= Solution
Triangulate the <sphere> by the boundary complex of the <cross-polytope> with <simplicial vertices> $\pm e_0,\ldots,\pm e_n$: a face contains at most one <simplicial vertex> from each antipodal pair. Take its <barycentric subdivision>. Its <simplicial vertices> are the nonempty faces $F$ of the original complex, and its simplices are strictly nested chains $F_0\subset\cdots\subset F_k$.
Form a quotient complex whose <simplicial vertices> are pairs $\{F,-F\}$, and whose simplices are the images of these chains. No chain contains both $F$ and $-F$. Moreover a chain of face-orbits lifts uniquely up to simultaneous negation: after choosing a largest face, every smaller face has at most one representative contained in it, because an original face contains no antipodal <simplicial vertices>. Thus the quotient is an honest <simplicial complex>, without two different simplices with the same <simplicial vertex> set. Its realization is homeomorphic to the <sphere> modulo its antipodal action. This provides \b[a triangulation of <Real projective space> $\mathbb{RP}^n$]. The subdivision avoids the duplicate-simplex problem of naively identifying <simplicial vertices> in the unsubdivided cross-polytope boundary.
Back to article page
