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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,^{[1]} 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 pushforward 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. Poincare duality implies that the manifold's 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 *H _{m}*(

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*). Namely, if , then

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 *x*_{i} the map *f* is a local diffeomorphism. Diffeomorphisms can be either orientation preserving or orientation reversing. Let *r* be the number of points *x*_{i} at which *f* is orientation preserving and *s* be the number at which *f* is orientation reversing. When the codomain 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** (deg_{2}(*f*)) the same way as before but taking the *fundamental class* in **Z**_{2} homology. In this case deg_{2}(*f*) is an element of **Z**_{2} (the field with two elements), the manifolds need not be orientable and if *n* is the number of preimages of *p* as before then deg_{2}(*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 Jacobian 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 topological degree can also be calculated using a surface integral over the boundary of ,^{[2]} and if is a connected *n*-polytope, then the degree can be expressed as a sum of determinants over a certain subdivision of its facets.^{[3]}

The degree satisfies the following properties:^{[4]}

- If , then there exists such that .
- for all .
- Decomposition property: if are disjoint parts of and .
*Homotopy invariance*: If and are homotopy equivalent via a homotopy such that and , then .- The function is locally constant on .

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+1*-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.^{[5]} An implementation of the algorithm is available in TopDeg - a software tool for computing the degree (LGPL-3).

- Covering number, a similarly named term. Note that it does not generalize the winding number but describes covers of a set by balls
- Density (polytope), a polyhedral analog
- Topological degree theory

**^**Brouwer, L. E. J. (1911). "Über Abbildung von Mannigfaltigkeiten".*Mathematische Annalen*.**71**(1): 97–115. doi:10.1007/bf01456931. S2CID 177796823.**^**Polymilis, C.; Servizi, G.; Turchetti, G.; Skokos, Ch.; Vrahatis, M. N. (May 2003). "Locating Periodic Orbits by Topological Degree Theory".*Libration Point Orbits and Applications*: 665–676. arXiv:nlin/0211044. doi:10.1142/9789812704849_0031. ISBN 978-981-238-363-1.**^**Stynes, Martin (June 1979). "A simplification of Stenger's topological degree formula" (PDF).*Numerische Mathematik*.**33**(2): 147–155. doi:10.1007/BF01399550. Retrieved 21 September 2024.**^**Dancer, E. N. (2000).*Calculus of Variations and Partial Differential Equations*. Springer-Verlag. pp. 185–225. ISBN 3-540-64803-8.**^**Franek, Peter; Ratschan, Stefan (2015). "Effective topological degree computation based on interval arithmetic".*Mathematics of Computation*.**84**(293): 1265–1290. arXiv:1207.6331. doi:10.1090/S0025-5718-2014-02877-9. ISSN 0025-5718. S2CID 17291092.

- Flanders, H. (1989).
*Differential forms with applications to the physical sciences*. Dover. - Hirsch, M. (1976).
*Differential topology*. Springer-Verlag. ISBN 0-387-90148-5. - Milnor, J.W. (1997).
*Topology from the Differentiable Viewpoint*. Princeton University Press. ISBN 978-0-691-04833-8. - Outerelo, E.; Ruiz, J.M. (2009).
*Mapping Degree Theory*. American Mathematical Society. ISBN 978-0-8218-4915-6.

- "Brouwer degree",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Let's get acquainted with the mapping degree, by Rade T. Zivaljevic.