The covering homotopy extension problem for compact transformation groups (Q1946439)

In the paper the category ISOV consisting of \(G\)-spaces with an action of a compact group \(G\) and isovariant \(G\)-maps is studied. The authors introduce a number of notions related to the category ISOV. In particular, the notions of absolute (neighborhood) isovariant extensors, so called Isov-A(N)E-spaces and Hurewicz Isov-bundles are defined. The fact that any orbit projection is a Hurewicz bundle plays a main role in the solution of interesting problems of equivariant topology. One of the main results of paper is that any \(G\)-space admits a closed \(G\)-embedding into an Isov-AE- space \(L\times\mathbb{J}\), where \(L\) is a linear normed space and \(\mathbb{J}\) is the countable power of the metric cone over the discrete union of all homogeneous equivariant absolute neighbourhood extensors of the form \(G/H\). Besides, theorems on the extension of covering homotopy for \(G\)-spaces and on an equivariant homotopy of the isovariant category are proved.