On the number of simplexes of subdivisions of finite complexes

M. L. Gromov

Matematičeskie zametki, Tome 3 (1968) no. 5, pp. 511-522.

Résumé

Combinatorial invariants of a finite simplicial complex $K$ are considered that are functions of the number $\alpha_i(K)$ of Simplexes of dimension $i$ of this complex. The main result is Theorem 2, which gives the necessary and sufficient condition for two complexes $K$ and $L$ to have subdivisions $K'$ and $L'$ such that $\alpha_i(K')=\alpha_i(L')$ for $0\le i\infty$. The theorem yields a corollary: if the polyhedra $|K|$ and $|L|$ are homeomorphic, then there exist subdivisions $K'$ and $L'$ such that $\alpha_i(K')=\alpha_i(L')$ for $i\ge0$.