On Product and Bündle Neighborhoods

If a nice space X is embedded in a euclidean space, it may fail to have product neighborhoods; i.e., neighborhoods which are products of X with a ball. However, if the euclidean space is a hyperplane of a higher-dimensional euclidean space one can sometimes guarantee the existence of product neighborhoods in the big euclidean space. For example, it follows from a lemma due to Klee [l ] that if X is a ¿-ball in R"QRn+k, then X has a neighborhood homeomorphic with XXBn in Rn+k (where Bn denotes an ra-ball). We obtain two results along these lines. Theorem 1 gives circumstances in which we get product neighborhoods and Theorem 2 yields ball-bundle neighborhoods. Theorem 2 has been used for smoothing combinatorial manifolds [3]. Definition. We say that a space X has a local multiplication into a space Z if there exists a neighborhood N of the diagonal A of XXX and a map : N—>Z such that : (i) (Nx) contains a fixed neighborhood W of zo, for all x in X. (Nx = pairs in N with first coordinate x.) Example 1. Any topological group G has a local multiplication into itself. Take N=GXG and define cp(g, h)=gh~\ Example 2. If X is a ¿-dimensional parallelizable manifold, then X has a local multiplication into 7?*. We take a Riemannian metric for X and choose N so that if (x, y)QN, then there is a unique geodesic from x to y. Let t(x, y) denote the vector tangent to this geodesic at x and having length equal to the length of the geodesic. Let c be a cross section of the ¿-frame bundle over X. (c exists since X is parallelizable.) We define <p(x, y) to be the point in Rk with coordinates equal to the dot products of t(x, y) with the vectors of c(x). It is easy to verify that ^ is a local multiplication. A space 5 is said to have the neighborhood extension property if for any closed subset 73 of a separable metric space F and any map /: B-+S, there exists an extension of / to some neighborhood of B.