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''degree of a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309001.png" /> between connected compact manifolds of identical dimension''
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An integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309002.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309003.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309004.png" /> are the fundamental classes (cf. [[Fundamental class|Fundamental class]]) of the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309006.png" /> over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309007.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309008.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d0309009.png" /> is the induced mapping. In the case of non-orientable manifolds, the degree of the mapping is uniquely defined modulo 2. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090010.png" /> is a differentiable mapping between closed differentiable manifolds, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090011.png" /> modulo 2 coincides with the number of inverse images of a regular value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090013.png" />. In the case of oriented manifolds
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090014.png" /></td> </tr></table>
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''degree of a continuous mapping  $  f: ( M, \partial  M) \rightarrow ( N, \partial  N) $
 +
between connected compact manifolds of identical dimension''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090015.png" /> is the sign of the Jacobian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090016.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090017.png" /> (the Browder degree).
+
An integer  $  \mathop{\rm deg}  f $
 +
such that  $  f _  \star  ( \mu _ {M} ) =  \mathop{\rm deg}  f \cdot \mu _ {N} $,
 +
where $  \mu _ {M} , \mu _ {N} $
 +
are the fundamental classes (cf. [[Fundamental class|Fundamental class]]) of the manifolds  $  M $
 +
and  $  N $
 +
over the ring  $  \mathbf Z $
 +
or  $  \mathbf Z _ {2} $,
 +
and  $  f _  \star  $
 +
is the induced mapping. In the case of non-orientable manifolds, the degree of the mapping is uniquely defined modulo 2. If  $  f:  M \rightarrow N $
 +
is a differentiable mapping between closed differentiable manifolds, then  $  \mathop{\rm deg}  f $
 +
modulo 2 coincides with the number of inverse images of a regular value  $  y $
 +
of  $  f $.  
 +
In the case of oriented manifolds
  
For a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090018.png" /> and an isolated point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090019.png" /> in the inverse image of zero, the concept of the local degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090020.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090021.png" /> is defined: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090023.png" /> is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090024.png" /> onto a small sphere
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$$
 +
\mathop{\rm deg}  f  = \sum _ {x \in f  ^ {-} 1 ( y) }  \mathop{\rm sign}  J _ {x} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090025.png" /></td> </tr></table>
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where  $  \mathop{\rm sign}  J _ {x} $
 +
is the sign of the Jacobian of  $  f $
 +
at a point  $  x $(
 +
the Browder degree).
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090026.png" /> is the projection from zero onto the unit sphere. In the case of a differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090027.png" />, the formula
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For a continuous mapping  $  f:  ( \mathbf R  ^ {n} , 0) \rightarrow ( \mathbf R  ^ {n} , 0) $
 +
and an isolated point  $  x $
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in the inverse image of zero, the concept of the local degree  $  \mathop{\rm deg} _ {x}  f $
 +
at the point  $  x $
 +
is defined:  $  \mathop{\rm deg} _ {x}  f = \mathop{\rm deg}  \pi \circ h $,  
 +
where  $  h $
 +
is the restriction of  $  f $
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onto a small sphere
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090028.png" /></td> </tr></table>
+
$$
 +
S _  \epsilon  ^ {n}  = \partial  B _  \epsilon  ^ {n} ,\ \
 +
B _  \epsilon  ^ {n} \cap f ^ { - 1 } ( 0= \
 +
x  \in  \mathop{\rm Int}  B _  \epsilon  ^ {n} ,
 +
$$
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090029.png" /> is the ring of germs (cf. [[Germ|Germ]]) of smooth functions at zero, factorized by the ideal generated by the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090030.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090031.png" /> is the maximal ideal of the quotient ring relative to the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090032.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090033.png" /> be the class of the Jacobian of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090034.png" />. Then for a linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090036.png" /> the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090037.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090038.png" /> is a symmetric bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090039.png" />.
+
and  $  \pi $
 +
is the projection from zero onto the unit sphere. In the case of a differentiable  $  f $,
 +
the formula
 +
 
 +
$$
 +
|  \mathop{\rm deg} _ {x}  f |  =  \mathop{\rm dim}  Q( f  ) - 2  \mathop{\rm dim}  I
 +
$$
 +
 
