Notes:[ilmath]\Delta[/ilmath]-complex

Sources

Hatcher

• [ilmath]\Delta^n:\eq\left\{(t_0,\ldots,t_n)\in\mathbb{R}^{n+1}\ \vert\ \sum_{i\eq 0}^nt_i\eq 1\wedge\forall i\in\{0,\ldots,n\}\subset\mathbb{N}[t_i\ge 0]\right\} [/ilmath]
  • Standard [ilmath]n[/ilmath]-simplex stuff, nothing special here.
• [ilmath]\sigma_\alpha:\Delta^{n(\alpha)}\rightarrow X[/ilmath] are maps that take the simplex into the topological space [ilmath](X,\mathcal{ J })[/ilmath]. Presumably these maps are continuous

[ilmath]\Delta[/ilmath]-complex

A collection [ilmath]\{\sigma_\alpha\}_{\alpha\in I} [/ilmath] that "cover" [ilmath]X[/ilmath] in the sense that:

• [ilmath]\forall x\in X\exists\alpha\in I\left[x\in\sigma_\alpha\vert_{(\Delta^n)^\circ}((\Delta^n)^\circ)\right][/ilmath] (modified from point 1 in hatcher, see point 4 below)

such that the following 3 properties hold:

1. [ilmath]\forall\alpha\in I\big[\sigma_\alpha\vert_{(\Delta^n)^\circ}:(\Delta^n)^\circ\rightarrow X\text{ is } [/ilmath][ilmath]\text{injective} [/ilmath][ilmath]\big][/ilmath]
   • Where [ilmath]\sigma_\alpha\vert_{(\Delta^n)^\circ}:(\Delta^n)^\circ\rightarrow X[/ilmath] is the restriction of [ilmath]\sigma_\alpha:\Delta^n\rightarrow X[/ilmath] to the interior of [ilmath]\Delta^n[/ilmath] (considered as a subset of [ilmath]\mathbb{R}^{n+1} [/ilmath])
2. For each [ilmath]\alpha\in I[/ilmath] there exists a [ilmath]\beta\in I[/ilmath] such that the restriction of [ilmath]\sigma_\alpha:\Delta^{n(\alpha)}\rightarrow X[/ilmath] to a face of [ilmath]\Delta^{n(\alpha)} [/ilmath] is [ilmath]\sigma_\beta:\Delta^{n(\alpha)-1\eq n(\beta)}\rightarrow X[/ilmath]
   • This lets us identify each face of [ilmath]\Delta^{n(\alpha)} [/ilmath] with [ilmath]\Delta^{n(\alpha)-1\eq n(\beta)} [/ilmath] by the canonical linear isomorphism between them that preserves the ordering of the vertices
     • This actually isn't to bad, as the restriction of [ilmath]\sigma_\alpha:\Delta^n\rightarrow X[/ilmath] to a face is equal to (as a map) some [ilmath]\sigma_\beta[/ilmath], so the linear map ... Caveat:there's a proof needed here
3. [ilmath]\forall U\in\mathcal{P}(X)[U\in\mathcal{J}\iff\forall\alpha\in I[\sigma_\alpha^{-1}(U)\text{ open in }\mathbb{R}^{n(\alpha)+1}][/ilmath] where we consider [ilmath]\mathbb{R}^{n(\alpha)+1} [/ilmath] with its usual topology (induced by the Euclidean metric)
4. [ilmath]\forall x\in X\exists\alpha\in I\big[x\in\sigma_\alpha\vert_{(\Delta^{n(\alpha)})^\circ}((\Delta^{n(\alpha)})^\circ)\wedge\forall\beta\in I[\alpha\neq\beta\implies x\notin \sigma_\beta\vert_{(\Delta^{n(\beta)})^\circ}((\Delta^{n(\beta)})^\circ)]\big][/ilmath]
   • In words: every point of [ilmath]x[/ilmath] occurs in exactly one of the (restrictions to the interior)'s images - we consider the interior as [ilmath]\Delta^n[/ilmath] being a subset of [ilmath]\mathbb{R}^{n+1} [/ilmath] with the usual Euclidean topology
   • TODO: What about the points - the [ilmath]0[/ilmath]-simplicies - these have empty interior considered as subsets of [ilmath]\mathbb{R}^1[/ilmath]
     - we probably just alter the definition a little to account for this.