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Brouwer degree
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(Definition)
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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 \begin{equation*} \operatorname{deg} f := \sum_{x \in f^{-1}(y)} \operatorname{sign} Df(x) . \end{equation*}
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.
- 1
- John W. Milnor. Topology From The Differentiable Viewpoint. The University Press of Virginia, Charlottesville, Virginia, 1969.
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"Brouwer degree" is owned by jirka.
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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 18 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 7690 times total.
Classification:
| AMS MSC: | 57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings) |
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Pending Errata and Addenda
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