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Rational obstruction theory and rational homotopy sets

M. Arkowitz·G. Lupton
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Abstract

We develop an obstruction theory for homotopy of homomorphisms f,g : M -> N between minimal differential graded algebras. We assume that 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 -> 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.

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M. Arkowitz, G. Lupton. 2000-10-12. Rational obstruction theory and rational homotopy sets. https://arxiv.org/abs/math/0010126

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