Some cellular subsets of the spheres
Vo Thanh Liem · Pacific Journal of Mathematics · 1977
Let H be a PL -homology sphere such that its multisuspension S k * H is topologically homeomorphic to the sphere.We prove that every cell-like subset of S k is cellular.Also, every non-compact PL -manifold of dimension greater than four accepts uncountably many simplicial triangulations each of which contains a non-cellular k -simplex for every k^ 0.Introduction.The double suspension problem introduces many different simplicial triangulations for a PL -manifold; in particular, the case of the sphere.It is easy to see that these strange triangulations are not locally flat, however, we ask whether their simplexes are cellular.A positive answer is given for every cell-like subset of S\ where S k is the suspension sphere in S k *ίf (Theorem 1), and for every cell-like subset of a codimension-2 simplex which properly meets every nontrivial face of this simplex (Theorem 2).But it is negative for general cases (Theorem 4).Finally, combining Theorem 4 and Theorem 5, it follows that there are uncountably many noncellular simplicial triangulations for a noncompact PL-manifold of dimension greater than four.The author wishes to thank J. C. Cantrell for his interesting discussions, R. J. Daverman for his enlightening idea in constructing a noncellular simplicial triangulation of a PL-manifold and the referee for his suggestions.