 +
holds, where $  Q( f  ) $
 +
is the ring of germs (cf. [[Germ|Germ]]) of smooth functions at zero, factorized by the ideal generated by the components of $  f $,  
 +
and $  I $
 +
is the maximal ideal of the quotient ring relative to the property $  I  ^ {2} = 0 $.  
 +
Let $  J _ {0} \in Q( f  ) $
 +
be the class of the Jacobian of the mapping $  f $.  
 +
Then for a linear form $  \phi : Q( f  ) \rightarrow \mathbf R $
 +
such that $  \phi ( J _ {0} ) > 0 $
 +
the formula $  \mathop{\rm deg} _ {x}  f = \mathop{\rm Index}  \langle  , \rangle _  \phi  $
 +
holds, where $  \langle  p, q\rangle _  \phi  = \phi ( p \cdot q) $
 +
is a symmetric bilinear form on $  Q( f  ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980) {{MR|0606196}} {{ZBL|0434.55001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , '''1''' , Birkhäuser (1985) (Translated from Russian) {{MR|777682}} {{ZBL|0554.58001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Eisenbud, H. Levine, "An algebraic formula for the degree of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090040.png" /> map germ" ''Ann. of Math.'' , '''106''' : 1 (1977) pp. 19–38 {{MR|467800}} {{ZBL|0398.57020}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) {{MR|0436148}} {{MR|0224103}} {{ZBL|0164.23805}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980) {{MR|0606196}} {{ZBL|0434.55001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , '''1''' , Birkhäuser (1985) (Translated from Russian) {{MR|777682}} {{ZBL|0554.58001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Eisenbud, H. Levine, "An algebraic formula for the degree of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030900/d03090040.png" /> map germ" ''Ann. of Math.'' , '''106''' : 1 (1977) pp. 19–38 {{MR|467800}} {{ZBL|0398.57020}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) {{MR|0436148}} {{MR|0224103}} {{ZBL|0164.23805}} </TD></TR></table>

Revision as of 17:32, 5 June 2020


degree of a continuous mapping $ f: ( M, \partial M) \rightarrow ( N, \partial N) $ between connected compact manifolds of identical dimension

An integer $ \mathop{\rm deg} f $ such that $ f _ \star ( \mu _ {M} ) = \mathop{\rm deg} f \cdot \mu _ {N} $, where $ \mu _ {M} , \mu _ {N} $ are the fundamental classes (cf. Fundamental class) of the manifolds $ M $ and $ N $ over the ring $ \mathbf Z $ or $ \mathbf Z _ {2} $, and $ f _ \star $ is the induced mapping. In the case of non-orientable manifolds, the degree of the mapping is uniquely defined modulo 2. If $ f: M \rightarrow N $ is a differentiable mapping between closed differentiable manifolds, then $ \mathop{\rm deg} f $ modulo 2 coincides with the number of inverse images of a regular value $ y $ of $ f $. In the case of oriented manifolds

$$ \mathop{\rm deg} f = \sum _ {x \in f ^ {-} 1 ( y) } \mathop{\rm sign} J _ {x} , $$

where $ \mathop{\rm sign} J _ {x} $ is the sign of the Jacobian of $ f $ at a point $ x $( the Browder degree).

For a continuous mapping $ f: ( \mathbf R ^ {n} , 0) \rightarrow ( \mathbf R ^ {n} , 0) $ and an isolated point $ x $ in the inverse image of zero, the concept of the local degree $ \mathop{\rm deg} _ {x} f $ at the point $ x $ is defined: $ \mathop{\rm deg} _ {x} f = \mathop{\rm deg} \pi \circ h $, where $ h $ is the restriction of $ f $ onto a small sphere

$$ S _ \epsilon ^ {n} = \partial B _ \epsilon ^ {n} ,\ \ B _ \epsilon ^ {n} \cap f ^ { - 1 } ( 0) = \ x \in \mathop{\rm Int} B _ \epsilon ^ {n} , $$

and $ \pi $ is the projection from zero onto the unit sphere. In the case of a differentiable $ f $, the formula

$$ | \mathop{\rm deg} _ {x} f | = \mathop{\rm dim} Q( f ) - 2 \mathop{\rm dim} I $$

holds, where $ Q( f ) $ is the ring of germs (cf. Germ) of smooth functions at zero, factorized by the ideal generated by the components of $ f $, and $ I $ is the maximal ideal of the quotient ring relative to the property $ I ^ {2} = 0 $. Let $ J _ {0} \in Q( f ) $ be the class of the Jacobian of the mapping $ f $. Then for a linear form $ \phi : Q( f ) \rightarrow \mathbf R $ such that $ \phi ( J _ {0} ) > 0 $ the formula $ \mathop{\rm deg} _ {x} f = \mathop{\rm Index} \langle , \rangle _ \phi $ holds, where $ \langle p, q\rangle _ \phi = \phi ( p \cdot q) $ is a symmetric bilinear form on $ Q( f ) $.

References

[1] A. Dold, "Lectures on algebraic topology" , Springer (1980) MR0606196 Zbl 0434.55001
[2] J.W. Milnor, "Toplogy from the differentiable viewpoint" , Univ. Virginia Press (1965)
[3] V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) (Translated from Russian) MR777682 Zbl 0554.58001
[4] D. Eisenbud, H. Levine, "An algebraic formula for the degree of a map germ" Ann. of Math. , 106 : 1 (1977) pp. 19–38 MR467800 Zbl 0398.57020
[5] A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) MR0436148 MR0224103 Zbl 0164.23805
How to Cite This Entry:
Degree of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degree_of_a_mapping&oldid=24409
This article was adapted from an original article by A.V. Khokhlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article