Vector fields on spheres

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the discussion of vector fields on spheres was a classical problem of <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, beginning with the <a href="/facts/Hairy_ball_theorem/pSX6zDBz">hairy ball theorem</a>, and early work on the classification of <a href="/facts/Division_algebra/xp5Bepx3">division algebras</a>.
</p><p>Specifically, the question is how many <a href="/facts/Linear_independence/8JrHfIIa">linearly independent</a> smooth nowhere-zero vector fields can be constructed on a <a href="/facts/Hypersphere/bFu2S7E2">sphere</a> in 
  
    
      
        n
      
    
    {\displaystyle n}
  
-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>. A definitive answer was provided in 1962 by <a href="/facts/Frank_Adams/fxgpN9Xk">Frank Adams</a>. It was already known, by direct construction using <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebras</a>, that there were at least 
  
    
      
        ρ
        (
        n
        )
        −
        1
      
    
    {\displaystyle \rho (n)-1}
  
 such fields (see definition below). Adams applied <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a> and <a href="/facts/Topological_K-theory/dNCXyK71">topological K-theory</a> to prove that no more independent vector fields could be found. Hence 
  
    
      
        ρ
        (
        n
        )
        −
        1
      
    
    {\displaystyle \rho (n)-1}
  
 is the exact number of pointwise linearly independent vector fields that exist on an (
  
    
      
        n
        −
        1
      
    
    {\displaystyle n-1}
  
)-dimensional sphere.
</p>

Vector fields on spheres open-in-new

Vector fields on spheres