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

Martin Scharlemann
1976 Transactions of the American Mathematical Society  
Necessary and sufficient conditions are given for the simplicial triangulation of all noncombinatorial manifolds in the dimension range 5 < n < 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. 0. Two of the oldest and most difficult problems arising in manifold theory are the following:
more » ... e the following: (i) Is every manifold homeomorphic to a simplicial complex? (ii) Is every simplicial triangulation of a manifold combinatorial (i.e. must the link of every simplex be a sphere)? Among the consequences of the fundamental breakthrough of Kirby-Siebenmann [9] was that at least one of these questions must be answered negatively, for there are topological manifolds without combinatorial (PL) triangulations. The existence of a counterexample to the second question is equivalent to the following conjecture: There is some homology m-sphere K, not PL equivalent to S™ such that the p-fold suspension SPAT is homeomorphic to Sm+P. Siebenmann shows that if the answer to question (i) is affirmative for manifolds of dimension n > 5, then the following hypothesis is true for m = n -3: Hypothesis H(m). There is a PL homology 3-sphere K such that ~LmK ** 5", and K bounds a PL manifold of index 8 (mod 16) (i.e. the Rochlin invariant of K is nontrivial). Furthermore, if hypothesis H(2) is true, then all orientable 5-manifolds are simplicially triangulable [15] . The purpose of this paper is to prove 0.1. Theorem. Let M" be a connected closed noncombinatorial manifold of dimension 5 < n < 8, and let kN G H4(N; Z2) be the obstruction to the Received by the editors February 26, 1975. AMS (tfOS) subject classifications (1970). Primary 57C15, 57C25; Secondary 55F10, 55F60.
doi:10.1090/s0002-9947-1976-0415629-5 fatcat:qdlvf4wlkjbilazj6atqgrtfna