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

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

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

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...

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...

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...

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...

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its...

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:...

X} to Y {\displaystyle Y} . Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be computable in practice for spaces...

invariants of a spectrum are its homotopy groups. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the...

{\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...

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...

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

continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less studied. The p-th cohomotopy set of a pointed topological...

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...

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

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...

themselves: called homotopies. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the...

this monoid is a group and is isomorphic to the group Θ n {\displaystyle \Theta _{n}} of h-cobordism classes of oriented homotopy n-spheres, which is...

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

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...

In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized...

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

{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...

/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...