Let
be a map between two compact, connected, oriented -dimensional manifolds without
boundary. Then
induces a homomorphism from the homology groups to , both canonically isomorphic to the integers,
and so
can be thought of as a homomorphism of the integers.
The integer to which the number 1 gets sent is called the degree of
the map .

There is an easy way to compute if the manifolds involved
are smooth. Let ,
and approximate
by a smooth map homotopic to such that is a "regular value" of (which exist and are everywhere dense
by Sard's theorem). By the implicit
function theorem, each point in has a neighborhood
such that
restricted to it is a diffeomorphism. If the diffeomorphism is orientation preserving, assign
it the number ,
and if it is orientation reversing, assign it the number . Add up all the numbers for all the points in , and that is the , the Brouwer degree of . One reason why the degree of a map is important is because
it is a homotopy invariant. A sharper result states
that two self-maps of the -sphere are homotopic iff they have
the same degree. This is equivalent to the result that the th homotopy group of the
-sphere is
the set
of integers. The isomorphism
is given by taking the degree of any representation.

One important application of the degree concept is that homotopy classes of maps from -spheres
to -spheres
are classified by their degree (there is exactly one
homotopy class of maps for every integer , and is the degree of those maps).