Simplicial triangulation of noncombinatorial manifolds of dimension less than 9

Scharlemann, Martin

doi:10.1090/S0002-9947-1976-0415629-5

Simplicial triangulation of noncombinatorial manifolds of dimension less than $9$
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Trans. Amer. Math. Soc. 219 (1976), 269-287 Request permission

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

W. Browder and J. Levine, Fibering manifolds over a circle, Comment. Math. Helv. 40 (1966), 153–160. MR 195104, DOI 10.1007/BF02564368
S. Cappell, R. Lashof, and J. Shaneson, A splitting theorem and the structure of $5$-manifolds, Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971) Academic Press, London, 1972, pp. 47–58. MR 0365586
Leslie C. Glaser, Shrinking decompositions of certain manifolds and the polyhedral Schoenflies conjecture, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N.Y., 1970) State Univ. of New York at Binghamton, Binghamton, N.Y., 1971, pp. 199–214. MR 0286110

Homotopy triangulations and topological manifolds

Deformation of smooth and piecewise-linear manifold structures

Deformation of sliced families of smooth and piecewise linear manifold structures

Some basic theorems for topological manifolds

R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742–749. MR 242166, DOI 10.1090/S0002-9904-1969-12271-8

Lectures on characteristic classes

J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744, DOI 10.1007/978-3-642-81735-9
Martin G. Scharlemann, Transversality theories at dimension four, Invent. Math. 33 (1976), no. 1, 1–14. MR 410756, DOI 10.1007/BF01425502
Martin Scharlemann, Equivalence of $5$-dimensional $s$-cobordisms, Proc. Amer. Math. Soc. 53 (1975), no. 2, 508–510. MR 380838, DOI 10.1090/S0002-9939-1975-0380838-5
L. C. Siebenmann, Are nontriangulable manifolds triangulable?, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 77–84. MR 0271956

The obstruction to finding a boundary for an open manifold of dimension

Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math. (2) 41 (1940), 809–824. MR 2545, DOI 10.2307/1968861