= Positive alternating simplex
{title2=$+e_{i_0},-e_{i_1},\ldots,(-1)^ke_{i_k},\quad i_0<\cdots<i_k$}
For a <simplicial map> to the <standard simplicial decomposition of a sphere>, a $k$-simplex is positive alternating when its image has the displayed distinct labels. Reversing all signs makes it negative alternating. A collapsed or nonalternating simplex is neutral. Counting positive facets modulo two is the local ingredient in the <antipodal alternating-simplex parity lemma>.
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