field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological...

mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted...

exotic spheres to singularities of complex manifolds. Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I" (PDF). Annals of Mathematics...

for n {\displaystyle n} sufficiently large. In particular, the homotopy groups of spheres π n + k ( S n ) {\displaystyle \pi _{n+k}(S^{n})} stabilize for...

topology as a Professor of Mathematics at the University of California, Los Angeles, known for computations of homotopy groups of spheres. Xu earned both his...

a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same...

applications of the Steenrod algebra were calculations by Jean-Pierre Serre of some homotopy groups of spheres, using the compatibility of transgressive...

{\displaystyle p} -local sphere spectrum. This is a key observation for studying stable homotopy groups of spheres using chromatic homotopy theory. Elliptic cohomology...

group. Adams spectral sequence Eilenberg–MacLane space CW complex Obstruction theory Stable homotopy theory Homotopy groups of spheres Higher group Hopf–Whitney...

research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated...

contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection...

homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda, who defined them and used them to compute homotopy groups...

topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored....

perfect field is isomorphic to the motivic stable homotopy group of spheres π0,0(S0,0) (see "A¹ homotopy theory"). Two fields are said to be Witt equivalent...

J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George W. Whitehead (1942)...

Associated bundle Fibration Hopf bundle Classifying space Cofibration Homotopy groups of spheres Plus construction Whitehead theorem Weak equivalence Hurewicz...

2-fold cover). Generally, the homotopy groups πk(O) of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard...

is known for constructing one of the first known infinite families of elements in the stable homotopy groups of spheres by showing that the classes h...

{\displaystyle \pi _{1}(X)} . Higher homotopy groups are sometimes difficult to compute. For instance, the homotopy groups of spheres are poorly understood and are...

homeomorphisms. The k-th homotopy group of a sphere spectrum is the k-th stable homotopy group of spheres. The localization of the sphere spectrum at a prime...

stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres, and...

spectrum are its homotopy groups. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the suspension maps...

{\displaystyle p} -torsion of the homotopy groups of the sphere spectrum, i.e. the stable homotopy groups of the spheres. Also, because for any CW-complex...

(unstable) homotopy groups of spheres. In a 1957 paper he showed the first non-existence result for the Hopf invariant 1 problem. This period of his work...

mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold that is a homotopy sphere is a sphere. More precisely...

spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space...

first to systematically calculate the homotopy groups of spheres. He is also central to the study of Stable homotopy theory, in particular making concrete...

The sequence of ideals I0, I1, I2, etc., is an ascending chain that does not terminate. The ring of stable homotopy groups of spheres is not Noetherian...

the stable homotopy groups of spheres, Academic Press 1986, 2nd edition, AMS 2003, online:[1] Nilpotency and periodicity in stable homotopy theory, Princeton...

Goro (1973), "The nilpotency of elements of the stable homotopy groups of spheres", Journal of the Mathematical Society of Japan, 25 (4): 707–732, doi:10...