Triangulations

The point of introducing orientation is to use $${n}$$-chains to describe arbitrary $${n}$$-dimensional surfaces in $${X}$$ composed of $${n}$$-simplices. By constructing such surfaces using adjacent simplices, internal boundaries can cancel when they consist of two boundaries in opposite directions. Constructing a surface out of simplices in this way is called a triangulation.

Many intuitive facts are accurately reproduced in this formal system, and extended to arbitrary dimension in a consistent way.