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Obstruction theory studies the extentability of Maps using algebraic Gadgets. While the terminology rapidly becomes technical and convoluted (as Iyanaga and Kawada note, ``It is extremely difficult to discuss higher obstructions in general since they involve many complexities''), the ideas associated with obstructions are very important in modern Algebraic Topology.
See also Algebraic Topology, Chern Class, Eilenberg-Mac Lane Space, Stiefel-Whitney Class
References
Iyanaga, S. and Kawada, Y. (Eds.). ``Obstructions.'' §300 in
Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 948-950, 1980.