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Here is a picture from Chapter 2 in Hatcher's Algebraic Topology: enter image description here

The context is that the space on the left does not have the structure of a $\Delta$-complex, whereas the shape on the right does. On the left he says that we identify the sides in such a way that we preserve a cyclic ordering of the vertices, and then on the right we subdivide the space to obtain something that does have the structure of a $\Delta$-complex.

Do the arrows in the right-hand diagram all represent side identification? Or do the ones on the 'outside' of the triangle represent side identification and the ones on the inside just tell us the ordering of the vertices? I'm a little bit confused to be honest, though I can't find any explanation anywhere else, so I'm assuming it must be pretty obvious!

Tim
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    The arrows represent how the edges are oriented. The edge gluings are as described in the text. The point is that there is no topological difference between the two examples, they represent the exact same space; the difference is in the subdivision. – Lee Mosher Oct 19 '15 at 15:28
  • @LeeMosher Brilliant, thanks. I saw that the gluings were in the text, but have always seen arrows used to indicate this too, so thought it was worth checking. If you want to submit that as an answer then I can accept it so this question is marked as answered! – Tim Oct 19 '15 at 15:42

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Copying the answer from my comment:

The arrows represent how the edges are oriented. The edge gluings are as described in the text. The point is that there is no topological difference between the two examples, they represent the exact same space; the difference is in the subdivision.

Lee Mosher
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