%
% David Blanc
% Mapping spaces and M-CW complexes
% July 9, 1996
%
abstract:
The concept of ``homotopy groups with coefficients'', in which spheres
are replaced by a Moore spaces as the representing objects, were first
studied by Peterson, and in greater detail by Neisendorfer.
Much of homotopy theory can be redone in this spirit, with an
arbitrary but fixed space $\M$ and its suspensions replacing the
spheres not only in the definition of homotopy groups, but also in
that of a $CW$-complex, loop space, and so on.
In particular, an M-CW complex is a space constructed inductively by
successively attaching M-cells. Some of the properties of
ordinary $CW$ complexes carry over to M-CW complexes - e.g., the
Whitehead theorem - but others do not.
In this note we address the question of recovering the space X from
the mapping space X^M, for a special class of ``self-map resolvable''
spaces M, a question analogous to the classical one of recovering X from
its n-fold loop space. Just as for loop spaces, one
needs some additional structure on X^M in order to do so.
Our procedure for recovering X is given recursively by a sequence
of homotopy colimits. We may also think of this procedure as another
construction of an M-CW approximation functor. Our approach can be made
more explicit in the case of the mod k Moore space.