mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted...
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In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other...
82 KB (7,972 words) - 00:30, 24 August 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
algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space...
53 KB (8,076 words) - 07:10, 6 August 2024
group Spin(2) is the unique connected 2-fold cover). Generally, the homotopy groups πk(O) of the real orthogonal group are related to homotopy groups...
56 KB (7,844 words) - 08:45, 30 June 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
(killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group...
27 KB (4,183 words) - 01:55, 27 July 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...
19 KB (3,645 words) - 16:45, 22 May 2024
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume...
3 KB (498 words) - 20:54, 13 March 2024
Homology (mathematics) (redirect from Homology group)
homology group. The nth homotopy group π n ( X ) {\displaystyle \pi _{n}(X)} of a topological space X {\displaystyle X} is the group of homotopy classes...
54 KB (8,218 words) - 14:50, 22 August 2024
Complex projective space (section Homotopy groups)
Moreover, by the long exact homotopy sequence, the second homotopy group is π2(CPn) ≅ Z, and all the higher homotopy groups agree with those of S2n+1:...
26 KB (3,915 words) - 23:24, 10 May 2024
Eilenberg–MacLane space (category Homotopy theory)
Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer. A connected topological space X is...
20 KB (3,349 words) - 21:33, 22 June 2024
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain...
4 KB (669 words) - 23:26, 17 August 2023
Algebraic topology (section Homotopy groups)
first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information...
19 KB (2,081 words) - 18:42, 13 April 2024
Whitehead theorem (category Theorems in homotopy theory)
between CW complexes X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence. This result was proved by J. H. C. Whitehead...
4 KB (605 words) - 08:26, 24 May 2022
Homotopical connectivity (category Homotopy theory)
connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial. Homotopical...
19 KB (3,209 words) - 01:06, 16 October 2023
{\displaystyle i\geq 2} , the i-th homotopy group with coefficients in an abelian group G of a based space X is the pointed set of homotopy classes of based maps from...
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{\displaystyle p\colon E\to B} satisfies the homotopy lifting property for a space X {\displaystyle X} if: for every homotopy h : X × [ 0 , 1 ] → B {\displaystyle...
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Spectrum (topology) (redirect from Stable homotopy category)
invariants of spectra are the homotopy groups of the spectrum. These groups mirror the definition of the stable homotopy groups of spaces since the structure...
21 KB (3,451 words) - 18:25, 26 March 2024
group π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} of a pointed topological space ( X , x ) {\displaystyle (X,x)} is defined as the group of homotopy classes...
11 KB (1,679 words) - 16:57, 1 August 2024
/2\mathbb {Z} .} But now since we killed off the lower homotopy groups of X (i.e., the groups in degrees less than 4) by using the iterated fibration...
12 KB (2,641 words) - 13:35, 29 February 2024
{O} (n)} It is obtained by killing the π 3 {\displaystyle \pi _{3}} homotopy group for Spin ( n ) {\displaystyle \operatorname {Spin} (n)} , in the same...
8 KB (1,136 words) - 11:22, 19 June 2023
Freudenthal suspension theorem (category Theorems in homotopy theory)
field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately...
4 KB (737 words) - 18:16, 27 July 2020
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic"...
3 KB (405 words) - 21:48, 9 January 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
Cohomology (redirect from Cohomology group)
the possible maps from X to Y. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be computable in practice for spaces...
43 KB (6,691 words) - 21:02, 23 March 2024
The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological...
35 KB (4,041 words) - 03:13, 28 July 2024
Real projective space (section Homotopy groups)
quotient bundle. The higher homotopy groups of RPn are exactly the higher homotopy groups of Sn, via the long exact sequence on homotopy associated to a fibration...
11 KB (1,634 words) - 21:23, 14 July 2023
Adams spectral sequence (category Homotopy theory)
sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational...
19 KB (3,283 words) - 13:36, 28 August 2023
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 homology groups as...
1 KB (172 words) - 18:09, 27 May 2024