A homotopy group generalizes the fundamental group to maps from higher dimensional hyperspheres, instead of from the circle.
For 
,
the 
th
homotopy group of a topological space 
with basepoint 
is the set of homotopy
classes of maps 
sending a fixed basepoint
of the n-sphere to 
, with homotopies fixing that
basepoint. It has a group
structure and is denoted 
,
or 
when the basepoint
is understood. The fundamental group is 
. For 
, the homotopy group 
is an Abelian group.
The group operations are not as simple as those for the fundamental group. Consider two based
maps 
and 
, which send their basepoint
to 
.
The product 
is given by mapping the equator to the basepoint 
. Then the northern hemisphere
is mapped to the hypersphere by collapsing the equator
to a point, and then it is mapped to 
by 
. The southern hemisphere is
similarly mapped to 
by 
.
The diagram above illustrates this product of based maps.
The identity element is represented by the constant map 
. The choice of direction of a loop
in the fundamental group corresponds to a manifold orientation of 
in a homotopy group. Hence the inverse
element of the homotopy class represented by
a map 
is obtained by composing 
with an orientation-reversing
map of the hypersphere that
fixes its basepoint. Choosing the north
pole of 
as that basepoint, reflection
of the first coordinate has these properties. Or
as a hypersurface, 
, switching orientation
reverses the roles of inside and outside. The above diagram shows that 
is homotopic to the constant
map, i.e., the identity element. It begins
by expanding the equator in 
, and then the resulting map is
contracted to the basepoint.
As with the fundamental group, the homotopy groups of a pathwise-connected topological
space are isomorphic for different choices
of basepoint, with an isomorphism
induced by a chosen path. But the higher homotopy groups
are always Abelian. The above diagram shows an example
of 
.
The basepoint is fixed, and because 
the map can be rotated. When
, i.e., the fundamental
group, it is impossible to rotate the map while keeping
the basepoint fixed.
A topological space with 
for all 
is called 
-connected. If 
is 
-connected, 
, then the Hurewicz group
homomorphism 
from the 
th-homotopy
group to the 
th-homology group is an isomorphism.
When 
is a continuous map, then 
is defined by composing each based map 
with 
, using 
as the basepoint of 
. The pushforward is natural, i.e., 
whenever
the composition of two maps
is defined. In fact, given a fibration,
 |
where 
is pathwise-connected, there is a long
exact sequence of homotopy groups, ending in pointed sets
in degree 0:
 |
