= Antipodal alternating-simplex parity lemma
{title2=$p_k\equiv1\pmod2$}
An antipodal <simplicial map> from a <regular antipodal triangulation of a sphere> $S^k$ to $F^n$ has an odd number of positive alternating $k$-simplices. For each $k$-simplex, the number of positive alternating facets is odd exactly when the simplex is positive or negative alternating; otherwise it is even, including collapsed images. Count these incidences in one <hemisphere>: interior facets contribute twice, boundary facets once. Antipodal pairing then gives $p_k\equiv p_{k-1}$, starting with one positive vertex on $S^0$. In particular, no such map exists when $k>n$.
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