Homotopy Group

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The th homotopy group of a topological space  is the set of homotopy classes of maps from the n-sphere to , with a group structure, and is denoted . The fundamental group is , and, as in the case of , the maps  must pass through a basepoint . For , the homotopy group  is an Abelian group.

The group operations are not as simple as those for the fundamental group. Consider two maps  and , which pass through . The product  is given by mapping the equator to the basepoint . Then the northern hemisphere is mapped to the sphere 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 shows the product of two spheres.

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 of a map  is given by switching orientation for the sphere. By describing the sphere in  coordinates, switching the first and second coordinate changes the orientation of the sphere. 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. 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 do not depend on the choice of basepoint. 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 space with  for all  is called -connected. If  is -connected, , then the Hurewicz homomorphism  from the th-homotopy group to the th-homology group is an isomorphism.

When  is a continuous map, then  is defined by taking the images under  of the spheres in . 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

REFERENCES:

Aubry, M. Homotopy Theory and Models. Boston, MA: Birkhäuser, 1995.