some cellular subsets of the spheres - Project Euclid
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PACIFIC JOURNAL OF MATHEMATICS Vol 68, No 1, 1977
SOME CELLULAR SUBSETS OF THE SPHERES V O - T H A N H LIEM
Let H be a PL -homology sphere such that its multisuspenk sion S * H is topologically homeomorphic to the sphere. We k prove that every cell-like subset of S 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 Sk is the suspension sphere in Sk *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. Let Rn denote the n -dimensional n +1 Euclidean space, S the boundary dΔ" of the standard (n + l)-simplex +1 Δ" . Let X be a compact subset of a metric space M. X satisfies the small loops condition (SLC) if UV(X CM) and there is a δ > 0 such that each δ-loop in V-X is null-homotopic in U - X (see [3]). For the notion of cellularity refer to Rushing [8] or McMillan [7]. A finite-dimensional compactum X is cell-like if X has the shape of a point (see [6]). A simplicial triangulation of a space M is a pair (T, φ) where T is a simplicial complex and φ is a homeomorphism from | Γ|, the underlying space of T, onto M. Two triangulations (T, φ) and (T", φ') are said to be equivalent if there is a PL -homeomorphism ψ: \T\-*\T'\ such that φ = φ'φ. (We refer to Hudson [5] for general notions in the PLcategory.) DEFINITIONS AND NOTATION.
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VO-THAN LIEM m
A triangulation (T, φ) of a manifold M is said to be locally flat, if
