Difference between revisions of "Degree of a mapping"
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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 | + | 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 | ||
− | + | $$ | |
+ | \mathop{\rm deg} f = \sum _ {x \in f ^ {-} 1 ( y) } \mathop{\rm sign} J _ {x} , | ||
+ | $$ | ||
− | + | where $ \mathop{\rm sign} J _ {x} $ | |
+ | is the sign of the Jacobian of $ f $ | ||
+ | at a point $ x $( | ||
+ | the Browder degree). | ||
− | and | + | 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 | ||
− | + | $$ | |
+ | S _ \epsilon ^ {n} = \partial B _ \epsilon ^ {n} ,\ \ | ||
+ | B _ \epsilon ^ {n} \cap f ^ { - 1 } ( 0) = \ | ||
+ | x \in \mathop{\rm Int} B _ \epsilon ^ {n} , | ||
+ | $$ | ||
− | holds, where | + | 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 ![]() |
[5] | A.H. Wallace, "Differential topology. First Steps" , Benjamin (1968) MR0436148 MR0224103 Zbl 0164.23805 |
Degree of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degree_of_a_mapping&oldid=24409