The homotopy groups are an infinite series of (covariant) functors indexed by non-negative integers from based topological spaces to groups for and sets for . as a set is the set of all homotopy classes of maps of pairs , that is, maps of the disk into , taking the boundary to the point . Alternatively, these can be thought of as maps from the sphere into , taking a basepoint on the sphere to . These sets are given a group structure by declaring the product of 2 maps to simply attaching two disks with the right orientation along part of their boundaries to get a new disk , and mapping by and by , to get a map of . This is continuous because we required that the boundary go to a , and well defined up to homotopy.
If , the groups we get are abelian.
Some examples are:
|Date of creation||2013-03-22 12:15:28|
|Last modified on||2013-03-22 12:15:28|
|Last modified by||bwebste (988)|
|Synonym||higher homotopy groups|