I am currently learning the basics of simplical homology and have been reading about gluing data and boundary maps. My question is when talking about choosing gluing data by defining map $\beta(\alpha,i)$ where $\alpha$ exists in the set of n-simplices and i describes which facet of that n simplex we are talking about, how does the orientation of our simplex effect the choice of which i value associates to which facet.
Furthermore, when applying a boundary map to an n simplex we achieve a formal sum of n-1 simplices of alternating sign. From this sum, how do i deduce what the orientation of this n-1 simplex is?