the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map η...

this time Hopf discovered the Hopf invariant of maps S 3 → S 2 {\displaystyle S^{3}\to S^{2}} and proved that the Hopf fibration has invariant 1. In the...

In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)...

M^{4m+2}} (for m ≠ 0 , 1 , 3 {\displaystyle m\neq 0,1,3} ) and the mod 2 Hopf invariant of maps S 4 m + 2 + k → S 2 m + 1 + k {\displaystyle S^{4m+2+k}\to S^{2m+1+k}}...

It is clear how to define a homotopy from [f][g] to [g][f]. Adams' Hopf invariant one theorem, named after Frank Adams, states that S0, S1, S3, S7 are...

classical case. He used this spectral sequence to attack the celebrated Hopf invariant one problem, which he completely solved in a 1960 paper by making a...

differential point of view Hopf invariant – Homotopy invariant of maps between n-spheres Kissing number – Geometric concept Writhe – Invariant of a knot diagram...

Topological K-theory has been applied in John Frank Adams’ proof of the “Hopf invariant one” problem via Adams operations. Adams also proved an upper bound...

1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8. It was...

The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of...

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly...

century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are...

Gromov–Witten invariant Arf invariant Hopf invariant Invariant theory Framed knot Chern–Simons theory Algebraic geometry Seifert surface Geometric invariant theory...

S^{15}\hookrightarrow S^{31}\rightarrow S^{16},} the first non-trivial case of the Hopf invariant one problem, because such a fibration would imply that the failed relation...

H_{n}(X).} Fibration Hopf fibration Hopf invariant Knot theory Homotopy class Homotopy groups of spheres Topological invariant Homotopy group with coefficients...

structure of a Hopf algebra is when considering all H-modules as a category. The additional structure is also used to define invariant elements of an...

They were introduced by J. Frank Adams (1960) in his solution to the Hopf invariant problem. Similarly one can define tertiary cohomology operations from...

In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994)...

appropriate Adem relations, was the solution by J. Frank Adams of the Hopf invariant one problem. One application of the mod 2 Steenrod algebra that is fairly...

In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (X, μ) with respect to an invertible...

Princeton in 1955. His other collaborators included; J. Frank Adams (Hopf invariant problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins...

In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots...

Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links. Such invariants of framed links also give rise to invariants of 3-manifolds...

pair of circular Hopf fibers which are not merely Clifford parallel and interlinked, but also completely orthogonal. The invariant great circles of the...

bi-invariant (that is, simultaneously left- and right-invariant). All left-invariant metrics have constant scalar curvature. Left- and bi-invariant metrics...

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander...

(flip) bifurcation Hopf bifurcation Neimark–Sacker (secondary Hopf) bifurcation Global bifurcations occur when 'larger' invariant sets, such as periodic...

defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935). Whitehead's original homomorphism is defined geometrically, and gives...

{\displaystyle \eta \colon S^{3}\to S^{2}} is the Hopf map. This can be shown by observing that the Hopf invariant defines an isomorphism π 3 ( S 2 ) ≅ Z {\displaystyle...

is the dual vector space. The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopf algebras are known as quantum...