Homotopy types within a rational homotopy type

Abstract

In addition. our results have an interesting interpretation in terms of rational equivalence. Let T2 be the category of simply-connected spaces with basepoint and let S, be a subcategory. Say that two spaces X and Y are rationally equivalent (in S,) if there exist mapsinSz,f:Z+Xandg:ZY (from a third space Z) which induce isomorphisms on rational cohomology. The equivalence classes of this equivalence relation are called rational homotopy types in Sz, and Quillen [7] has characterized the rational homotopy types of T, in terms of an algebraic invariant. If we restrict our attention to the subcategory P of simply-connected finite CW complexes we find that two spaces in P are rationally equivalent in P if they are rationally equivalent in T2. In P each rational homotopy type consists of a countable set of homotopy types and each of these countable sets possesses a canonical metric space distance.