In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations.
The degree of a map was first defined by Brouwer,who showed that the degree is homotopy invariant (invariant among homotopies), and used it to prove the Brouwer fixed point theorem. In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number.
The simplest and most important case is the degree of a continuous map from the -sphere to itself (in the case , this is called the winding number):
Let be a continuous map. Then induces a homomorphism , where is the th homology group. Considering the fact that , we see that must be of the form for some fixed . This is then called the degree of .
Let X and Y be closed connected oriented m-dimensional manifolds. Orientability of a manifold implies that its top homology group is isomorphic to Z. Choosing an orientation means choosing a generator of the top homology group.
A continuous map f : X→Y induces a homomorphism f* from Hm(X) to Hm(Y). Let [X], resp. [Y] be the chosen generator of Hm(X), resp. Hm(Y) (or the fundamental class of X, Y). Then the degree of f is defined to be f*([X]). In other words,
If y in Y and f−1(y) is a finite set, the degree of f can be computed by considering the m-th local homology groups of X at each point in f−1(y).
In the language of differential topology, the degree of a smooth map can be defined as follows: If f is a smooth map whose domain is a compact manifold and p is a regular value of f, consider the finite set
By p being a regular value, in a neighborhood of each xi the map f is a local diffeomorphism (it is a covering map). Diffeomorphisms can be either orientation preserving or orientation reversing. Let r be the number of points xi at which f is orientation preserving and s be the number at which f is orientation reversing. When the domain of f is connected, the number r − s is independent of the choice of p (though n is not!) and one defines the degree of f to be r − s. This definition coincides with the algebraic topological definition above.
The same definition works for compact manifolds with boundary but then f should send the boundary of X to the boundary of Y.
One can also define degree modulo 2 (deg2(f)) the same way as before but taking the fundamental class in Z2 homology. In this case deg2(f) is an element of Z2 (the field with two elements), the manifolds need not be orientable and if n is the number of preimages of p as before then deg2(f) is n modulo 2.
Integration of differential forms gives a pairing between (C∞-)singular homology and de Rham cohomology: , where is a homology class represented by a cycle and a closed form representing a de Rham cohomology class. For a smooth map f : X→Y between orientable m-manifolds, one has
where f* and f* are induced maps on chains and forms respectively. Since f*[X] = deg f · [Y], we have
for any m-form ω on Y.
If is a bounded region, smooth, a regular value of and , then the degree is defined by the formula
where is the Jacobi matrix of in . This definition of the degree may be naturally extended for non-regular values such that where is a point close to .
The degree satisfies the following properties:
These properties characterise the degree uniquely and the degree may be defined by them in an axiomatic way.
In a similar way, we could define the degree of a map between compact oriented manifolds with boundary.
The degree of a map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps are homotopic if and only if .
In other words, degree is an isomorphism between and .
Moreover, the Hopf theorem states that for any -dimensional closed oriented manifold M, two maps are homotopic if and only if
A self-map of the n-sphere is extendable to a map from the n-ball to the n-sphere if and only if . (Here the function F extends f in the sense that f is the restriction of F to .)
There is an algorithm for calculating the topological degree deg(f, B, 0) of a continuous function f from an n-dimensional box B (a product of n intervals) to , where f is given in the form of arithmetical expressions. An implementation of the algorithm is available in TopDeg - a software tool for computing the degree (LGPL-3).
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.
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, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of a normal vector allows one to use the right-hand rule to define a "clockwise" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a "clockwise" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure in the space cannot be moved continuously on that surface and back to it starting point so that it looks like its own mirror image.
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 topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation.
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 mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.
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, 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, then the kth cohomology group of M is isomorphic to the th homology group of M, for all integers k
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace.
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension . The surgery structure set of a compact -dimensional manifold is a pointed set which classifies -dimensional manifolds within the homotopy type of .
In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology. The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology.
This is a glossary of properties and concepts in algebraic topology in mathematics.