Divide the 1-dimensional sphere $S^1$ into arcs. Label each vertex between two arcs as either 1 or 2. Count the number of "full arcs" - arcs with both 1 and 2 at their endpoints. Divide these full arcs to two kinds: "positive" (1-2 going clockwise) and "negative" (2-1). Then, it is clear that the number of positive full arcs equals the number of negative full arcs: each time we go from 1 to 2, we must at some point go back from 2 to 1, since we are on a circle.

Is the following generalization to multi-dimensional spheres correct?

Triangulate the $n$-dimensional sphere $S^n$.
Label each vertex of the triangulation by a label from $\{1,\ldots,n+1\}$.
Count the number of "full simplices" - simplices with all $n+1$ different labels at their vertices.
Divide labeled simplices into two equivalence classes: "positive" and "negative", such that for each two simplices in the same class, there is an orientation-preserving map from one to the other agreeing on the labels. To relabel a simplex keeping it in the same class, we can perform any *even permutation* on the labels (for example, if a simplex labeled 1-2-3-4 is positive, then the simplex labeled 2-3-1-4 is also positive, but the simplex labeled 2-1-3-4 is negative).

Conjecture: the number of positive simplices equals the number of negative simplices.

Is this conjecture correct?

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Elements of Combinatorial and Differential Topology, Amer. Math. Soc., 2006. It answers your question if you take into account my previous comment on orientations. $\endgroup$