Here is the boundary map in case of simplicial homology(AT pg.105):
And here is the boundary map in case of singular homology(AT pg.108):
My question is:
I know that the difference between the 2 homologies is that in the first case we are using simplicial complexes as basis but in the second case we are using continuous maps as basis. My question is, why $\alpha$ exists in the definition of the boundary map in case of simplicial homology and it is absent in singular homology definition?
Could anyone help me answer this question, please?
These pictures are from Hatcher yes? You can actually find the explanation there, but I'll try explain it a little more clearly than he does.
So, a $\Delta$-complex structure on $X$ is defined as a collection of continuous maps $\sigma_\alpha:\Delta^n \to X$ subject to various conditions. Then $\Delta_n(X)$ are the free abelian groups generated by these maps. And from the example that is given in Hatcher after the definition of a $\Delta$-complex is given, we see that for a torus this can be given by $6$ such maps, $2$ from $\Delta^2$, $3$ from $\Delta^1$ and $1$ from $\Delta^0$. So for the torus $\Delta_0(X)$ is generated by $1$ map, and so on. Therefore when you talk about the boundary map and how it acts on the basis of $\Delta_n(X)$ it makes sense to include the index $\alpha$.
On the other hand each $C_n(X)$ is generated by all continuous maps $\sigma:\Delta^n\to X$. There are no other conditions on these maps, and there are likely uncountably many of them, and therefore no natural way to index them.
Besides that the boundary maps are basically the same, as mentioned in the comments below the question. The appearance of an index in the first definition is just cosmetic; there to remind you that you're defining it for an easily indexed basis.