Rational obstruction theory and rational homotopy sets

Abstract.

We develop an obstruction theory for homotopy of homomorphisms $f,g : {\mathcal M }\to{\mathcal N }$ between minimal differential graded algebras. We assume that ${\mathcal M }=\Lambda V$ has an obstruction decomposition given by $V=V_0\oplus V_1$ and that f and g are homotopic on $\Lambda V_0$. An obstruction is then obtained as a vector space homomorphism $V_1\to H^*({\mathcal N})$. We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes $[{\mathcal M},{\mathcal N }]$. This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps.