topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, for

i \ge 2

, the ''i''-th homotopy group with coefficients in an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

''G'' of a based space ''X'' is the

pointed set In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a Set (mathematics), set and x_0 is an element of X called the base point (also spelled basepoint). Map (mathematics), Maps between pointed sets ...

homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

classes of based maps from the Moore space of type

(G, i)

to ''X'', and is denoted by

\pi_i(X; G)

. For

i \ge 3

\pi_i(X; G)

is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

. The groups

\pi_i(X; \Z)

are the usual

homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...

s of ''X''.

References

* Algebraic topology Homotopy theory {{topology-stub ko:호모토피 군#계수가 있는 호모토피