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...
82 KB (7,972 words) - 00:30, 24 August 2024
mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted...
20 KB (3,417 words) - 21:07, 23 November 2023
is open whether non-trivial smooth homotopy spheres exist in dimension 4. Homology sphere Homotopy groups of spheres Poincaré conjecture A., Kosinski,...
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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...
4 KB (669 words) - 23:26, 17 August 2023
exotic spheres to singularities of complex manifolds. Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I" (PDF). Annals of Mathematics...
29 KB (3,844 words) - 19:54, 8 August 2024
Postnikov system (category Homotopy theory)
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...
19 KB (3,645 words) - 16:45, 22 May 2024
{\displaystyle p} -local sphere spectrum. This is a key observation for studying stable homotopy groups of spheres using chromatic homotopy theory. Elliptic cohomology...
3 KB (405 words) - 21:48, 9 January 2024
Bott periodicity theorem (category Theorems in homotopy theory)
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...
13 KB (1,836 words) - 15:27, 3 May 2024
Toda bracket (category Homotopy theory)
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...
7 KB (1,141 words) - 19:48, 5 January 2024
Steenrod algebra (section Connection to the Adams spectral sequence and the homotopy groups of spheres)
applications of the Steenrod algebra were calculations by Jean-Pierre Serre of some homotopy groups of spheres, using the compatibility of transgressive...
30 KB (5,578 words) - 11:34, 13 January 2024
Zhouli Xu (category Fellows of the American Mathematical Society)
Associate Professor of Mathematics at the University of California, San Diego, known for computations of homotopy groups of spheres. Xu earned both his...
7 KB (595 words) - 14:51, 2 April 2024
Eilenberg–MacLane space (category Homotopy theory)
contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection...
20 KB (3,349 words) - 21:33, 22 June 2024
J-homomorphism (redirect from Stable fibre homotopy type)
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)...
8 KB (918 words) - 21:06, 22 August 2023
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...
3 KB (540 words) - 20:29, 16 February 2023
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...
56 KB (7,844 words) - 08:45, 30 June 2024
homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with...
23 KB (3,271 words) - 09:41, 30 July 2024
Adams spectral sequence (category Homotopy theory)
{\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...
19 KB (3,283 words) - 13:36, 28 August 2023
Generalized Poincaré conjecture (category Homotopy theory)
mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely...
10 KB (1,285 words) - 04:55, 26 May 2024
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...
21 KB (3,169 words) - 09:30, 25 April 2024
spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space...
53 KB (8,076 words) - 07:10, 6 August 2024
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...
12 KB (2,641 words) - 13:35, 29 February 2024
Mark Mahowald (category University of Minnesota alumni)
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...
6 KB (618 words) - 09:58, 7 April 2024
topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored....
25 KB (3,945 words) - 18:51, 26 January 2024
Associated bundle Fibration Hopf bundle Classifying space Cofibration Homotopy groups of spheres Plus construction Whitehead theorem Weak equivalence Hurewicz...
4 KB (311 words) - 12:17, 30 October 2023
Hopf fibration (category Homotopy theory)
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...
36 KB (4,799 words) - 18:21, 21 August 2024
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...
8 KB (1,174 words) - 14:02, 27 August 2024
sphere Homology sphere – Topological manifold whose homology coincides with that of a sphere Homotopy groups of spheres – How spheres of various dimensions...
37 KB (7,130 words) - 10:34, 22 August 2024
Homology (mathematics) (redirect from Homology groups)
{\displaystyle \pi _{1}(X)} . Higher homotopy groups are sometimes difficult to compute. For instance, the homotopy groups of spheres are poorly understood and are...
54 KB (8,218 words) - 14:50, 22 August 2024
Frederick R. Cohen (category Fellows of the American Mathematical Society)
class groups. This was followed by a series of influential papers on unstable homotopy groups of spheres with John Moore and Joseph Neisendorfer. Late...
6 KB (488 words) - 20:30, 19 August 2024
Douglas Ravenel (category Massachusetts Institute of Technology School of Science faculty)
the stable homotopy groups of spheres, Academic Press 1986, 2nd edition, AMS 2003, online:[1] Nilpotency and periodicity in stable homotopy theory, Princeton...
9 KB (909 words) - 21:38, 9 April 2024