A criterion, cohomology or extension class that records the failure of a proposed extension of objects or structures to exist or to split; it encodes incompatibilities preventing an object defined on a subdomain from being extended to a larger domain with the desired properties.

Principle

Extension obstructions quantify the obstruction to glue or extend local data into a global extension; in algebraic contexts they typically live in Ext groups or derived functor groups and control existence and classification of extensions and whether an extension splits.

Demonstration

For modules A and B, an exact sequence 0 -> A -> E -> B -> 0 defines an extension class in Ext^1(B,A); a nonzero class is an obstruction to splitting into A ⊕ B, and higher obstruction classes can detect failure to extend deformations or filtrations to larger orders or lengths.

Misapplication

Assuming that vanishing of a single low-degree obstruction guarantees full extendability in contexts with higher obstructions; or conflating the nonexistence of split extensions with lack of any extension at all when in fact nontrivial extensions may exist but not split.

Consequence

Obstruction classes organize extension problems into computable algebraic invariants: vanishing yields existence or splitting (up to equivalence), while nonvanishing pinpoints the exact obstruction and guides either modification of data or the search for alternative extension types.

Reversal

Viewed oppositely, a trivial extension class signifies that the extension problem imposes no obstruction and admits a split model; studying reversals focuses on when global decompositions reduce to direct sums or products rather than genuine extensions.

Boundary

Pertains to algebraic and homological extension problems (modules, sheaves, group extensions, etc.); for non-abelian or highly non-linear categories the nature of obstruction classes can change and may require different cohomological tools, so the standard Ext-language may be insufficient.

Semantic Tension

The term can be confused with lifting obstruction: extension obstructions address enlarging or gluing an object outwards, while lifting obstructions concern producing preimages under projections; both use similar cohomological machinery but correspond to opposite gluing directions.

Synthesis

An extension obstruction is the invariant or class that measures why a proposed enlargement or splitting of an object fails; it packages existence and splitting questions into algebraic data whose vanishing is equivalent to the existence of the intended extension.