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...

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...

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

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...

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

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...

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

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

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

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...

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

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...

{\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...

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)...

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...

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

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

sphere Homology sphere – Topological manifold whose homology coincides with that of a sphere Homotopy groups of spheres – How spheres of various dimensions...

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

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

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...

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...

term of the Adams spectral sequence for Brown–Peterson cohomology, which is in turn used for calculating the stable homotopy groups of spheres. Chromatic...

a rational homotopy n {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homotopy groups as the n {\displaystyle...

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...

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

computational tool (e.g., for the homotopy groups of spheres). Cobordism theories are represented by Thom spectra MG: given a group G, the Thom spectrum is composed...

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