ar X iv : m at h / 00 10 12 6 v 1 [ m at h . A T ] 1 2 O ct 2 00 0 RATIONAL OBSTRUCTION THEORY AND RATIONAL HOMOTOPY SETS

We develop an obstruction theory for homotopy of homomorphisms f, g : M → N between minimal differential graded algebras. We assume that M = ΛV has an obstruction decomposition given by V = V0⊕V1 and that f and g are homotopic on ΛV0. An obstruction is then obtained as a vector space homomorphism V1 → H(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 [M,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] when A 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. Introduction. A basic object of study in unstable homotopy theory, perhaps the basic object of study, is the set of homotopy classes of maps from one topological space to another. For arbitrary spaces, it is difficult to describe this homotopy set fully, and the best that can be hoped for is some partial information. In this paper we use methods from rational homotopy theory to study this set for rational spaces (i.e., spaces whose homotopy groups are rational vector spaces). This can be regarded as an approximation to homotopy sets for finite complexes. As is well known, the homotopy theory of rational spaces is equivalent to the homotopy theory of minimal, differential, graded commutative algebras over the rationals (minimal algebras, for short). Minimal algebras provide an effective algebraic setting to work in, and we begin by considering homomorphisms of minimal algebras. We develop an obstruction theory for homotopy of homomorphisms of minimal algebras and present a few applications of this theory. These include a simple proof that there are no non-trivial phantom maps between minimal algebras and results on a conjecture of Copeland-Shar that the homotopy set [M,N ] for minimal algebras M and N is either trivial or infinite. In addition, we give examples of minimal algebras that have few self-maps, including an elliptic minimal algebra with trivial group of homotopy classes of self-equivalences. Because of the categorical equivalence mentioned above, the results on homotopy classes of homomorphisms of minimal algebras, obtained by the obstruction theory, translate immediately into corresponding results about homotopy classes of maps of rational spaces. Thus we obtain results, of interest in their own right, about the set of homotopy classes of maps [A,B], where A and B are rational spaces. 1991 Mathematics Subject Classification. Primary 55P62, 55Q05, 55S35. Secondary 55P10.