Obstructy

Obstructy is a term used in obstruction theory, a branch of algebraic topology, to denote a specific obstruction class that arises when attempting to extend a partial structure from a subspace to the entire space. The obstructy is an element of a cohomology group determined by the homotopy groups of a target fibration or the associated fiber, and it vanishes exactly when the desired extension (or lifting) is possible, subject to suitable connectivity hypotheses.

In standard obstruction theory, extending a map or section through the skeleta of a CW complex leads to a sequence of obstructions, each living in a cohomology group H^{n+1}(X, A; π_n(F)). The obstructy is the representative of these obstructions at a given stage; if it is zero, the extension can continue to the next stage. The concrete computation of obstructy often involves tools such as Postnikov towers, spectral sequences, and cohomology with local coefficients.

Typical use cases for obstructy include proving the existence of global sections of fibrations, solving lifting problems, and classifying fiber bundles. In computational topology, obstructy can be encountered when algorithms attempt to determine whether local data can be assembled into a global structure.

Variations and terminology differ: some authors describe the same idea directly as a cohomology class or an obstruction element, without the -y suffix. The term obstructy is not universally standardized and is often used informally or in specific papers. See also obstruction theory, cohomology, and spectral sequences.