mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted...

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other...

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

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...

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...

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...

(killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group...

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 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...

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...

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:...

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...

In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain...

first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information...

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...

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...

{\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...

{\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...

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...

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...

/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...

{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...

field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately...

In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic"...

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

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...

The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological...

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...

sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational...

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...