The problem and its solution are given below:
My questions are:
1-Why the edge $c$ in the drawing takes this direction?
2-Are we considering the sequence of chain complexes exact?
3-In the third line from below,in the calculation of $H_{1}(X)$ in the numerator, why is $<a,b,c> = <a,b, a+b-c>$, specifically why the third coordinate became like this? Also why $<a,b> = \mathbb{Z}^2$, what is the meaning of $<a,b>$, are they numbers?
Could anyone help me answer these questions, please?
No.
1-Why the edge c in the drawing takes this direction?
There is a standard way of assigning the direction of arrow, which is from lower index to hight. The vertex where two arrows are coming can be thought $v2$ of the simplex $[v0 v1 v2]$
In the third line from below, in the calculation of H1(X) in the numerator, why is =, specifically why the third coordinate became like this? Also why =Z2, what is the meaning of , are they numbers?
For drawing part: Let [$v_0 v_1 v_2$] is a 2-simplex. Then on every edge, we assign a direction/orientation from lower index to higher index. In fact, it's a consequence of the definition of the boundary map. Now, choose orientation for the $2$ simplex [$v_0 v_1 v_2$]. Let's denote the edge [$v_0 v_1$] , [$v_1v_2$], [$v_0 v_2$] by $ a, b, c$ respectively. Then the edge $a$ gets orrientation from the vertex $v_0$ to $v_1$, and $b$ gets orrintation from $v_1$ to $v_2$ and $c$ gets orrientaion from $v_0$ to $v_2$. Then Identify the three edges according to problem statement. It'll give the right picture. I hope this helps.
Think about the basis of a vector space. It will give you the main motivation. I hope this helps.