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.
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Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000
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Arkowitz, M., Lupton, G. Rational obstruction theory and rational homotopy sets. Math Z 235, 525–539 (2000). https://doi.org/10.1007/s002090000144
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DOI: https://doi.org/10.1007/s002090000144