# Homology groups of a square

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.

• 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? – Lee Mosher Jan 19 '20 at 15:09
• There are 4 0-simplices. You got the square by taking two triangles and identifying two vertices from each diagonal – Shai Deshe Jan 19 '20 at 15:11
• 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. – kam Jan 19 '20 at 15:11
• @ShaiDeshe I see, thank you! If you submit that answer I shall accept it. – kam Jan 19 '20 at 15:13
• 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 $[v_0,v_2,v_3]$. – Lee Mosher Jan 19 '20 at 15:19