#jsDisabledContent { display:none; } My Account |  Register |  Help

# Obstruction theory

Article Id: WHEBN0002179639
Reproduction Date:

 Title: Obstruction theory Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Obstruction theory

In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.

In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a cross-section of a bundle.

## Contents

• In homotopy theory 1
• Obstruction to extending a section of a principal bundle 1.1
• Construction 1.1.1
• Applications 1.1.2
• In geometric topology 2
• In surgery theory 3
• References 5

## In homotopy theory

The older meaning for obstruction theory in homotopy theory relates to a procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. Traditionally called Eilenberg obstruction theory, after Samuel Eilenberg. It involves cohomology groups with coefficients in homotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complex X to another, Y, defined initially on the 0-skeleton of X (the vertices of X), an extension to the 1-skeleton will be possible whenever Y is sufficiently path-connected. Extending from the 1-skeleton to the 2-skeleton means filling in the images of the solid triangles from X, given the image of the edges. However, further extending to 3-skeleton involves the opposite—i.e. the solid triangle images are removed from X.

### Obstruction to extending a section of a principal bundle

#### Construction

Suppose that B is a simply connected simplicial complex and that p : EB is a fibration with fiber F. Furthermore, assume that we have a partially defined section σn : BnE on the n-skeleton of B.

For every (n + 1)-simplex Δ in B, σn can be restricted to its boundary (which is a topological n-sphere). Because p send each of these back to each Δ, we have a map from an n-sphere to p-1(Δ). Because fibrations satisfy the homotopy lifting property, and Δ is contractible; p-1(Δ) is homotopy equivalent to F. So this partially defined section assigns an element of πn(F) to every (n + 1)-simplex. This is precisely the data of a πn(F)-valued simplicial cochain of degree n + 1 on B, i.e. an element of Cn + 1(B; πn(F)). This cochain is called the obstruction cochain because it being the zero means that all of these elements of πn(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 each Δ) and the constant map.

The fact that this cochain came from a partially defined section (as opposed to an arbitrary collection of maps from the 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 section σn 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 group Hn + 1(B; πn(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 as homotopy sections, i.e. a map σ : BE such that pσ is homotopic (as opposed to equal) to the identity map on B. Thus it provides a complete invariant of the existence of sections up to homotopy on the (n + 1)-skeleton.

#### Applications

• By inducting over n, one can construct a first 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 of principal bundles.
• Because any 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 into B to a map into E even if p : EB is not a fibration.
• It is crucial to the construction of Postnikov systems.

## In geometric topology

In geometric topology, obstruction theory is concerned with when a topological manifold has a piecewise linear structure, and when a piecewise linear manifold has a differential 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.

## In surgery theory

The two basic questions of surgery theory are whether a topological space with n-dimensional Poincaré duality is homotopy equivalent to an n-dimensional manifold, and also whether a homotopy equivalence of n-dimensional manifolds is homotopic to a diffeomorphism. In both cases there are two obstructions for n>9, a primary topological K-theory obstruction to the existence of a vector bundle: if this vanishes there exists a normal map, allowing the definition of the secondary surgery obstruction in algebraic L-theory to performing surgery on the normal map to obtain a homotopy equivalence.

## References

• Husemöller, Dale (1994), Fibre Bundles, Springer Verlag,
• Scorpan, Alexandru (2005). The wild world of 4-manifolds. American Mathematical Society.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.