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Consider the unit square with a $\Delta$- complex structure obtained by the two $2$-simplices $[v_0,v_1,v_2], [v_2,v_0,v_3]$. Is $\Delta{}_0(X)\cong\mathbb{Z}^6$ or $\mathbb{Z}^4$ where $X$ is the space of the square.

My confusion lies with the diagonal 1-simplex $[v_0,v_2]$. Its clear there are 5 1 simplices, but are there 4 or 6 0-simplices? surely if there are 6, that contradicts the requirements for a $\Delta$-complex structure?

edit: $v_i$ are the $i^{th}$ corner of the square.

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  • $\begingroup$ You've listed four 0-simplices in your question, namely $v_0,v_1,v_2,v_3$. What other two 0-simplices are you contemplating? $\endgroup$ – Lee Mosher Jan 19 '20 at 15:09
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    $\begingroup$ There are 4 0-simplices. You got the square by taking two triangles and identifying two vertices from each diagonal $\endgroup$ – Shai Deshe Jan 19 '20 at 15:11
  • $\begingroup$ What I confused with is, there are the 5 'sides', 4 are the sides of the square, and 1 is the diagonal. Since the endpoints are the faces of the sides, they must also be in the structure, but that would mean $v_0$ and $v_2$ are counted twice. $\endgroup$ – kam Jan 19 '20 at 15:11
  • $\begingroup$ @ShaiDeshe I see, thank you! If you submit that answer I shall accept it. $\endgroup$ – kam Jan 19 '20 at 15:13
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    $\begingroup$ By the way, your specification of this problem has an error. As written, one face of the 2-simplex $[v_0,v_1,v_2]$ is the 1-simplex $[v_0,v_2]$, whereas one face of the 2-simplex $[v_2,v_0,v_3]$ is the 1-simplex $[v_2,v_0]$, and $[v_0,v_2] \ne [v_2,v_0]$. This error could be corrected by writing the second 2-simplex as $[v_0,v_2,v_3]$. $\endgroup$ – Lee Mosher Jan 19 '20 at 15:19
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There are 4 0-simplices. You got the square by taking two triangles and identifying two vertices from each diagonal

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  • $\begingroup$ Thank you. After thinking about it, surely you would actually need to identify the two 1-simplicies from each triangle instead of just the endpoints $\endgroup$ – kam Jan 19 '20 at 15:16

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