View source for Notes:Potentially invalid vertex scheme

__TOC__
==Instance==
<div style="float:right;margin:0px;margin-left:0.2em;">
{| class="wikitable" border="1" style="overflow:hidden;max-width:30em;margin:0px;"
|-
| <center><m>\xymatrix{ a \ar@{-}[r] \ar@{-}[d] & b \ar@{-}[r] \ar@{-}[d] & c \ar@{-}[r] \ar@{-}[d] & a \ar@{-}[d] \\ d \ar@{-}[ur] \ar@{-}[r] \ar@{-}[d] & e \ar@{-}[r] \ar@{-}[d] \ar@{-}[ur] & f \ar@{-}[r] \ar@{-}[d] \ar@{-}[ur] & d  \ar@{-}[d] \\ a \ar@{-}[r] \ar@{-}[ur] & b \ar@{-}[r] \ar@{-}[ur] & c \ar@{-}[r]  \ar@{-}[ur] & a }</m></center>
|-
! The labelling scheme in question
|}
</div>
This is done in the context of Munkres - Elements of Algebraic Topology - page 16:
* What is the ''geometric realisation'' of the complex shown to the right?
** Surely {{M|\cong\mathbb{S}^1\times I}} where {{M|I:\eq[0,1]\subset\mathbb{R} }}?
*: intuitively, if one folds along {{C|defd}} to join {{c|abca}} to itself, we get a rectangle. We then have to bend it round to join {{C|ada}} to {{C|ada}}, thus making a tube.
*: However, we do have to identify {{C|be}} to {{C|eb}} and ditto for {{C|cf}}... I'm not sure what these do
*** But in the case described the {{C|ada}}-to-{{C|ada}} join is not ambiguous, dealing with that first we see that it is very much so. Do we twist or not? 
{{Caveat|Is this even a "valid" vertex labelling scheme?}}
'''Problems:'''
# The left and right edges, {{c|ada}} are ambiguous, do we twist or not (mobius band vs tube)
# {{C|be}}-to-{{C|be}} join
# {{C|cf}}-to-{{C|cf}} join

'''"Solution"''' 
* This should perform "more identifications than one would think" and not be a torus. 



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