Degree of a continuous mapping

In <a href="/facts/Topology/2EqW47cU">topology</a>, the degree of a <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous mapping</a> between two <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Orientability/1SEQM2Pt">oriented</a> <a href="/facts/Manifold/rXPERJtu">manifolds</a> of the same <a href="/facts/Dimension/0ktmUyac">dimension</a> is a number that represents the number of times that the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> manifold wraps around the <a href="/facts/Range_of_a_function/66CeTV6u">range</a> manifold under the mapping. The degree is always an <a href="/facts/Integer/PUwwrawx">integer</a>, but may be positive or negative depending on the orientations.
The degree of a map between general manifolds was first defined by <a href="/facts/Luitzen_Egbertus_Jan_Brouwer/6q6ULf3b">Brouwer</a>, who showed that the degree is <a href="/facts/Homotopy_invariant/1nWrPxdj">homotopy invariant</a> and used it to prove the <a href="/facts/Brouwer_fixed_point_theorem/xP2p03lQ">Brouwer fixed point theorem</a>. Less general forms of the concept existed before Brouwer, such as the <a href="/facts/Winding_number/kaquyUL2">winding number</a> and the Kronecker characteristic (or Kronecker integral).
In modern mathematics, the degree of a map plays an important role in topology and <a href="/facts/Geometry/wTQf5ffQ">geometry</a>. In <a href="/facts/Physics/a7aH2y08">physics</a>, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a <a href="/facts/Topological_quantum_number/rSa0TewR">topological quantum number</a>.

Degree of a continuous mapping open-in-new

Degree of a continuous mapping