The homotony classification of mans, and closely related to it the study of extension and lifting nroblerns, is a central topic in algebraic topology. Steenrod writes "Many of the basic theorems of topology, and some of its most successful applications in other areas of mathematics, are solutions of particular extension problems. The deepest results of this kind have been obtained by the method of algebraic topology. The essence of the method is a conversion of the geometric problem into an algebraic problem which is sufficiently complex to embody the essential features of the geometric problem, yet sufficiently simole to be solvable by standard algebraic methods. Many extension problems remain unsolved, and much of the current development of algebraic topology is inspired by the hope of finding a truly general solution."
Read more at http://ebookee.org/Obstruction-Theory-On-Homotopy-Classification-of-Maps_1297749.html#AbZmh0ya9ratyc8O.99