An Introduction to the Theory of Obstructions: Notes from the Obstruction Theory Seminar at West Virginia University

The generalization of everyday concepts, and applying rigor thereunto, has long intrigued the mathematician. The topic of an obstruction theory arises from the question of deforming continuous maps, and what, exactly, holes are. Intuitively, the fundamental difference between an orange and a doughnut is that a doughnut has a hole, and an orange does not. When shrinking a rubber band around an orange, the rubber band will vanish; whereas the rubber band that passes through the hole of the doughnut, or the one that encircles the doughnut at its widest point, cannot vanish without cutting the doughnut or snapping the rubber band. When we apply the mathematical rigor that transforms oranges, doughnuts, and rubber bands into spheres, tori, and copies of S, we raise the question of what the “empty space” in the torus actually represents. The shrinking of the rubber band represents continuous deformations of circles and spheres. What, then, will obstruct our continuous deformation? The answer to this question forms the foundation of an obstruction theory.