Some cellular subdivisions of simplicial complexes

In a previous paper the author has associated with every inverse system of compact CW-complexes X with limit X and every simplicial complex K with geometric realization |K| a resolution of X × |K|, which consists of spaces having the homotopy type of polyhedra. In a subsequent paper it is shown that this construction is functorial. The proof depends essentially on particular cellular subdivisions of K. The purpose of this paper is to describe in detail these subdivisions and establish their relevant properties. In particular, one defines two subdivisions L(K) and N(K) of K. Each cell from L(K), respectively from N(K), is contained in a simplex σ K and it is the direct sum a b, respectively c d, of certain simplices contained in σ. One defines new subdivisions L\u27(K) and N\u27(K) of K by taking for their cells the direct sums L(a) b, respectively c N(d). The main result asserts that there is an isomorphism of cellular complexes : L\u27(K) → N\u27(K), which induces a selfhomeomorphism θ : |K| → |K|