Hopf theorem

<h2 id="formal-statement">Formal statement</h2>
Let M be an n-dimensional <a href="/facts/Closed_manifold/jMG6G6Bl">compact</a> <a href="/facts/Connected_space/5CUTxfoA">connected</a> <a href="/facts/Manifold/rXPERJtu">oriented manifold</a> and 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S^{n}}
 
 the <a href="/facts/N-sphere/bFu2S7E2">n-sphere</a> and 
 
 
 
 f
 ,
 g
 :
 M
 →
 
 S
 
 n
 
 
 
 
 {\displaystyle f,g\colon M\to S^{n}}
 
 be continuous. Then 
 
 
 
 deg
 ⁡
 (
 f
 )
 =
 deg
 ⁡
 (
 g
 )
 
 
 {\displaystyle \deg(f)=\deg(g)}
 
 if and only if f and g are <a href="/facts/Homotopic/1nWrPxdj">homotopic</a>.

<ul><li><a href="/facts/John_Milnor/LTf24paN">Milnor, John W.</a> (1997). Topology from the Differentiable Viewpoint. <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-04833-8.</li></ul>

Hopf theorem open-in-new

Hopf theorem