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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. Radó, 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:
 [1] R Fintushel, R J Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. $$(3)$$ 61 (1990) 109 · Zbl 0705.57009 [2] M Furuta, Homology cobordism group of homology 3-spheres, Invent. Math. 100 (1990) 339 · Zbl 0716.55008 [3] D E Galewski, R J Stern, Classification of simplicial triangulations of topological manifolds, Ann. of Math. $$(2)$$ 111 (1980) 1 · Zbl 0441.57017 [4] D Galewski, R Stern, A universal 5-manifold with respect to simplicial triangulations, Academic Press (1979) 345 · Zbl 0477.57010 [5] D E Galewski, R J Stern, Simplicial triangulations of topological manifolds, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc. (1978) 7 · Zbl 0511.57015 [6] , Problems in low-dimensional topology, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35 [7] R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press (1977) · Zbl 0361.57004 [8] T Matumoto, Triangulation of manifolds, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc. (1978) 3 · Zbl 0509.57010 [9] R J Milgram, Some remarks on the Kirby-Siebenmann classottingen, 1987)”, Lecture Notes in Math. 1361, Springer (1988) 247 · Zbl 0674.57015 [10] D Randall, On 4-dimensional bundle theories, Contemp. Math. 161, Amer. Math. Soc. (1994) 217 · Zbl 0840.57017 [11] A A Ranicki, On the Hauptvermutung, $$K$$-Monogr. Math. 1, Kluwer Acad. Publ. (1996) 3 · Zbl 0871.57023 [12] Y B Rudyak, On Thom spectra, orientability, and cobordism, Springer Monographs in Mathematics, Springer (1998) · Zbl 0906.55001 [13] L C Siebenmann, Are nontriangulable manifolds triangulable?, Markham, Chicago, Ill. (1970) 77 · Zbl 0297.57012
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