Define a piecewise linear sphere to be a simplicial complex that admits a subdivision, which is simplicially isomorphic to a subdivision of the boundary of a simplex.
What's the current state of the theory of piecewise linear spheres? In particular, I'm interested in
- The classification of all piecewise linear spheres, up to PL-isomorphisms.
Denote the space of piecewise linear spheres of dimension $n$ by $T_n$. Given a PL sphere $S$ of dimension $n$ and a vertex $v \in S$. The link of $v$ in $S$ is another PL sphere $s$, which I call local spheres. Let $C(S)$ be the subset (of $T_{n-1}$) that consists of possible local spheres $s$.
- Is $C(S) = C(S')$, provided that $S$ and $S'$ are isomorphic?