Difference between revisions of "Notes:Delta complex"
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===={{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}} | ||
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#** 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
Contents
[hide]Sources
Hatcher
- Δn:={(t0,…,tn)∈Rn+1 | ∑ni=0ti=1∧∀i∈{0,…,n}⊂N[ti≥0]}
- Standard n-simplex stuff, nothing special here.
- σα:Δn(α)→X are maps that take the simplex into the topological space (X,J). Presumably these maps are continuous
Δ-complex
A collection {σα}α∈I that "cover" X in the sense that:
- ∀x∈X∃α∈I[x∈σα|(Δn)∘((Δn)∘)] (modified from point 1 in hatcher, see point 4 below)
such that the following 3 properties hold:
- ∀α∈I[σα|(Δn)∘:(Δn)∘→X is injective][Note 1]
- Where σα|(Δn)∘:(Δn)∘→X is the restriction of σα:Δn→X to the interior of Δn (considered as a subset of Rn+1)
- For each α∈I there exists a β∈I such that the restriction of σα:Δn(α)→X to a face of Δn(α) is σβ:Δn(α)−1=n(β)→X
- This lets us identify each face of Δn(α) with Δn(α)−1=n(β) by the canonical linear isomorphism between them that preserves the ordering of the vertices
- This actually isn't to bad, as the restriction of σα:Δn→X to a face is equal to (as a map) some σβ, so the linear map ... Caveat:there's a proof needed here
- This lets us identify each face of Δn(α) with Δn(α)−1=n(β) by the canonical linear isomorphism between them that preserves the ordering of the vertices
- ∀U∈P(X)[U∈J⟺∀α∈I[σ−1α(U) open in Rn(α)+1] where we consider Rn(α)+1 with its usual topology (induced by the Euclidean metric)
- ∀x∈X∃α∈I[x∈σα|(Δn(α))∘((Δn(α))∘)∧∀β∈I[α≠β⟹x∉σβ|(Δn(β))∘((Δn(β))∘)]]
- In words: every point of x occurs in exactly one of the (restrictions to the interior)'s images - we consider the interior as Δn being a subset of Rn+1 with the usual Euclidean topology
- TODO: What about the points - the 0-simplicies - these have empty interior considered as subsets of R1- we probably just alter the definition a little to account for this.
Notes
- Jump up ↑ Hatcher combines points one and four into one
