collections of vertices such that the vertices span a simplex in [ilmath]K[/ilmath]


  1. [ilmath]n\in\mathbb{N}_0[/ilmath] here so [ilmath]n[/ilmath] may be zero, we are expressing our interest in only those finite members of [ilmath]\mathcal{P}(V_K)[/ilmath] here, and that are non-empty.
    • TODO: This needs to be rewritten!


  1. Elements of Algebraic Topology - James R. Munkres