View source for Notes:Delta complex/Formal attempt

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==Formal attempt==
</noinclude>We try and keep everything combinatorial, so keep an [[abstract simplicial complex]] in the back of your mind, and a simplex as being like {{M|\{a,b,c\} }} for a triangle and such.

'''Notations:'''
* Let {{M|\#(n):\eq\{1,\ldots,n\}\subset\mathbb{N} }} - I did want to use {{M|C(n)}} for "count" or "consecutive" but given the context that'd be a poor choice!
** Consider {{M|\#(n)}} as a [[poset]] in its own right (in fact a total order is in play) with the "usual" ordering on {{M|\mathbb{N} }} it inherits. This is a standard [[substructure construction]].
* Let {{M|K}} be our Delta complex, let us sidestep defining exactly what this is now, as a tuple of sets.
* Let {{M|S_n(K)}} be the set of {{n|{{plural|simpl|ex|ices}}}} of {{M|K}}
* Let {{M|I(m,n)}} be defined to be equal the collection of all ''[[injective]]'' [[monotonic]] ''[[functions]]'' of the form {{M|f:\#(m+1)\rightarrow\#(n+1)}}<ref group="Note">This basically means:
* {{M|\forall x,y\in \#(m+1)[x < y\implies f(x)<f(y)]}} - notice the strict ordering used here. This ensures that it is 1-to-1. We can never have equality of {{M|f(x)}} and {{M|f(y)}} 
** {{Caveat|Not proved yet}} {{XXX|Do the proof!}}</ref>
** The {{M|+1}} comes from the definition: {{M|\text{Dim}(\sigma):\eq\vert\sigma\vert - 1\in\mathbb{N} }} - we take care with the case {{M|\sigma\eq\emptyset}} as I'm developing a framework including this and come up with 2 "null objects" that do not alter the theory, for now {{M|\text{Dim}(\emptyset)\eq -1}} will do. It wont matter.
* {{M|\Delta^m}} be the [[standard n-simplex|standard {{M|m}}-simplex]] in {{M|\mathbb{R}^{m+1} }}
* {{M|G(n,m)}} - this is our goal, it's a collection of a bunch of maps of the form {{M|G:S_n(K)\rightarrow S_m(K)}} {{Caveat|Notice the flip of {{M|n}} and {{M|m}}) with certain properties. 
** Our goal is to find a [[bijection]], say {{M|F:I(m,n)\rightarrow G(n,m)}}
===First stab===
'''Definition: '''
* The "gluing data" of a {{M|\Delta}}-complex corresponds to two parts:
*# {{M|S_n(K)}} - the set of {{M|n}}-simplices of {{M|K}}
*# The "gluing maps", {{M|G_f}}, which can be enumerated as follows:
*#* Let {{M|m,\ n\in\mathbb{N}_0}} be given and be such that {{M|m\le n}}
*#** Then for each {{M|f\in I(m,n)}} there exists a {{M|G_f:S_n(K)\rightarrow S_m(K)}} such that:
*#**# If {{M|f\eq \text{Id}_{\#(n+1)} }} then {{M|G_f\eq\text{Id}_{S_n(K)} }}, and
*#**# If {{M|f\in I(m,n)}} and {{M|g\in I(n,j)}} then {{M|G_{g\circ f}\eq G_f\circ G_g}}
That's it!
====Problems====
# I need to form a statement (and then prove it) which shows that we need only consider {{M|m\eq k}} and {{M|n\eq k+1}} cases (for {{M|k\in\mathbb{N}_0}}) we don't need all of them, that statement 2 of the {{M|G_f}} function definition ensures the result is consistent. It's pretty obvious but I'm not sure how to phrase it. 
# I need to show that we have a Hatcher-{{M|\Delta}}-complex {{iff}} we have one of these. 
====Gluing process====
* Let {{M|m,n\in\mathbb{N} }} be given such that {{M|m\le n}}.
** Let {{M|f\in I(m,n)}} be given, so {{M|f:\#(m+1)\rightarrow\#(n+1)}} is an [[injection]] and is [[monotonic]] - as per the definition of {{M|I(m,n)}}.
*** We associate {{M|f}} with {{M|L_f:\mathbb{R}^{m+1}\rightarrow\mathbb{R}^{n+1} }} which is a [[linear map]] defined by its action on a basis as {{M|L_f(e_i):\eq e_{f(i)} }} where {{M|e_i\in\mathbb{R}^\text{whatever} }} is a [[tuple]] that has {{m|0}} in every entry except the {{M|i^\text{th} }} which has {{M|1}}; as usual.<ref group="Note">There's some [[abuse of notation]] going on here, as if {{M|e_i\in\mathbb{R}^n}} then {{M|e_i\notin\mathbb{R}^m}} with {{M|m\neq n}} of course. We identify {{M|\mathbb{R}^m}} with a subspace of {{M|\mathbb{R}^n}} where {{M|n\ge m}} spanned by the first {{M|m}} basis vectors. It's not that big of a leap, so shouldn't require any more discussion</ref>
**** It is fairly easy to see that {{M|\text{Ker}(M_f)\eq\{0\} }}, then by "''[[a linear map is injective if and only if its kernel is trivial]]''" and "''[[the image of a linear map is a vector subspace of the codomain]]''" wee see that:
***** {{M|L_f':\mathbb{R}^{m+1}\rightarrow L_f(\mathbb{R}^{m+1})}} is a [[linear isomorphism]]
**** As {{M|\mathbb{R}^{m+1} }} is finite dimensional we see that {{M|L_f'}} is a [[continuous map]], so forth. As would be {{M|L_f}} itself of course.
**** Notice that {{M|L_f'\vert_{\Delta^m}:\Delta^m\rightarrow \text{Some }m\text{-face of }\Delta^n }} 
***** and that this is a [[homeomorphism]] onto its image. 
**** This is the idea of our "gluing map" we see we glue some {{M|m}}-face of an {{M|n}}-simplex to some {{M|m}}-simplex that we already have.
***** Define {{M|G_f:S_n(K)\rightarrow S_m(K)}} by {{M|G_f:\sigma\mapsto\text{the }m\text{-simplex to which the }m\text{-face of }\sigma\text{ given by }f\text{ corresponds to} }}
(see paper notes. Will write this again later)


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==Notes==
<references group="Note"/>
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