|
|
|
|
degree mod 2 of a mapping
|
(Definition)
|
|
|
Suppose that $M$ and $N$ are two 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. If $y \in N$ is
a regular value of $f$ , then we denote by $\# f^{-1}(y)$ the number of points in $M$ that map to $y$ .
Definition 1 Let $y \in N$ be a regular value, then we define the degree mod 2 of $f$ by \begin{equation*} \operatorname{deg}_2 f := \# f^{-1}(y) \pmod{2} . \end{equation*}
It can be shown that the degree mod 2 does not depend on the regular value $y$ that we pick so that $\operatorname{deg}_2 f$ is well defined.
This is similar to the Brouwer degree but does not require oriented manifolds. In fact $\operatorname{deg}_2 f = \operatorname{deg} f \pmod{2}$ .
- 1
- John W. Milnor. Topology From The Differentiable Viewpoint. The University Press of Virginia, Charlottesville, Virginia, 1969.
|
"degree mod 2 of a mapping" is owned by jirka.
|
|
(view preamble | get metadata)
Cross-references: oriented manifolds, Brouwer degree, similar, well defined, map, points, number, regular value, differentiable mapping, connected, compact, boundary, dimension, differentiable manifolds
There is 1 reference to this entry.
This is version 3 of degree mod 2 of a mapping, born on 2004-12-10, modified 2005-03-07.
Object id is 6555, canonical name is DegreeMod2OfAMapping.
Accessed 3702 times total.
Classification:
| AMS MSC: | 57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
