Isotopy invariants of topological spaces

Abstract

A few decades ago, topologists had already emphasized the difference between homotopy and isotopy. However, recent developments in algebraic topology are almost exclusively on the side of homotopy. Since a complete system of homotopy invariants has been obtained by Postnikov, it seems that hereafter we should pay more attention to isotopy invariants and new efforts should be made to attack the classical problems. The purpose of this paper is to introduce and study new algebraic isotopy invariants of spaces. A general method of constructing these invariants is given by means of a class of functors called isotopy functors. Special isotopy functors are constructed in this paper, namely, the m th residual functor R m and the m th. enveloping functor E m . Applications of these isotopy invariants to linear graphs are given in the last two sections. It turns out that these invariants can distinguish various spaces belonging to the same homotopy type.