degree (map of spheres)

Given a non-negative integer $n$ , let $S^{n}$ denote the $n$ -dimensional sphere. Suppose $f\\colon S^{n}\\to S^{n}$ is a continuous map. Applying the $n^{th}$ reduced homology functor $\\widetilde{H}_{n}(\\_)$ , we obtain a homomorphism $f_{*}\\colon\\widetilde{H}_{n}(S^{n})\\to\\widetilde{H}_{n}(S^{n})$ . Since $\\widetilde{H}_{n}(S^{n})\\approx\\mathbbmss{Z}$ , it follows that $f_{*}$ is a homomorphism $\\mathbbmss{Z}\\to\\mathbbmss{Z}$ . Such a map must be multiplication by an integer $d$ . We define the degree of the map $f$ , to be this $d$ .

0.1 Basic Properties

1.
If $f,g\\colon S^{n}\\to S^{n}$ are continuous, then $\\deg(f\\circ g)=\\deg(f)\\cdot\\deg(g)$ .
2.
If $f,g\\colon S^{n}\\to S^{n}$ are homotopic, then $\\deg(f)=\\deg(g)$ .
3.
The degree of the identity map is $+1$ .
4.
The degree of the constant map is $0$ .
5.
The degree of a reflection through an $(n+1)$ -dimensional hyperplane through the origin is $-1$ .
6.
The antipodal map, sending $x$ to $-x$ , has degree $(-1)^{n+1}$ . This follows since the map $f_{i}$ sending $(x_{1},\\ldots,x_{i},\\ldots,x_{n+1})\\mapsto(x_{1},\\ldots,-x_{i},\\ldots,x_{n+1})$ has degree $-1$ by (4), and the compositon $f_{1}\\circ\\cdots\\circ f_{n+1}$ yields the antipodal map.

0.2 Examples

If we identify $S^{1}\\subset\\mathbbmss{C}$ , then the map $f:S^{1}\\to S^{1}$ defined by $f(z)=z^{k}$ has degree $k$ . It is also possible, for any positive integer $n$ , and any integer $k$ , to construct a map $f\\colon S^{n}\\to S^{n}$ of degree $k$ .

Using degree, one can prove several theorems, including the so-called ’hairy ball theorem’, which that there exists a continuous non-zero vector field on $S^{n}$ if and only if $n$ is odd.

单词	DegreemapOfSpheres
释义	degree (map of spheres) Given a non-negative integer $n$ , let $S^{n}$ denote the $n$ -dimensional sphere. Suppose $f\\colon S^{n}\\to S^{n}$ is a continuous map. Applying the $n^{th}$ reduced homology functor $\\widetilde{H}_{n}(\\_)$ , we obtain a homomorphism $f_{}\\colon\\widetilde{H}_{n}(S^{n})\\to\\widetilde{H}_{n}(S^{n})$ . Since $\\widetilde{H}_{n}(S^{n})\\approx\\mathbbmss{Z}$ , it follows that $f_{}$ is a homomorphism $\\mathbbmss{Z}\\to\\mathbbmss{Z}$ . Such a map must be multiplication by an integer $d$ . We define the degree of the map $f$ , to be this $d$ . 0.1 Basic Properties 1. If $f,g\\colon S^{n}\\to S^{n}$ are continuous, then $\\deg(f\\circ g)=\\deg(f)\\cdot\\deg(g)$ . 2. If $f,g\\colon S^{n}\\to S^{n}$ are homotopic, then $\\deg(f)=\\deg(g)$ . 3. The degree of the identity map is $+1$ . 4. The degree of the constant map is $0$ . 5. The degree of a reflection through an $(n+1)$ -dimensional hyperplane through the origin is $-1$ . 6. The antipodal map, sending $x$ to $-x$ , has degree $(-1)^{n+1}$ . This follows since the map $f_{i}$ sending $(x_{1},\\ldots,x_{i},\\ldots,x_{n+1})\\mapsto(x_{1},\\ldots,-x_{i},\\ldots,x_{n+1})$ has degree $-1$ by (4), and the compositon $f_{1}\\circ\\cdots\\circ f_{n+1}$ yields the antipodal map. 0.2 Examples If we identify $S^{1}\\subset\\mathbbmss{C}$ , then the map $f:S^{1}\\to S^{1}$ defined by $f(z)=z^{k}$ has degree $k$ . It is also possible, for any positive integer $n$ , and any integer $k$ , to construct a map $f\\colon S^{n}\\to S^{n}$ of degree $k$ . Using degree, one can prove several theorems, including the so-called ’hairy ball theorem’, which that there exists a continuous non-zero vector field on $S^{n}$ if and only if $n$ is odd.
随便看	equivalence of Hausdorff’s maximum principle, Zorn’s lemma and the well-ordering theorem EquivalenceOfKuratowskisLemmaAndZornsLemma equivalence of Kuratowski’s lemma and Zorn’s lemma Equivalence of path induction and based path induction EquivalenceOfZornsLemmaAndTheAxiomOfChoice equivalence of Zorn’s lemma and the axiom of choice EquivalenceRelation equivalence relation Equivalences Equivalences EquivalentAutomata equivalent automata EquivalentCharacterizationsOfDedekindDomains equivalent characterizations of Dedekind domains EquivalentConditionForBeingAFundamentalSystemOfEntourages equivalent condition for being a fundamental system of entourages EquivalentConditionForTheTranslatesOfAnL2FunctionToFormARieszSequenceAn equivalent condition for the translates of an L_2 function to form a Riesz sequence, an EquivalentConditionsForNormalityOfAFieldExtension equivalent conditions for normality of a field extension EquivalentConditionsForTriangles equivalent conditions for triangles EquivalentConditionsForUniformIntegrability equivalent conditions for uniform integrability EquivalentDefiningConditionsOnANoetherianRing