Antipodal alternating-simplex parity lemma ($p_k\equiv 1(\mathrm{mod}2)$)

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 $k>n$.