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Brouwer Degree

Suppose that and are two oriented differentiable manifolds of dimension (without boundary) with compact and connected and suppose that $f \colon M \to N$ is a differentiable mapping. Let denote the differential mapping at the point $x \in M$ , that is the linear mapping $Df(x) \colon T_x(M) \to T_{f(x)}(N)$ . Let $\operatorname{sign} Df(x)$ denote the sign of the determinant of . That is the sign is positive if preserves orientation and negative if reverses orientation.

Definition 1 Let $y \in N$ be a regular value, then we define the Brower degree (or just degree) of

$\displaystyle \operatorname{deg} f := \sum_{x \in f^{-1}(y)} \operatorname{sign} Df(x) .$

It can be shown that the degree does not depend on the regular value that we pick so that $\operatorname{deg} f$ is well defined.

Note that this degree coincides with the degree as defined for maps of spheres.

Bibliography

1: John W. Milnor. Topology From The Differentiable Viewpoint. The University Press of Virginia, Charlottesville, Virginia, 1969.

1	Manifold
1	Point
1	Linear Transformation
1	Differntiable Function
1	Compact
1	Well Defined
1	Positive
1	Dimension
1	Function
1	Mapping
2	Sphere
2	Determinant
2	Boundary In Topology
2	Connected Space
2	Preservation And Reflection
3	Orientation
5	Sards Theorem
8	Degree Map Of Spheres