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.
The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is 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 the image of the 0-skeleton will belong to the same path-connected 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 πn-1(Y) 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 πn-1(Y). Amazingly, this cochain turns out to be a cocycle and so defines a cohomology class in the nth cohomology group of X with coefficients in πn-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 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.
Suppose that B is a simply connected simplicial complex and that p : E → B is a fibration with fiber F. Furthermore, assume that we have a partially defined section σn : Bn → E 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 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 : B → E such that p ∘ σ 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.
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 (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.
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.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space. A fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic; rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a more technical property.
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology a line bundle is defined as a vector bundle of rank 1.
In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry.
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944.
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
In mathematics, specifically in homotopy theory, a classifying spaceBG of a topological group G is the quotient of a weakly contractible space EG by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.
In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
In mathematics, topology generalizes the notion of triangulation in a natural way as follows:
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site taking values in simplicial sets. Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.
This is a glossary of properties and concepts in algebraic topology in mathematics.