Rational obstruction theory and rational homotopy sets

Details Rational obstruction theory and rational homotopy sets Rational obstruction theory and rational homotopy sets

2000-10-12

arxiv.org/abs/math/0010126v1

Math. Zeit., Vol. 235 (2000), 525--539.

Mathematics

Algebraic Topology

To appear, Math. Zeit

Scientific paper

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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