nLab degree of a continuous function




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory



Given a continuous function between two connected closed oriented topological manifolds of the same dimension, its degree is a measure for how often the function “wraps its domain around its codomain”.


For XX is a connected closed oriented manifold of dimension nn, its top homology group H n(X)=H n(X;)H_n(X) = H_n(X; \mathbb{Z}) is isomorphic to \mathbb{Z}, where the generator 11 \in \mathbb{Z} is identified with the orientation class [ω X][\omega_X] of XX, the fundamental class of XX.


Given a continuous map f:XYf \colon X \to Y between two such manifolds, the homomorphism f *=H n(f):H n(X)H n(Y)f_\ast = H_n(f) \colon H_n(X) \to H_n(Y) is therefore specified by the integer nn such that f *[ω X]=n[ω Y]f_\ast [\omega_X] = n [\omega_Y]. This integer is called the degree of ff.

Computing the degree

We suppose throughout that XX and YY are connected closed oriented manifolds of the same dimension nn. The degree of a continuous function g:XYg \colon X \to Y is frequently computed according to the following considerations:

  • The space of continuous functions g:XYg \colon X \to Y has a dense subspace consisting of smooth functions f:XYf \colon X \to Y, and in particular every continuous function gg is homotopic to a smooth function ff. It therefore suffices to compute the degree of ff.

  • By Sard's theorem, the set of singular values? of a smooth function ff has measure zero (using for example the orientation on YY to define a volume form and hence a measure). Accordingly, we may choose a regular value yYy \in Y.

  • The inverse image f 1(y)f^{-1}(y) is a compact 00-dimensional manifold, hence consists of finitely many (possibly zero) points x 1,,x rXx_1, \ldots, x_r \in X. Since these are regular points, ff restricts to a diffeomorphism

    f i:U iVf_i \colon U_i \to V

    where U iU_i is a small neighborhood of x ix_i and VV is a small neighborhood of yy. The diffeomorphism f if_i either preserves or reverses the orientation of U iU_i, i.e., the sign of the determinant as a mapping between differential n-forms

    Ω n(U i)Ω n(V)\Omega^n(U_i) \to \Omega^n(V)

    is either +1+1 or 1-1.

  • By a straightforward application of the excision axiom in homology, it follows that the degree of ff is the sum of these signs:

    deg(f)= i=1 rsign(Ω n(f i))\deg(f) = \sum_{i=1}^r sign(\Omega^n(f_i))

    and this quantity is independent of the choice of regular value yy.


Hopf degree theorem


(Hopf degree theorem)

Let nn \in \mathbb{N} be a natural number and XMfdX \in Mfd be a connected orientable closed manifold of dimension nn. Then the nnth cohomotopy classes [XcS n]π n(X)\left[X \overset{c}{\to} S^n\right] \in \pi^n(X) of XX are in bijection to the degree deg(c)deg(c) \in \mathbb{Z} of the representing functions, hence the canonical function

π n(X)S nK(,n)H n(X,) \pi^n(X) \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow} H^n(X,\mathbb{Z}) \;\simeq\; \mathbb{Z}

from nnth cohomotopy to nnth integral cohomology is a bijection.

(e.g. Kosinski 93, IX (5.8), Kobin 16, 7.5)

Poincaré–Hopf theorem

See at Poincaré–Hopf theorem.

Generalization to the Adams d-invariant

The Hopf degree of a map is a special case of its Adams d-invariant; see there for more.



Texbook accounts:

  • B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, section 13 of: Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

See also:

Last revised on July 19, 2021 at 15:30:36. See the history of this page for a list of all contributions to it.