Search results

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...

homotopy groups using an inverse system of topological spaces whose homotopy type at degree k {\displaystyle k} agrees with the truncated homotopy type of...

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...

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...

K(G,n)} , if it has n-th homotopy group π n ( X ) {\displaystyle \pi _{n}(X)} isomorphic to G and all other homotopy groups trivial. Assuming that G is...

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

H^{n}(X)=Z^{n}(X)/B^{n}(X),} in analogy with the nth homology group. Homotopy groups are similar to homology groups in that they can represent "holes" in a topological...

{\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...

Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which proved to be...

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...

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...

a key link between homotopy groups and homology groups. For any path-connected space X and positive integer n there exists a group homomorphism h ∗ :...

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

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...

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