The self-equivalence groups in certain coherent homotopy categories

Abstract

We study the self-equivalence groups associated with objects in (i) the track homotopy category over a fixed space $B$, (ii) the track homotopy category under a fixed space $A$ and (iii) the category of homotopy pairs. In each case a short exact sequence decomposition of the self-equivalence group is available. In the case of (i) the group is isomorphic to the group of fibre-homotopy self-equivalences of an associated fibration, the decomposition (in other form) is known and has been used as the basis of computations. We make sample computations in the simplest situations for (i), (ii), and (iii), in each case solving the extension problem that arises by considering secondary operations and determining the Toda-Hopf invariant of relevant tracks. We indicate that in certain cases such computations can be used to determine the self-equivalence group of a mapping cone.