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Equivalences to the triangulation conjecture. (English) Zbl 1032.57022
The Triangulation Conjecture, which states that every closed \(n\)-dimensional topological manifold can be triangulated, holds for \(n=2\) [T. Rad√≥, Acta Szeged 2, 101-121 (1925; JFM 51.0273.01)] and for \(n=3\) [E. Moise, Ann. of Math. (2) 56, 96-114 (1952; Zbl 0048.17102)], but is false for \(n=4\) [S. Akbulut and J. D. McCarthy, Casson’s invariant for oriented homology 3-spheres. An exposition (Math. Notes 36, Princeton Univ. Press, Princeton) (1990; Zbl 0695.57011)] and still open for \(n\geq 5\).
In this paper, the author employs the obstruction theory of D. E. Galewski and R. J. Stern [Ann. of Math. (2) 111, 1-34 (1980; Zbl 0441.57017)] and T. Matumoto [Proc. Symp. Pure Math. 32, 3-6 (1978; Zbl 0509.57010)] to derive eight equivalent formulations of the Triangulation Conjecture for \(n\geq 5\). For example, the Triangulation Conjecture is true for \(n\geq 5\) if and only if the two distinct combinatorial triangulations of \(\mathbb{R} P^5\) are simplicially concordant. The author moreover gives new results on the triangulability of certain classes of manifolds. In particular, he shows that \(k\)-fold Cartesian products of closed \(4\)-manifolds are triangulable for \(k\geq 2\).
MSC:
57Q15 Triangulating manifolds
55S35 Obstruction theory in algebraic topology
57N16 Geometric structures on manifolds of high or arbitrary dimension
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References:
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