mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, obstruction theory is a name given to two different mathematical theories, both of which yield

cohomological In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

invariants. In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a

cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...

of a

bundle Bundle or Bundling may refer to: * Bundling (packaging), the process of using straps to bundle up items Biology * Bundle of His, a collection of heart muscle cells specialized for electrical conduction * Bundle of Kent, an extra conduction path ...

In homotopy theory

The older meaning for obstruction theory in

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...

relates to the procedure, inductive with respect to dimension, for extending a

continuous mapping In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...

defined on a

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

, or

CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...

. It is traditionally called ''Eilenberg obstruction theory'', after

Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...

. It involves

cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

s with coefficients in

homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...

s to define obstructions to extensions. For example, with a mapping from a simplicial complex ''X'' to another, ''Y'', defined initially on the

0-skeleton In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other wo ...

of ''X'' (the vertices of ''X''), an extension to the 1-skeleton will be possible whenever the image of the 0-skeleton will belong to the same

path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...

component of ''Y''. Extending from the 1-skeleton to the 2-skeleton means defining the mapping on each solid triangle from ''X'', 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 of ''X'', given the mapping already defined on its boundary. At some point, say extending the mapping from the (n-1)-skeleton of ''X'' to the n-skeleton of ''X'', this procedure might be impossible. In that case, one can assign to each n-simplex the homotopy class of the mapping already defined on its boundary, (at least one of which will be non-zero). These assignments define an n-cochain with coefficients in . Amazingly, this cochain turns out to be a cocycle and so defines a

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...

class in the nth cohomology group of ''X'' with coefficients in . 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 of ''X'' so that the mapping may be extended to the n-skeleton of ''X''. 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.

Obstruction to extending a section of a principal bundle

Construction

Suppose that is a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...

simplicial complex and that is a

fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all ma ...

with fiber . Furthermore, assume that we have a partially defined

section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

on the -skeleton of . For every -simplex in , can be restricted to the boundary (which is a topological -sphere). Because sends each back to , defines a map from the -sphere to . Because fibrations satisfy the homotopy lifting property, and is

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...

; is

homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

to . So this partially defined section assigns an element of to every -simplex. This is precisely the data of a -valued simplicial cochain of degree on , i.e. an element of . This cochain is called the obstruction cochain because it being the zero means that all of these elements of are trivial, which means that our partially defined section can be extended to the -skeleton by using the homotopy between (the partially defined section on the boundary of each ) 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 -simplices) can be used to prove that this cochain is a cocycle. If one started with a different partially defined section that agreed with the original on the -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 such that if a partially defined section on the -skeleton exists that agrees with the given choice on the -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 such that is homotopic (as opposed to equal) to the identity map on . Thus it provides a complete invariant of the existence of sections up to homotopy on the -skeleton.

Applications

* By inducting over , 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 to a map into even if is not a fibration. * It is crucial to the construction of

Postnikov system In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with t ...

In geometric topology

geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originate ...

, obstruction theory is concerned with when a

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...

has a piecewise linear structure, and when a piecewise linear manifold has a

differential structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...

. In dimension at most 2 (Rado), and 3 (Morse), 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 In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...

are whether a topological space with ''n''-dimensional

Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...

to an ''n''-dimensional

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

, and also whether a

homotopy equivalence In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...

of ''n''-dimensional manifolds is

homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

to a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...

. In both cases there are two obstructions for ''n>9'', a primary topological K-theory obstruction to the existence of a

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...

: 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

References

* * *{{cite book , last= Scorpan , first= Alexandru , title= The wild world of 4-manifolds , publisher= American Mathematical Society , year= 2005 , isbn= 0-8218-3749-4 Homotopy theory Differential topology Surgery theory Theories