Degree of a mapping

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 ) $.

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