Notes:$\Delta$-complex

Sources

Hatcher

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

$\Delta$-complex

A collection $\{\sigma_\alpha\}_{\alpha\in I}$ that "cover" $X$ in the sense that:

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

such that the following 3 properties hold:

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