Compatibility of imposed differentiable structures
James R. Munkres · Illinois Journal of Mathematics · 1968
JAMES Ro /UNKRESIn an earlier paper 11, we constructed an obstruction theory for the problem of imposing a derentiable structure on a combinatorial manifold .At the tme, we dd not know whether the structures obtained by means of this theory could be chosen to be compatible wth the piecewseolinear structure of K. Our purpose here is to prove that they can be so chosen; the proof is a corollary of our recent work on the concordance problem I3.For convenience in exposition, we restrict ourselves to the case of a non-bounded manifold ; extension to the case where M has a boundary is not dh.cult.Compactness is not assumed.The proof involves the notion of a smooth cel[ complex in a differentiable n-manifold M.This is a collectior C of dosed ceils imbedded in M with dis- joint interiors such that(1) For each cell c, Bd c is the union of finitely many cells of lower di- mension.(2) Each point of M has a neighborhood intersecting only finitely many cells of C.(3) Each m-cell c is smooth, in the sense that it lies in a smooth m-di- mersional non-bounded submanifold N, of M.(4) The submanifolds hr can be chosen to intersect transversally.If the union C of the cells of C equals M, we call C a smooth cell decomposi- t/on of M.A triangulation of C is a triangulation f L -C of C] which induces a triangulation of each cell of C; it is said to be smooth if f is a smooth imbedding of each simplex of L into M. Now any smooth cell complex in M has a smooth triangulation, uniquely determined up to a piecewise linear homeomorphism (Lemma 1).In particu- lar, the subcomplex of C consisting of a single cell c and its faces has a smooth triangulation f'L -c !; if L is a combinatorial ball (a piecewise-linear homeomorph of a simplex) then we call c a smooth combinatorial cell.(This is actually a restrictio o c only in those dimensions where a triangulated topological ball need not be a combinatorial ball.)If every cell in C is combi- natorial, we call C a smooth combinatorial cell complex.For example, let us consider a manifold M with a smooth triangulation