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...

map (see homotopy groups of spheres for this and more complicated examples of homotopy groups). Hence the torus is not homeomorphic to the sphere. In the...

is open whether non-trivial smooth homotopy spheres exist in dimension 4. Homology sphere Homotopy groups of spheres Poincaré conjecture A., Kosinski,...

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...

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...

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

homotopy groups using an inverse system of topological spaces whose homotopy type at degree k {\displaystyle k} agrees with the truncated homotopy type of the...

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...

homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with...

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)...

Associate Professor of Mathematics at the University of California, San Diego, known for computations of homotopy groups of spheres. Xu earned both his...

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...

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

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...

sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime p. It is described in more detail in Ravenel...

{\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...

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

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...

(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...

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

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

which one can use in order to deduce information about the higher homotopy groups of spheres. Consider the following fibration which is an isomorphism on π...

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

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic...

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...

higher-homotopy groups (k ≥ 4) are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres. The...

bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying...