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...

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...

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...

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

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

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...

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

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...

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)...

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

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...

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...

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

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

class groups. This was followed by a series of influential papers on unstable homotopy groups of spheres with John Moore and Joseph Neisendorfer. Late...

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

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

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...

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

(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...

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

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...

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

is the mathematical result of which Archimedes was reportedly most proud: Given a sphere and a circumscribed cylinder of the same height and diameter...

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

the first of these two papers, the authors explore the stable homotopy groups of spheres by analyzing the E 2 {\displaystyle E^{2}} -term of the Adams–Novikov...

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

original on 2016-11-08.[self-published source?] "A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF). p. 16. Archived (PDF) from the...