engleski

Extension theory and the &Psi^&infin operator

We are going to define for each simplicial complex K, an operator \Psi^\infty on the subcomplexes of K. If one is given a collection of spaces, closed subspaces of them, and maps of the closed subspaces to a subpolyhedron of |K| that extend to maps into |K|, then we are going to use the \Psi^\infty operator to help determine a subcomplex of minimal cardinality into which the maps can be extended simultaneously. The question (raised by A. Dranishnikov and J. Dydak) of whether the extension dimension, extdim_{; ; (C, T)}; ; (X), has a countable representative when X is compact and metrizable, C is the class of compact metrizable spaces, and T is the class of CW-complexes is an unsolved problem. We shall define an "anti-basis" for a CW-complex and use this along with the \Psi^\infty operator to allow one to view this problem from another perspective.

extension dimension; extending maps

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