A geometric condition for smoothability of combinatorial manifolds

Let us commence with the terminology. For a complex Y, \ Y\ will denote the polyhedron covered by Y and Y will stand for the first barycentric subdivision of Y. We say that a subcomplex X of Y is complete if the intersection of a (closed) simplex of Y and | X \ is either empty or a simplex of X. A combinatorial manifold is a polyhedron with a distinguished class of simplicial subdivisions which are formal manifolds, [5, p. 825]. For a combinatorial manifold P, the boundary of P is written dP and the interior P—dPΊs written IntP, and a closed combinatorial manifold will be a compact combinatorial manifold without boundary. Let X be a subcomplex of Y where | Y\ is a combinatorial manifold. (Note that X' is a complete subcomplex of Y'.} Then N(X, Y) denotes the star neighborhood of X in y, that is, the polyhedron consists of simplices of F, which contain simplices of X. It is well known that dN(K', L') (that is, the boundary of the star neighborhood of the first barycentric subdivision of K in the first barycentric subdivision of L) is a closed combinatorial (m—1) -manifold if the polyhedron \L\ is a combinatorial m-manifold without boundary and K is a finite complete subcomplex of L; [4, p. 293]. For convenience, we say that a polyhedron Q is imbedded piecewise linearly in euclidean space R if there are (linear) simplicial subdivisions X and L of Q and R respectively such that X is a subcomplex of L, where it may be assumed without loss of generality that X is a complete subcomplex of L. Now let us explain the condition for smoothability. DEFINITION 1. Let M be a closed combinatorial ^-manifold imbedded piecewise linearly in euclidean (n+r)-space R, r^l. We say that M is in smoothable position in R if the following is satisfied. Let K0 and L0 be simplicial subdivisions of M and R respectively, where K0 is a complete subcomplex of L0. Then there exist piecewise linear homeomorphisms pi. Mί-^dN(Kτ', Z '̂) for each O^a'^r— 1, where MQ=M and for l^z^r, Mτ=pi-ι(Mι-ι) and where Kl and LL are simplicial subdivisions of Mi and dN(Kτ-ι', L l_/) respectively such that Kt is a complete subcomplex of Li. In the text, however, Wτ stands for dN(Kι-ι, Ll-l ) and Lτ will be the subcomplex of L t_/ covering W% for each l<^'^r. Note that M^ is a closed combinatorial ^-manifold, which is combinatorially equivalent to M", and Wι is a closed combinatorial (n+r—ϊ) -manifold, for each l^i^r, satisfying MtCTFt and WΊD W2 =) D TFr.