Difference between revisions of "Notes:Delta complex"
From Maths
(Saving work) |
(Changed rules, clarified a little) |
||
| Line 8: | Line 8: | ||
===={{M|\Delta}}-complex==== | ===={{M|\Delta}}-complex==== | ||
A collection {{M|\{\sigma_\alpha\}_{\alpha\in I} }} that "cover" {{M|X}} in the sense that: | A collection {{M|\{\sigma_\alpha\}_{\alpha\in I} }} that "cover" {{M|X}} in the sense that: | ||
| − | * {{M|\forall x\in X\exists\alpha\in I\left[x\in\sigma_\alpha\vert_{(\Delta^n)^\circ}((\Delta^n)^\circ)\right]}} (modified from point | + | * {{M|\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: | such that the following 3 properties hold: | ||
| − | # {{M|\forall\alpha\in I\big[\sigma_\alpha\vert_{(\Delta^n)^\circ}:(\Delta^n)^\circ\rightarrow X\text{ is } }}[[injective|{{M|\text{injective} }}]]{{M|\big]}} | + | # {{M|\forall\alpha\in I\big[\sigma_\alpha\vert_{(\Delta^n)^\circ}:(\Delta^n)^\circ\rightarrow X\text{ is } }}[[injective|{{M|\text{injective} }}]]{{M|\big]}}<ref group="Note">Hatcher combines points one and four into one</ref> |
#* Where {{M|\sigma_\alpha\vert_{(\Delta^n)^\circ}:(\Delta^n)^\circ\rightarrow X}} is the [[restriction]] of {{M|\sigma_\alpha:\Delta^n\rightarrow X}} to the {{link|interior}} of {{M|\Delta^n}} (considered as a [[subset of]] {{M|\mathbb{R}^{n+1} }}) | #* Where {{M|\sigma_\alpha\vert_{(\Delta^n)^\circ}:(\Delta^n)^\circ\rightarrow X}} is the [[restriction]] of {{M|\sigma_\alpha:\Delta^n\rightarrow X}} to the {{link|interior}} of {{M|\Delta^n}} (considered as a [[subset of]] {{M|\mathbb{R}^{n+1} }}) | ||
# For each {{M|\alpha\in I}} there exists a {{M|\beta\in I}} such that the restriction of {{M|\sigma_\alpha:\Delta^{n(\alpha)}\rightarrow X}} to a face of {{M|\Delta^{n(\alpha)} }} is {{M|\sigma_\beta:\Delta^{n(\alpha)-1\eq n(\beta)}\rightarrow X}} | # For each {{M|\alpha\in I}} there exists a {{M|\beta\in I}} such that the restriction of {{M|\sigma_\alpha:\Delta^{n(\alpha)}\rightarrow X}} to a face of {{M|\Delta^{n(\alpha)} }} is {{M|\sigma_\beta:\Delta^{n(\alpha)-1\eq n(\beta)}\rightarrow X}} | ||
| Line 16: | Line 16: | ||
#** This actually isn't to bad, as the restriction of {{M|\sigma_\alpha:\Delta^n\rightarrow X}} to a face is ''equal to'' (as a [[map]]) some {{M|\sigma_\beta}}, so the linear map ... {{caveat|there's a proof needed here}} | #** This actually isn't to bad, as the restriction of {{M|\sigma_\alpha:\Delta^n\rightarrow X}} to a face is ''equal to'' (as a [[map]]) some {{M|\sigma_\beta}}, so the linear map ... {{caveat|there's a proof needed here}} | ||
# {{M|\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 {{M|\mathbb{R}^{n(\alpha)+1} }} with its usual topology ([[topology induced by a metric|induced]] by the [[Euclidean metric]]) | # {{M|\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 {{M|\mathbb{R}^{n(\alpha)+1} }} with its usual topology ([[topology induced by a metric|induced]] by the [[Euclidean metric]]) | ||
| + | # {{M|\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 {{M|x}} occurs in exactly one of the (restrictions to the interior)'s images - we consider the interior as {{M|\Delta^n}} being a subset of {{M|\mathbb{R}^{n+1} }} with the usual [[Euclidean metric|Euclidean]] topology | ||
| + | #* {{XXX|What about the points - the {{M|0}}-simplicies - these have empty interior considered as subsets of {{M|\mathbb{R}^1}}}} - we probably just alter the definition a little to account for this. | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
Revision as of 15:00, 24 January 2017
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:
- [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][Note 1]
- 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])
- 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
- 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
- [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)
- [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.
Notes
- ↑ Hatcher combines points one and four into one
