Inmathematics,obstruction theory is a name given to two differentmathematical theories, both of which yieldcohomologicalinvariants.

In the original work ofStiefel andWhitney,characteristic classes were defined as obstructions to the existence of certain fields of linear independentvectors. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing across-section of abundle.

The older meaning for obstruction theory inhomotopy theory relates to the procedure, inductive with respect to dimension, for extending acontinuous mapping defined on asimplicial complex, orCW complex. It is traditionally calledEilenberg obstruction theory, afterSamuel Eilenberg. It involvescohomology groups with coefficients inhomotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complexX to another,Y, defined initially on the0-skeleton ofX (the vertices ofX), an extension to the 1-skeleton will be possible whenever the image of the 0-skeleton will belong to the samepath-connected component ofY. Extending from the 1-skeleton to the 2-skeleton means defining the mapping on each solid triangle fromX, given the mapping already defined on its boundary edges. Likewise, then extending the mapping to the 3-skeleton involves extending the mapping to each solid 3-simplex ofX, given the mapping already defined on its boundary.

At some point, say extending the mapping from the (n-1)-skeleton ofX to the n-skeleton ofX, this procedure might be impossible. In that case, one can assign to each n-simplex the homotopy class${\displaystyle {\pi }_{n-1}(Y)}$ of the mapping already defined on its boundary, (at least one of which will be non-zero). These assignments define ann-cochain with coefficients in${\displaystyle {\pi }_{n-1}(Y)}$. Amazingly, this cochain turns out to be acocycle and so defines acohomology class in the nth cohomology group ofX with coefficients in${\displaystyle {\pi }_{n-1}(Y)}$. When this cohomology class is equal to 0, it turns out that the mapping may be modified within its homotopy class on the (n-1)-skeleton ofX so that the mapping may be extended to the n-skeleton ofX. If the class is not equal to zero, it is called the obstruction to extending the mapping over the n-skeleton, given its homotopy class on the (n-1)-skeleton.

Suppose thatB is asimply connected simplicial complex and thatp :E →B is afibration with fiberF. Furthermore, assume that we have a partially definedsectionσ_n :B_n →E on then-skeleton ofB.

For every(n + 1)-simplexΔ inB,σ_n can be restricted to the boundary∂Δ (which is a topologicaln-sphere). Becausep sends eachσ_n(∂Δ) back to∂Δ,σ_n defines a map from then-sphere top⁻¹(Δ). Because fibrations satisfy the homotopy lifting property, andΔ iscontractible;p⁻¹(Δ) ishomotopy equivalent toF. So this partially defined section assigns an element ofπ_n(F) to every(n + 1)-simplex. This is precisely the data of aπ_n(F)-valuedsimplicialcochain of degreen + 1 onB, i.e. an element ofC^{n + 1}(B;π_n(F)). This cochain is called theobstruction cochain because it being the zero means that all of these elements ofπ_n(F) are trivial, which means that our partially defined section can be extended to the(n + 1)-skeleton by using the homotopy between (the partially defined section on the boundary of eachΔ) and the constant map.

The fact that this cochain came from a partially defined section (as opposed to an arbitrary collection of maps from all the boundaries of all the(n + 1)-simplices) can be used to prove that this cochain is a cocycle. If one started with a different partially defined sectionσ_n that agreed with the original on the(n − 1)-skeleton, then one can also prove that the resulting cocycle would differ from the first by a coboundary. Therefore we have a well-defined element of the cohomology groupH^{n + 1}(B;π_n(F)) such that if a partially defined section on the(n + 1)-skeleton exists that agrees with the given choice on the(n − 1)-skeleton, then this cohomology class must be trivial.

The converse is also true if one allows such things ashomotopy sections, i.e. a mapσ :B →E such thatp ∘σ is homotopic (as opposed to equal) to the identity map onB. Thus it provides a complete invariant of the existence of sections up to homotopy on the(n + 1)-skeleton.

• By inducting overn, one can construct afirst obstruction to a section as the first of the above cohomology classes that is non-zero.
• This can be used to find obstructions to trivializations ofprincipal bundles.
• Becauseany map can be turned into a fibration, this construction can be used to see if there are obstructions to the existence of a lift (up to homotopy) of a map intoB to a map intoE even ifp :E →B is not a fibration.
• It is crucial to the construction ofPostnikov systems.

Ingeometric topology, obstruction theory is concerned with when atopological manifold has apiecewise linear structure, and when a piecewise linear manifold has adifferential structure.

In dimension at most 2 (Rado), and 3 (Moise), the notions of topological manifolds and piecewise linear manifolds coincide. In dimension 4 they are not the same.

In dimensions at most 6 the notions of piecewise linear manifolds and differentiable manifolds coincide.

The two basic questions ofsurgery theory are whether a topological space withn-dimensionalPoincaré duality ishomotopy equivalent to ann-dimensionalmanifold, and also whether ahomotopy equivalence ofn-dimensional manifolds ishomotopic to adiffeomorphism. In both cases there are two obstructions forn>9, a primarytopological K-theory obstruction to the existence of avector bundle: if this vanishes there exists anormal map, allowing the definition of the secondarysurgery obstruction inalgebraic L-theory to performing surgery on the normal map to obtain ahomotopy equivalence.

• Kirby–Siebenmann class
• Wall's finiteness obstruction

• Husemöller, Dale (1994),Fibre Bundles, Springer Verlag,ISBN 0-387-94087-1
• Steenrod, Norman (1951),The Topology of Fibre Bundles, Princeton University Press,ISBN 0-691-08055-0
• Scorpan, Alexandru (2005).The wild world of 4-manifolds. American Mathematical Society.ISBN 0-8218-3749-4.

• Homotopy theory
• Differential topology
• Surgery theory
• Theories