A Local to Global Selection Theorem for Simplex-valued Functions
by
Leonard R. Rubin
University of Oklahoma
Coauthors: Ivan Ivansic

Suppose that s is a function from a space X to a simplicial complex K, f is a map (continuous function) from X to |K|, and for each x Î X, f(x) Î s(x). Then f is called a selection of s. This work deals in part with finding a selection of such a s when a collection of local selections for the restrictions of s is given. Spaces X are assumed to be paracompact throughout.

Another aspect of our research is to provide a reverse type theorem. That is, suppose we are given a collection of locally defined maps of X to |K| meeting certain "closeness" requirements on the overlaps of their domains. Then, under some mild conditions, we can determine a s from X to K so that these maps are local selections of s. Moreover, there will also exist a selection f of s.

We make use of a version of E. Michael's "local to global" theorem that is stronger than the one stated in his original paper (1954) on the subject.

Date received: February 20, 2005