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Brouwer degree (Definition)

Suppose that $ M$ and $ N$ are two oriented differentiable manifolds of dimension $ n$ (without boundary) with $ M$ compact and $ N$ connected and suppose that $ f \colon M \to N$ is a differentiable mapping. Let $ Df(x)$ 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 $ Df(x)$. That is the sign is positive if $ f$ preserves orientation and negative if $ f$ reverses orientation.

Definition 1   Let $ y \in N$ be a regular value, then we define the Brower degree (or just degree) of $ f$ by
$\displaystyle \operatorname{deg} f := \sum_{x \in f^{-1}(y)} \operatorname{sign} Df(x) .$    

It can be shown that the degree does not depend on the regular value $ y$ 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.



"Brouwer degree" is owned by jirka.
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See Also: degree mod 2 of a mapping

Other names:  degree
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Cross-references: spheres, well defined, regular value, negative, orientation, preserves, positive, determinant, linear mapping, point, mapping, differentiable mapping, connected, compact, boundary, dimension, differentiable manifolds, oriented
There are 4 references to this entry.

This is version 4 of Brouwer degree, born on 2004-12-10, modified 2005-03-05.
Object id is 6554, canonical name is BrouwerDegree.
Accessed 5647 times total.

Classification:
AMS MSC: 57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings)
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