Math 2.0

Algebraic topology Definition

Degree of a continuous map

Builds on Continuous mapn-sphereSingular homologyTopological manifoldOrientation of a manifoldFundamental class

Definition

Let n≥1n \ge 1 and let f ⁣:Sn→Snf \colon S^n \to S^n be a continuous map of the nn-sphere. The induced homomorphism f∗ ⁣:Hn(Sn;ℤ)→Hn(Sn;ℤ)f_\ast \colon H_n(S^n; \Z) \to H_n(S^n; \Z) on singular homology is an endomorphism of a group isomorphic to ℤ\Z. So f∗(a)=d af_\ast(a) = d \, a for a unique integer dd and all aa. This integer is the degree of ff, written deg⁡f\deg f.

More generally, let MM and NN be closed connected oriented topological nn-manifolds with fundamental classes [M][M] and [N][N], and let f ⁣:M→Nf \colon M \to N be continuous. The degree of ff is the integer deg⁡f\deg f with f∗[M]=(deg⁡f)[N]f_\ast [M] = (\deg f) [N] in Hn(N;ℤ)H_n(N; \Z).

Remarks

Homotopic maps have the same degree, and deg⁡(g∘f)=deg⁡g⋅deg⁡f\deg (g \circ f) = \deg g \cdot \deg f.

Example: on S1⊂ℂS^1 \subset \C, the map f(z)=zkf(z) = z^k has degree kk. For a reflection rr of SnS^n in a hyperplane, deg⁡r=−1\deg r = -1; the antipodal map x↦−xx \mapsto -x is a composite of n+1n + 1 reflections, so its degree is (−1)n+1(-1)^{n+1}.

Non-example: a map Sm→SnS^m \to S^n with m≠nm \neq n has no degree in this sense, since the source and target manifolds must have the same dimension.

If ff is smooth and yy is a regular value, then deg⁡f=∑x∈f−1(y)sgn⁡det⁡Df(x)\deg f = \sum_{x \in f^{-1}(y)} \sgn \det Df(x), where the sign is computed in oriented charts. For n=1n = 1, the degree of a map S1→S1S^1 \to S^1 equals its winding number about 00.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Continuous map, concept, no evidence levelContinuous mapconceptFundamental class, concept, no evidence levelFundamental classconceptOrientation of a manifold, concept, no evidence levelOrientation of a mani…conceptSingular homology, concept, no evidence levelSingular homologyconceptTopological manifold, concept, no evidence levelTopological manifoldconceptn-sphere, object, no evidence leveln-sphereobjectDegree of a continuous map, definition, no evidence levelDegree of a continuou…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Continuous map concept
  2. Fundamental class concept
  3. Orientation of a manifold concept
  4. Singular homology concept
  5. Topological manifold concept
  6. n-sphere object
  7. Degree of a continuous map definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping
About this page
TagDX3DJY91QH
URIm2://DX3DJY91QH
Kinddefinition
WritingW0 stub
MSC55M25
DefinesDegree of a continuous map
UsesContinuous map, n-sphere, Singular homology, Topological manifold, Orientation of a manifold, Fundamental class
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.