A Useful Functor and Three Famous Examples in Topology
Robert Francis Williams · Transactions of the American Mathematical Society · 1963
The purpose of this note is to describe a functor which provides a framework for certain constructions in topology.It is related to the sets (£, n, B, X, q) described in [7] and is particularly adapted to discussing the limit of repeated modifications of triangulable spaces.Roughly speaking, one forms a space XAK by replacing each top dimensional simplex of a complex K with a copy of a space X.If in addition there are mappings on the spaces X, K, these induce a mapping on the new space XAK.It has been called to my attention that several authors have considered analogous functors (though not as far as I know, in written form).This is not surprising inasmuch as XAK is defined just as the Whitney sum of two bundles.Though the principal applications of this functor are to be found elsewhere, in a paper by Frank Raymond and the author [8; 7] and a forthcoming paper by the author, three famous examples, Pontrjagin [6], Boltyanskii [2], and Kolmogoroff [5] are given as applications in the last section.Two of these are in dimension theory proper, but the third is essentially about transformation groups.This was pointed out by Professor Deane Montgomery whom the author would like to thank for his considerable aid and encouragement.It is hoped that the reader will find our description of Boltyanskii's example easier than the original, as a simpler, more homogeneous version is given.In addition, in our version of Kolmogoroff's example, the group acts without fixed points.This answers a question raised by Anderson [1].