# Download Algebraic Geometry. Proc. conf. Tokyo, Kyoto, 1982 by M. Raynaud, T. Shioda PDF By M. Raynaud, T. Shioda

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One way is trivial; for if J is a combinatorial manifold , then the closed vertex stars of J give a covering of M by balls, such that each point of M has some ball as a neighbourhood . x he a vertex of J in 8 . By the definition of manifold (polymanifold), there is a pieoewise linear embedding f : A —> J covering a neighbourhood of where A is an n-simplex , such that f~''oc € A f : A ' —^J' A , . Subdivide so that is simplicial; we have pieoewise linear homeomorphisms y lk(f X , A ' ) _^,lk(x,J') lk(s:,J) Where the middle arrow is an isomorphism and the other two arrows are pseudo radial projections .

Let This gives f level preserving on ( A x l)' . Define f level preserving on A X I by mapping the centre of the prism to itself, and joining to the boundary linearly . Then f is the desired isotopy . ^Lemma 17 . Suppose M^C Q,^ are manifolds . and that m" is a closed subset of . Then Q"" - M"' is a manifold . Proof . Let M^ = q"' - m"' . We have to show that every point 'XI M^ has a ball neighbourhood in M^ . If x G Q. - M^ , then -x has a ball neighbourhood in Q^ that is contained in M^ , because m'^ is 0 closed in Q .

However such factorization is only true for manifolds , and not true for polyhedra in general . For instance X \\ 0 Y ) X D Y Consider the following example . Let x y z be a triangle , and let y', z' be two interior points not concurrent with x . Let X be the space obtained by identifying the intervals x y = xy' , xz of y z in X . xz' , and let Y be the image - 26 - X Then X 0 conewise , axid Y 0 because Y is an arc . But X Y because any initial elementary simplicial collapse of any triangulation of X must have its free face in Y , and so must remove part of B .