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Simplicial triangulation of noncombinatorial manifolds of dimension less than $ 9$


Author: Martin Scharlemann
Journal: Trans. Amer. Math. Soc. 219 (1976), 269-287
MSC: Primary 57D05; Secondary 57C15
DOI: https://doi.org/10.1090/S0002-9947-1976-0415629-5
MathSciNet review: 0415629
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Abstract: Necessary and sufficient conditions are given for the simplicial triangulation of all noncombinatorial manifolds in the dimension range $ 5 \leqslant n \leqslant 7$, for which the integral Bockstein of the combinatorial triangulation obstruction is trivial. A weaker theorem is proven in case $ n = 8$.

The appendix contains a proof that a map between PL manifolds which is a TOP fiber bundle can be made a PL fiber bundle.


References [Enhancements On Off] (What's this?)

  • [1] W. Browder and J. Levine, Fibering manifolds over a circle, Comment. Math. Helv. 40 (1966), 153-160. MR 33 #3309. MR 0195104 (33:3309)
  • [2] S. Cappell, R. Lashof and J. Shaneson, A splitting theorem and the structure of 5-manifolds, Symposia Math. 10 (1972), 47-58. MR 0365586 (51:1838)
  • [3] L. C. Glaser, Shrinking decompositions of certain manifolds and the polyhedral Schoenflies conjecture, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, New York, 1970), State Univ. of New York at Binghamton, 1971, pp. 199-214. MR 44 #3326a. MR 0286110 (44:3326a)
  • [4] J. Hollingsworth and J. Morgan, Homotopy triangulations and topological manifolds, Preprint, 1970.
  • [5] H. King, Thesis, Univ. of Calif. at Berkeley, 1974.
  • [6] R. C. Kirby and L. C. Siebenmann, Deformation of smooth and piecewise-linear manifold structures (to appear).
  • [7] -, Deformation of sliced families of smooth and piecewise linear manifold structures (to appear).
  • [8] -, Some basic theorems for topological manifolds (to appear).
  • [9] -, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742-749. MR 39 #3500. MR 0242166 (39:3500)
  • [10] J. W. Milnor, Lectures on characteristic classes, Mimeo Notes, Princeton Univ., Princeton, N. J., 1957.
  • [11] -, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 33 #4922. MR 0196736 (33:4922)
  • [12] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der math. Wissenschaften, Band 69, Springer-Verlag, New York, 1972. MR 0350744 (50:3236)
  • [13] M. Scharlemann, Transversality theories at dimension four (to appear). MR 0410756 (53:14502)
  • [14] -, Equivalence of 5-dimensional s-cobordisms, Proc. Amer. Math. Soc. 53 (1975), 508-510. MR 0380838 (52:1735)
  • [15] L. C. Siebenmann, Are non-triangulable manifolds triangulable? Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 77-84. MR 42 #6837. MR 0271956 (42:6837)
  • [16] -, The obstruction to finding a boundary for an open manifold of dimension $ \geqslant 5$, Thesis, Princeton Univ., 1965.
  • [17] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [18] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. MR 15, 890. MR 0061823 (15:890a)
  • [19] C. T. C. Wall, Surgery on compact manifolds, Academic Press, New York, 1971. MR 0431216 (55:4217)
  • [20] J. H. C. Whitehead, On $ {C^1}$ complexes, Ann. of Math. (2) 41 (1940), 804-824. MR 2, 73. MR 0002545 (2:73d)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0415629-5
Keywords: Noncombinatorial triangulation, PL triangulation obstruction, integral Bockstein homomorphism, manifold category (DIFF, TOP, PL)
Article copyright: © Copyright 1976 American Mathematical Society

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