SECOND EDIT: I start to understand (a bit) what is going on, but still an answer is much needed. Below I edited the question (along with the title) to point out the new little things I got.
FIRST EDIT: I started a bounty on this question, for various reasons. First of all, because I really would like to get an asnwer. Also, I feel that it is really basic and many students could miss the same point. Finally, I feel all the books I consulted were not able to actually give me a proper answer.
I was watching this video on youtube by Wildberger on Homology with the hope of understanding what this is all about and, as a matter of fact, I felt things where working fine.
However, I do have a question, which arose in my mind at around 32:16 of the video in the link.
Question: do cells of different order have to behave somewhat nicely with respect to each others in their directions or not?
Now, this is the question, but I feel more misunderstandings are hidden there. Thus, in the hope of uncover all of them, I add the following picture from the video.
Thus, Wildbgerger wants to study $X_4$, in which we have cells of dimensions up to $3$: this is simply a directed graph. In particular, $C$ is the addition he addresses that happens to be the solid $3$-dimensional ball comprised between the two planes $A$ and $B$ (where both $A$ and $B$ are oriented clockwise). Also, we have that $x, y, z$ are vertices and $a, b, c, d$ are oriented edges.
As a matter of fact, in this lecture Wildenberg introduces homology by focusing first on graph homology.
At 32:16 he points out that (of course) we have to give a direction to $C$. But it seems we cannot do it as we like: rather we have to do it in such a way that the orientation of $A$ and $B$ is respected (which, from what I got, should give us that $\partial (C) = 0$). To explain this point, he draws the simplex in the picture and says that in that case the orientation of the plane inside it has to respect the direction of the arrows. Fair enough, but I have the following question:
- how does this work with the fact that both $c$ and $d$ as segments in $X_4$ are directed upwards?
Thus, I think the question above is related to this one I have, which concerns the relation between $c$ and $d$.
- How do we make coexist the fact that we come up with a cycle on $c$ and $d$ where they both go in the same direction? That is, how does it get reconciled with simplicial homology (that, it seems to me, is somewhat of a generalization of graph homology)?
I know I am making a mess out of it, but I am missing the basic rules behind this whole direction thing in homology (on the contrary, I think I basically got the rules of orientation behind simplicial homology).
As always, any feedbacks will be greatly appreciated!