Homotopy Groups of Spheres - Applications

Applications

  • The winding number (corresponding to an integer of π1(S1) = Z) can be used to prove the fundamental theorem of algebra, which states that every non-constant complex polynomial has a zero.
  • The fact that πn−1(Sn−1) = Z implies the Brouwer fixed point theorem that every continuous map from the n-dimensional ball to itself has a fixed point.
  • The stable homotopy groups of spheres are important in singularity theory, which studies the structure of singular points of smooth maps or algebraic varieties. Such singularities arise as critical points of smooth maps from Rm to Rn. The geometry near a critical point of such a map can be described by an element of πm−1(Sn−1), by considering the way in which a small m − 1 sphere around the critical point maps into a topological n − 1 sphere around the critical value.
  • The fact that the third stable homotopy group of spheres is cyclic of order 24, first proved by Vladimir Rokhlin, implies Rokhlin's theorem that the signature of a compact smooth spin 4-manifold is divisible by 16 (Scorpan 2005).
  • Stable homotopy groups of spheres are used to describe the group Θn of h-cobordism classes of oriented homotopy n-spheres (for n ≠ 4, this is the group of smooth structures on n-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by exotic spheres). More precisely, there is an injective map
    where bPn+1 is the cyclic subgroup represented by homotopy spheres that bound a parallelizable manifold, πnS is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. This is an isomorphism unless n is of the form 2k−2, in which case the image has index 1 or 2 (Kervaire & Milnor 1963).
  • The groups Θn above, and therefore the stable homotopy groups of spheres, are used in the classification of possible smooth structures on a topological or piecewise linear manifold (Scorpan 2005).
  • The Kervaire invariant problem, about the existence of manifolds of Kervaire invariant 1 in dimensions 2k − 2 can be reduced to a question about stable homotopy groups of spheres. For example, knowledge of stable homotopy groups of degree up to 48 has been used to settle the Kervaire invariant problem in dimension 26 − 2 = 62 (Barratt, Jones & Mahowald 1984). (This was the smallest value of k for which the question was open at the time.)
  • The Barratt–Priddy theorem says that the stable homotopy groups of the spheres can be expressed in terms of the plus construction applied to the classifying space of the symmetric group, leading to an identification of K-theory of the field with one element with stable homotopy groups (Deitmar 2006).

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