The homotopy groups
are an infinite series of (covariant) functors πn indexed by non-negative integers from based topological spaces to groups for n>0 and sets for n=0. πn(X,x0) as a set is the set of all homotopy classes of maps of pairs (Dn,∂Dn)→(X,x0), that is, maps of the disk into X, taking the boundary to the point x0. Alternatively, these can be thought of as maps from the sphere Sn into X, taking a basepoint on the sphere to x0. These sets are given a group structure by declaring the product of 2 maps f,g to simply attaching two disks D1,D2 with the right orientation along part of their boundaries to get a new disk D1∪D2, and mapping D1 by f and D2 by g, to get a map of D1∪D2. This is continuous because we required that the boundary go to a , and well defined up to homotopy.
If f:X→Y satisfies f(x0)=y0, then we get a homomorphism of homotopy groups f*:πn(X,x0)→πn(Y,y0) by simply composing with f. If g is a map Dn→X, then f*([g])=[f∘g].
More algebraically, we can define homotopy groups inductively byπn(X,x0)≅πn-1(ΩX,y0), where ΩX is the loop space of X, and y0 is the constant path sitting at x0.
If n>1, the groups we get are abelian.
Homotopy groups are invariant under homotopy equivalence, and higher homotopy groups (n>1) are not changed by the taking of covering spaces.
Some examples are:
πn(Sn)=ℤ.
πm(Sn)=0 if m<n.
πn(S1)=0 if n>1.