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The Euler class is a topological invariant associated with a vector bundle. It is an obstruction to the existence of a nowhere vanishing section. Example: The Euler class of the tangent bundle of a sphere is non-zero, indicating no non-vanishing vector fields exist on it.

Cohomology is a mathematical concept used in algebraic topology to study spaces in terms of algebraic invariants. Example: De Rham cohomology can detect the inability to extend a non-vanishing vector field from the punctured plane to the entire plane.

Steenrod squares are cohomological operations used to detect obstructions and provide information on the secondary cohomology operations. Example: They can detect the impossibility of finding a section of a fiber bundle when no primary obstruction exists.

A fiber bundle is a space that locally looks like a product space but globally may have a different topological structure. Example: The Möbius strip is a fiber bundle over the circle with a fiber that is a line segment.

The homotopy lifting property pertains to a map's ability to lift homotopies from one space to another. Example: A covering map exhibits the homotopy lifting property, which ensures that a homotopy of paths in the base can be lifted to the total space.

Obstruction theory is a branch of algebraic topology that studies the question of whether a given partial mathematical structure can be extended to a larger structure. Example: Determining if a vector field on a manifold can be extended from a hemisphere to the entire sphere.

Postnikov towers are a tool in algebraic topology that decomposes a space into layers, each of which captures information about homotopy groups at a certain dimension. Example: Analyzing a Postnikov tower can help identify where obstructions to lifts of maps may occur.